Area and perimeter of composite shapes | TPT - Free Printable

Let’s solve each compound shape one by one. We’ll find the perimeter (total distance around the outside) and area (space inside) for each.

We’ll go row by row, left to right.

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Shape 1: L-shaped figure (top left)

Dimensions:
- Left side: 10 cm
- Bottom: 15 cm
- Right bottom part: 6 cm high
- Top horizontal part: 7 cm long

To find perimeter: walk around the outside.

Start at bottom-left corner:

→ Go up 10 cm
→ Go right ? — wait, we need to figure out missing sides.

Actually, let’s break it into rectangles or use “add all outer edges”.

Better method: For compound shapes, sometimes it’s easier to imagine completing the rectangle and subtracting, but for perimeter, just trace the outline.

Let me label the points mentally.

From bottom-left:

1. Up 10 cm → top-left
2. Right ? — actually, from top-left, you go right until you hit the inner corner. The total width is 15 cm, and the lower part sticks out 7 cm on the right? Wait, no.

Looking again:

The full height on left is 10 cm. On the right, the vertical part is only 6 cm tall. So the top part must be 10 - 6 = 4 cm tall.

Also, the bottom is 15 cm wide. The top horizontal segment is labeled 7 cm — that’s the length of the top arm.

So, the shape looks like a big rectangle with a bite taken out of the top-right? Or more like an L lying on its back.

Actually, standard way: add all outer sides.

Let’s list all outer edges clockwise:

Start at bottom-left:

- Up: 10 cm
- Right: ? — from top-left, how far right before turning down? The top horizontal piece is 7 cm, so from left edge, you go right 7 cm? But total width is 15 cm, so the remaining part on the right is 15 - 7 = 8 cm? That doesn’t match.

Wait — perhaps better to think:

The shape has:

- Left side: 10 cm
- Bottom: 15 cm
- Right side: 6 cm (vertical)
- Then from there, going left along the bottom of the top part: this should be 15 - 7 = 8 cm? No.

I think I’m confusing myself.

Alternative approach: Use the fact that for such L-shapes, perimeter can be found by adding all visible sides, and the "inner" corners don't reduce perimeter — they just change direction.

Let me draw it mentally:

Imagine a rectangle 15 cm wide and 10 cm tall. Now cut out a rectangle from the top-right that is (15 - 7) = 8 cm wide and (10 - 6) = 4 cm tall? Let's check.

If original rectangle is 15x10, area = 150.

Cut out a rectangle from top-right: width = 15 - 7 = 8 cm? But the top horizontal part is 7 cm, meaning the left part of the top is 7 cm, so the cut-out is on the right, starting after 7 cm from left.

Height of cut-out: since the right side is only 6 cm tall, and total height is 10 cm, so cut-out height is 10 - 6 = 4 cm.

So cut-out rectangle is 8 cm wide (from x=7 to x=15) and 4 cm tall (from y=6 to y=10).

Then area of shape = 15*10 - 8*4 = 150 - 32 = 118 cm².

Now perimeter: when you cut out a rectangle from the corner, you remove two sides but add two new sides of same length, so perimeter remains the same as original rectangle!

Original rectangle perimeter: 2*(15+10) = 50 cm.

After cutting out a rectangle from the corner, the perimeter doesn't change because you lose two segments but gain two identical ones.

For example, if you have a rectangle and cut a smaller rectangle from the corner, the outer path now goes around the cut, but the total length added equals what was removed.

In this case, yes — perimeter should still be 50 cm.

But let me verify by tracing:

Start at bottom-left:

- Up 10 cm (left side)
- Right 7 cm (top of left part)
- Down 4 cm (right side of the cut-out? Wait no)

Actually, from top-left, after going right 7 cm, you turn down — but how far? Since the total height is 10 cm, and the right part is only 6 cm tall, that means from the top, you go down 4 cm to reach the level of the right part.

Then from there, you go right to the end: 15 - 7 = 8 cm? But that would be along the top of the lower part.

Then down 6 cm (right side)

Then left 15 cm (bottom)

Then up... but we started at bottom-left, so after left 15 cm, we're back at start? No.

Let's list segments in order:

1. Start at bottom-left corner.
2. Move up: 10 cm → to top-left.
3. Move right: 7 cm → to the inner corner (where the L bends).
4. Move down: 4 cm (because 10 - 6 = 4) → to the top of the right vertical part.
5. Move right: 8 cm (15 - 7 = 8) → to top-right corner.
6. Move down: 6 cm → to bottom-right corner.
7. Move left: 15 cm → to bottom-left corner.

Now sum: 10 + 7 + 4 + 8 + 6 + 15 = let's calculate: 10+7=17, +4=21, +8=29, +6=35, +15=50 cm. Yes! Perimeter is 50 cm.

Area: as above, 15*10 - 8*4 = 150 - 32 = 118 cm².

But is the cut-out 8x4? Width: from x=7 to x=15 is 8 cm, height from y=6 to y=10 is 4 cm, yes.

Alternatively, divide into two rectangles:

Left rectangle: 7 cm wide, 10 cm tall → area 70

Bottom-right rectangle: 8 cm wide (15-7), 6 cm tall → area 48

Total area: 70 + 48 = 118 cm². Same.

Good.

So Shape 1: Perimeter = 50 cm, Area = 118 cm²

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Shape 2: Another L-shape (top right)

Dimensions:

Top: 10 cm

Right side: 15 cm

Left bottom part: 6 cm high

Inner horizontal: 2 cm (labeled on the left side of the cut)

This looks like a rectangle with a bite taken out of the bottom-left.

Full height 15 cm, full width 10 cm.

The cut-out is on the bottom-left: it says 2 cm on the left side, and probably the height of the cut is 15 - 6 = 9 cm? Let's see.

Label:

- Total width: 10 cm
- Total height: 15 cm
- The bottom part on the right is 6 cm high, so the top part is 15 - 6 = 9 cm high.
- The inner horizontal segment is labeled 2 cm — this is likely the width of the cut-out on the left.

So, the shape is like a rectangle 10x15, with a rectangle cut out from bottom-left that is 2 cm wide and 9 cm tall? Because the top part starts after 2 cm from left, and goes full height minus 6 cm?

Actually, from the diagram: the left side has a step: from bottom, up 6 cm, then right 2 cm? No.

Looking at labels:

It says "2cm" on the left side, vertically? Or horizontally?

In the image description: "2cm" is written near the left side, probably indicating the width of the indentation.

Assume: the shape has a rectangular protrusion or cut.

Standard interpretation: it's a large rectangle 10 cm wide and 15 cm tall, but with a smaller rectangle removed from the bottom-left corner.

The removed part: width = 2 cm (since it says 2cm on the left), and height = 15 - 6 = 9 cm (because the right part is only 6 cm tall from bottom).

So cut-out is 2 cm wide, 9 cm tall.

Then area = 10*15 - 2*9 = 150 - 18 = 132 cm².

Perimeter: again, cutting a rectangle from the corner doesn't change perimeter, so same as original rectangle: 2*(10+15) = 50 cm.

Verify by tracing:

Start at bottom-left of the actual shape — but where is the bottom-left? After the cut.

Actually, the outer path:

Start at bottom-right corner.

Move left: 10 cm → to bottom-left of the main part, but there's a cut.

Better: start at the bottom of the right part.

Define points.

From bottom-right:

- Left: 10 cm → but wait, the bottom is not straight.

The bottom consists of: from right, left 10 cm? No, because there's a step on the left.

Actually, the bottom edge: from bottom-right, move left 10 cm? But the left part is indented.

Let's think:

The shape has:

- Right side: 15 cm down
- Bottom: from bottom-right, move left 10 cm? But then at left, instead of going up, there's a step.

From the diagram: the bottom has a segment of 10 cm? No.

Labels: top is 10 cm, right is 15 cm, and on the left, it says "2cm" which is probably the horizontal part of the cut, and "6cm" is the height of the bottom part on the right.

So, the bottom edge: from bottom-right, move left 10 cm? But that would include the cut area.

Actually, the bottom is only under the right part. The left part is higher.

So, outer path:

Start at bottom-right corner.

1. Move left: 10 cm? No — the width is 10 cm, but the bottom is not full width because of the cut.

I think I have it backward.

Perhaps the "2cm" is the depth of the cut on the left.

Let me assume the shape is composed of:

- A top rectangle: 10 cm wide, 9 cm tall (since 15-6=9)
- A bottom rectangle: 8 cm wide (10-2=8), 6 cm tall, attached to the right side.

Yes, that makes sense.

So, top part: 10x9

Bottom part: 8x6, aligned to the right.

Then area = 10*9 + 8*6 = 90 + 48 = 138 cm²? But earlier I said 132, inconsistency.

If cut-out is 2x9, area 18, total 150-18=132, but 10*9 + 8*6 = 90+48=138, not matching.

Mistake.

If the bottom part is 8 cm wide and 6 cm tall, and top is 10 cm wide and 9 cm tall, but they overlap or something? No, in an L-shape, they share a common region? No.

In this configuration, if top is full width 10 cm, height 9 cm, and bottom is only on the right, width 8 cm, height 6 cm, then the total height on the right is 9+6=15 cm, good, and on the left, only 9 cm, so the cut is on the bottom-left, size 2 cm wide (10-8=2) and 6 cm tall? But the height of the cut should be the difference in height, which is 6 cm, not 9 cm.

I think I confused myself.

Let's look at the labels given:

In the image: for shape 2, it has "10cm" on top, "15cm" on right, "6cm" on the bottom-right vertical, and "2cm" on the left side, probably indicating the horizontal extent of the cut.

Typically, in such diagrams, the "2cm" is the width of the indentation on the left.

So, the shape can be seen as a rectangle 10 cm wide and 15 cm tall, with a rectangle of size 2 cm (width) by h cm (height) removed from the bottom-left.

What is h? The height of the removed part is the amount by which the left side is shorter. Since the right side is 15 cm, and the bottom part on the right is 6 cm, that means from the bottom, the left side starts at height 6 cm? No.

Actually, the vertical dimension: the total height is 15 cm. The bottom part on the right is 6 cm high, so the top part is 9 cm high. The left side has a step: from the bottom, it goes up 6 cm, then right 2 cm, then up 9 cm? That doesn't make sense.

Perhaps it's better to calculate perimeter by adding all outer sides.

List the outer edges:

Start at bottom-right corner.

- Move up: 15 cm (right side)
- Move left: 10 cm (top side)
- Move down: ? — from top-left, how far down? The left side is not full height; there's a cut.

From top-left, move down 9 cm (since the top part is 9 cm tall? But total height is 15, and bottom is 6, so if the cut is on bottom-left, then from top-left, you go down 15 cm? No.

I recall that for such shapes, the perimeter can be calculated as the perimeter of the bounding box plus twice the depth of the cut, but let's think differently.

Notice that in both directions, the total "span" is the same.

For perimeter, when you have a rectilinear shape, you can sum all horizontal and vertical segments.

Horizontal segments:

- Top: 10 cm
- Bottom: the bottom has two parts: from left, there is a segment, then a gap, then the right part.

From the diagram, the bottom edge: on the right, there is a 6 cm high part, so the bottom is at y=0 for x from 2 to 10 (assuming left is x=0), and at y=6 for x from 0 to 2? No.

Assume coordinates.

Set bottom-left of the bounding box as (0,0).

Then the shape occupies:

- From x=0 to x=10, y=6 to y=15 (top part)
- From x=2 to x=10, y=0 to y=6 (bottom part)

Is that correct? Let's see the labels.

If top is 10 cm, so x from 0 to 10 at top.

Right side 15 cm, so y from 0 to 15.

Bottom-right part is 6 cm high, so from y=0 to y=6 on the right.

The "2cm" is likely the width of the cut on the left, so the bottom part starts at x=2.

So yes, the shape is:

- Rectangle A: x=0 to 10, y=6 to 15 → size 10x9
- Rectangle B: x=2 to 10, y=0 to 6 → size 8x6

They overlap in x=2 to 10, y=6 to 6? No, at y=6, it's the boundary, so no overlap.

Area = area A + area B = 10*9 + 8*6 = 90 + 48 = 138 cm².

But earlier I thought cut-out, but here it's additive.

In this case, there is no cut-out; it's two rectangles joined.

The bounding box is 10x15, but the bottom-left 2x6 is empty? Let's see: from x=0 to 2, y=0 to 6 is not included, so yes, it's like a rectangle with a bite taken out of bottom-left, size 2x6.

Bounding box 10x15 = 150 cm².

Cut-out: 2 cm wide, 6 cm tall = 12 cm².

Area = 150 - 12 = 138 cm². Yes, matches.

Perimeter: now, when you cut out a rectangle from the corner, the perimeter increases by twice the dimensions of the cut, because you add two new sides.

Original perimeter of bounding box: 2*(10+15) = 50 cm.

When you cut out a 2x6 rectangle from the bottom-left corner, you remove the two sides that were on the boundary, but add the two new sides of the cut.

Specifically, you remove the bottom side from x=0 to 2 (length 2) and the left side from y=0 to 6 (length 6), but you add the top of the cut (x=0 to 2 at y=6, length 2) and the right of the cut (x=2 from y=0 to 6, length 6).

So net change: -2 -6 +2 +6 = 0. So perimeter unchanged? But that can't be right because visually, the path is longer.

Let's trace the outer path.

Start at (0,6) — top-left of the cut, but let's start at a corner.

Start at (0,15) — top-left.

- Move right to (10,15): 10 cm
- Move down to (10,0): 15 cm
- Move left to (2,0): 8 cm (since from x=10 to x=2)
- Move up to (2,6): 6 cm
- Move left to (0,6): 2 cm
- Move up to (0,15): 9 cm

Sum: 10 + 15 + 8 + 6 + 2 + 9 = let's calculate: 10+15=25, +8=33, +6=39, +2=41, +9=50 cm. Again 50 cm.

Oh, so even though we have a cut, the perimeter is still 50 cm. Is that always true for corner cuts? In this case, yes, because the cut is rectangular and from the corner, the added paths compensate.

So for Shape 2: Perimeter = 50 cm, Area = 138 cm².

But let me confirm area: 10*9 = 90 for top, 8*6 = 48 for bottom-right, total 138, yes.

Or bounding box 150 minus cut-out 2*6=12, 150-12=138.

Good.

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Shape 3: Middle left — another L-shape

Dimensions:

Left side: 5 cm

Bottom: 2 cm (wait, labeled "2cm" at bottom-left)

Then "6cm" on the bottom of the right part

"3cm" on the right side

And the top is not labeled, but we can infer.

From the diagram: it's like a rectangle with a bite on the bottom-left.

Labels: left side 5 cm, bottom has "2cm" which is probably the width of the cut, "6cm" is the length of the bottom of the right part, "3cm" is the height of the right part.

So, similar to before.

Assume bounding box: width = 2 + 6 = 8 cm? Because bottom has left part 2 cm (cut), right part 6 cm, so total width 8 cm.

Height: left side 5 cm, right side 3 cm, so the top part is higher on left.

So, the shape is:

- Left part: x=0 to 2, y=3 to 5? Let's define.

From bottom: at x=0 to 2, y=0 to ? The left side is 5 cm, but the right side is only 3 cm, so probably the bottom is not flat.

Typically, the "2cm" at bottom-left is the horizontal extent of the cut, and "5cm" is the full height on left, "3cm" is the height on right, "6cm" is the width of the bottom-right part.

So, the shape can be divided as:

- Top rectangle: x=0 to 8, y=3 to 5? Height 2 cm? 5-3=2 cm.

Width: total width should be 2 + 6 = 8 cm, since bottom has 2 cm cut and 6 cm right part.

So top part: 8 cm wide, 2 cm tall (from y=3 to y=5)

Bottom part: x=2 to 8, y=0 to 3, size 6x3

Then area = 8*2 + 6*3 = 16 + 18 = 34 cm²

Bounding box: 8 cm wide, 5 cm tall = 40 cm²

Cut-out: bottom-left, x=0 to 2, y=0 to 3, size 2x3 = 6 cm²

Area = 40 - 6 = 34 cm², good.

Perimeter: trace outer path.

Start at (0,5) top-left.

- Right to (8,5): 8 cm
- Down to (8,0): 5 cm? But at x=8, y from 0 to 5, but the shape only goes down to y=0 at x>2, but at x=8, it's full height? No.

At x=8, the right side is from y=0 to y=3 for the bottom part, and from y=3 to y=5 for the top part, so yes, continuous from y=0 to y=5 at x=8.

Similarly, at x=0, from y=3 to y=5.

So path:

Start at (0,5)

- Right to (8,5): 8 cm
- Down to (8,0): 5 cm
- Left to (2,0): 6 cm (since from x=8 to x=2)
- Up to (2,3): 3 cm
- Left to (0,3): 2 cm
- Up to (0,5): 2 cm

Sum: 8 + 5 + 6 + 3 + 2 + 2 = 26 cm

Calculate: 8+5=13, +6=19, +3=22, +2=24, +2=26 cm.

Using the formula: bounding box perimeter 2*(8+5)=26 cm, and since cut-out is from corner, perimeter unchanged, yes.

So Shape 3: Perimeter = 26 cm, Area = 34 cm²

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Shape 4: Middle right — U-shape or frame

Dimensions:

Top: 11 cm

Right side: 5 cm

Bottom: 3 cm on each side, and 5 cm in the middle? Labels: "5cm" on the bottom of the inner part, "3cm" on the bottom-right, and "5cm" on the right side.

Also, the inner width is 5 cm, and the sides are 3 cm wide? Let's see.

It looks like a rectangle with a rectangular hole in the bottom, but not quite.

From the diagram: it's like a U-shape standing on its base.

Outer dimensions: width 11 cm, height 5 cm.

Inner cut-out: width 5 cm, and depth? The bottom has "3cm" on the sides, so the cut-out is centered or something.

Labels: "11cm" on top, "5cm" on right side, "3cm" on bottom-right, "5cm" on the bottom of the inner part.

Probably, the shape has outer width 11 cm, outer height 5 cm.

The bottom has two legs of 3 cm each, and a gap of 5 cm in between, so total width 3+5+3=11 cm, good.

The height of the legs is 5 cm, but the inner part is open at the bottom.

Actually, it's like a rectangle 11x5 with a rectangle cut out from the bottom center.

The cut-out: width 5 cm, height? Since the legs are 5 cm tall, and the cut is at the bottom, the height of the cut is the full height? No.

If the outer height is 5 cm, and the legs go full height, then the cut-out is from y=0 to y=h, but what is h?

The label "5cm" on the right side is the full height, so the legs are 5 cm tall.

The inner width is 5 cm, so the cut-out is 5 cm wide, and since it's at the bottom, and the top is solid, the cut-out height is less than 5 cm.

But there's no label for the height of the cut-out. Perhaps it's implied that the cut-out goes up to the top? But that would make it not a U-shape.

Looking back at the user's description: "5cm" is written on the bottom of the inner part, which might mean the width of the inner rectangle.

And "3cm" on the bottom-right, which is the width of the leg.

Also, the right side is 5 cm, full height.

So, likely, the shape is composed of three parts: left leg, right leg, and top bar.

Left leg: width 3 cm, height 5 cm

Right leg: width 3 cm, height 5 cm

Top bar: width 11 cm, height? But if legs are 5 cm, and top bar connects them, then the top bar must be at the top, so height of top bar is additional? No, that would make total height more than 5 cm.

Perhaps the top bar is part of the 5 cm height.

Standard interpretation: the outer rectangle is 11 cm wide and 5 cm tall.

A rectangular hole is cut out from the bottom, centered, with width 5 cm, and height h.

But what is h? Not specified.

Perhaps from the context, the "5cm" on the bottom of the inner part is the width, and the height of the cut is such that the legs are 3 cm wide, but the height is full 5 cm for the legs, so the cut-out height is 5 cm? But then it would be cut all the way up, but the top is still there.

I think I have it: in such diagrams, for a U-shape, the height of the legs is given, and the top bar has thickness.

But here, the right side is labeled 5 cm, which is likely the full height, so the legs are 5 cm tall, and the top bar is at the top, so the cut-out is from y=0 to y= d, but d is not given.

Perhaps the "5cm" on the inner bottom is not the width, but the height? But it's labeled on the bottom.

Another possibility: the shape is like a picture frame, but only on three sides.

Let's read the labels carefully from the user's text: "11cm" on top, "5cm" on right side, "3cm" on bottom-right, "5cm" on the bottom of the inner part.

Probably, "5cm" on the bottom of the inner part means the width of the inner rectangle is 5 cm, and "3cm" is the width of the side walls.

So, outer width 11 cm, so the two side walls are each (11 - 5)/2 = 3 cm wide, which matches the "3cm" label.

Now, the height: the right side is 5 cm, which is the full height of the shape.

The inner part has a bottom, but in a U-shape, the bottom is open, so the inner rectangle is not closed at the bottom.

For area and perimeter, we need to know how high the cut is.

Perhaps the cut goes all the way to the top? But then it wouldn't be a U-shape; it would be two separate rectangles.

I think for a U-shape, the cut is only at the bottom, and the top is solid.

But the height of the cut is not specified. Unless the "5cm" on the right side is the height of the legs, and the top bar has additional height, but that doesn't make sense.

Perhaps the 5 cm is the height of the entire shape, and the cut-out height is the same as the leg height, but that would mean the cut is full height, which is not typical.

Let's assume that the shape has outer dimensions 11 cm wide and 5 cm tall.

A rectangular region is removed from the bottom center, with width 5 cm, and height h.

But h is not given. However, in many such problems, if not specified, the cut might be full height, but then the top would be disconnected.

Perhaps from the context, the "5cm" on the inner bottom is the width, and the height of the cut is implied by the side walls.

Another idea: the "5cm" on the right side is the height, and the "3cm" on the bottom-right is the width of the leg, and the inner "5cm" is the width of the opening, so the height of the opening is the same as the leg height, which is 5 cm, but then the top is not connected.

I think there's a mistake in my reasoning.

Let me search for standard problems.

Perhaps for this shape, it is a rectangle with a rectangular notch at the bottom.

But to resolve, let's look at the last shape or think differently.

Notice that in the user's description, for this shape, it has "11cm" top, "5cm" right, "3cm" bottom-right, "5cm" on the bottom of the inner part.

Probably, the "5cm" on the bottom of the inner part is the length of the bottom of the inner rectangle, i.e., the width of the cut-out.

And the height of the cut-out is not specified, but perhaps it is the full height minus something, but no.

Another thought: in some diagrams, the "5cm" on the inner bottom might be the depth, but it's labeled on the bottom, so likely width.

Perhaps the shape is symmetric, and the cut-out height is the same as the side wall width or something.

Let's calculate the area as the area of the outer rectangle minus the area of the cut-out rectangle.

Outer rectangle: 11 cm * 5 cm = 55 cm²

Cut-out rectangle: width 5 cm, height h.

What is h? If the legs are 3 cm wide, and the cut is at the bottom, and the top is solid, then the cut-out height must be less than 5 cm.

But no value is given. Unless the "5cm" on the right side is not the full height, but the height of the leg.

Perhaps the 5 cm is the height of the vertical parts, and the top bar has thickness, but not specified.

I recall that in some problems, for a U-shape, the height of the legs is given, and the top bar has the same thickness as the legs, but here the leg width is 3 cm, so perhaps the top bar is 3 cm thick.

Assume that the top bar is 3 cm thick (height), and the legs are 5 cm tall, but then total height would be 5 + 3 = 8 cm, but the right side is labeled 5 cm, which contradicts.

Perhaps the 5 cm is the total height, and the top bar is included in it.

Let's assume that the cut-out height is k, then the area is 11*5 - 5*k = 55 - 5k.

But k is unknown.

Perhaps from the diagram, the "5cm" on the inner bottom is not the width, but the height of the cut-out. But it's labeled on the bottom, so unlikely.

Another idea: in the user's text, for this shape, it says "5cm" on the bottom of the inner part, and "3cm" on the bottom-right, and "5cm" on the right side.

Perhaps "5cm" on the right side is the height, "3cm" is the width of the bottom-right leg, and "5cm" on the inner bottom is the width of the inner rectangle, so the cut-out is 5 cm wide, and its height is the full 5 cm, but then the top is not there, which is impossible.

I think I found the issue: in a U-shape standing on its base, the "height" usually refers to the height of the legs, and the top bar is additional, but here the right side is labeled 5 cm, which might be the height of the leg, and the top bar has its own height, but not specified.

Perhaps for this problem, the shape is meant to be a rectangle with a rectangular hole, but the hole is not specified.

Let's look at the last shape for clue, or assume that the cut-out height is the same as the side width, but that's arbitrary.

Perhaps from the context, the "5cm" on the inner bottom is the width, and the height of the cut is 5 cm, but then the shape would have the top missing, which is not the case.

Let's calculate the perimeter first.

For perimeter, if it's a U-shape with outer width 11 cm, outer height 5 cm, and inner width 5 cm, and if the cut is full height, then it's not connected, but if the cut is only at the bottom, we need the depth.

I recall that in some textbooks, for such a shape, the height of the cut is given by the difference, but here it's not.

Another thought: the "5cm" on the right side is the full height, and the "3cm" on the bottom-right is the width of the leg, and the "5cm" on the inner bottom is the width of the opening, so the height of the opening is the same as the leg height, which is 5 cm, but then the top is not present, so perhaps the shape is only the three sides: left, right, and top, but then it's not closed.

I think there's a standard interpretation: for a U-shape like this, the dimensions given are for the outer rectangle, and the inner rectangle is cut out from the bottom, with width 5 cm, and height equal to the leg height, but the leg height is not the full 5 cm.

Perhaps the 5 cm on the right side is the height of the vertical part, and the top bar is separate.

Let's assume that the total height is 5 cm, and the top bar has thickness t, then the leg height is 5 - t.

But t is not given.

Perhaps from the symmetry, and the leg width is 3 cm, and the top bar has the same thickness as the leg width, so t = 3 cm, then leg height = 5 - 3 = 2 cm.

That could be.

Let me try that.

So, outer width 11 cm, outer height 5 cm.

Top bar: width 11 cm, height 3 cm (thickness)

Then below it, two legs: each width 3 cm, height 2 cm (since 5-3=2), and they are at the sides, so the gap between them is 11 - 3 - 3 = 5 cm, which matches the "5cm" on the inner bottom.

Perfect.

So, the shape consists of:

- Top rectangle: 11 cm x 3 cm
- Left leg: 3 cm x 2 cm
- Right leg: 3 cm x 2 cm

But the legs are attached to the top bar, so no overlap.

Area = 11*3 + 3*2 + 3*2 = 33 + 6 + 6 = 45 cm²

As a single shape: outer rectangle 11x5 = 55 cm², minus the cut-out rectangle in the bottom center: width 5 cm, height 2 cm (since legs are 2 cm tall, so cut-out is 5x2 = 10 cm²), so area = 55 - 10 = 45 cm², good.

Perimeter: now, this is not a simple corner cut; it's a U-shape, so perimeter will be larger.

Trace the outer path.

Start at top-left corner (0,5) assuming y=0 at bottom.

- Right to (11,5): 11 cm
- Down to (11,3): 2 cm (since top bar is 3 cm thick, so from y=5 to y=3)
- Left to (8,3): 3 cm? No, at y=3, from x=11 to x=8? Let's define.

At y=3, the top bar ends, and the legs start.

From (11,5) down to (11,3): 2 cm (height of the right part of the top bar? No.

The top bar is from y=3 to y=5, full width 11 cm.

Then below, from y=0 to y=3, only the legs: left leg x=0 to 3, y=0 to 3? But we said leg height is 2 cm, so from y=0 to y=2 for legs, and y=2 to y=5 for top bar? I think I messed up.

If total height is 5 cm, and top bar thickness is 3 cm, then top bar is from y=2 to y=5 (height 3 cm), and legs from y=0 to y=2 (height 2 cm).

Yes.

So, coordinates:

- Top bar: x=0 to 11, y=2 to 5
- Left leg: x=0 to 3, y=0 to 2
- Right leg: x=8 to 11, y=0 to 2 (since gap is 5 cm, from x=3 to x=8)

Now, outer perimeter:

Start at (0,5) top-left.

- Right to (11,5): 11 cm
- Down to (11,2): 3 cm (through the top bar)
- Left to (8,2): 3 cm (along the top of the right leg? At y=2, from x=11 to x=8, but this is the bottom of the top bar, and above the right leg.

From (11,2), if we go left, we are on the line y=2, which is the boundary between top bar and legs.

But for the outer path, after coming down to (11,2), we need to go down the right leg.

So:

From (11,5) down to (11,2): 3 cm (this is the right side of the top bar)

Then from (11,2) down to (11,0): 2 cm (right side of the right leg)

Then left to (8,0): 3 cm (bottom of the right leg)

Then up to (8,2): 2 cm (left side of the right leg? But at x=8, y from 0 to 2, this is the inner side, not outer.

Mistake.

The outer path should follow the outside.

From (11,0) , after moving left to (8,0), then we are at the bottom-right of the gap, but to continue, we need to go up the left side of the right leg, but that is internal if we consider the whole shape.

Actually, for the U-shape, the outer perimeter includes the outer sides and the inner sides of the U.

So, let's list all outer edges.

Start at (0,5)

- Right to (11,5): 11 cm (top)
- Down to (11,0): 5 cm? But at x=11, from y=5 to y=0, but between y=2 and y=0, it's the leg, so yes, continuous down to (11,0): 5 cm
- Left to (8,0): 3 cm (bottom of right leg)
- Up to (8,2): 2 cm (left side of right leg — this is the inner side of the U)
- Left to (3,2): 5 cm (bottom of the top bar, or top of the gap)
- Down to (3,0): 2 cm (right side of left leg — inner side)
- Left to (0,0): 3 cm (bottom of left leg)
- Up to (0,5): 5 cm (left side)

Now sum these segments:

1. 11 cm (top)
2. 5 cm (right side down)
3. 3 cm (bottom right leg)
4. 2 cm (up left side of right leg)
5. 5 cm (left along y=2 from x=8 to x=3)
6. 2 cm (down right side of left leg)
7. 3 cm (bottom left leg)
8. 5 cm (up left side)

Sum: 11+5=16, +3=19, +2=21, +5=26, +2=28, +3=31, +5=36 cm.

We can also calculate as: the outer rectangle would be 11x5, perimeter 32 cm, but with the U-cut, we add the two vertical sides of the cut and the bottom of the cut, but remove the bottom of the outer rectangle where the cut is.

Standard way for U-shape: perimeter = 2*height + 2*width + 2*depth_of_cut, but let's see.

In this case, the cut is 5 cm wide and 2 cm deep (since legs are 2 cm tall, so the cut depth is 2 cm).

When you have a U-shape, compared to the outer rectangle, you remove the bottom segment of length 5 cm (where the cut is), but add two vertical segments of length 2 cm each (the sides of the cut), and the bottom of the cut is not added because it's open, but in our path, we have the bottom of the legs.

In our calculation, we have 36 cm.

Outer rectangle perimeter: 2*(11+5) = 32 cm.

When you cut out a rectangle of 5x2 from the bottom center, you remove the bottom side of length 5 cm, but add the two vertical sides of length 2 cm each, and the top of the cut is already there, but in this case, since the cut is at the bottom, and we're removing material, for the perimeter of the remaining shape, you add the two new vertical sides.

So, original perimeter 32 cm.

Remove the bottom segment of length 5 cm (which was part of the outer perimeter).

Add two vertical segments of length 2 cm each ( the sides of the cut).

So net change: -5 + 2 + 2 = -1 cm, so perimeter = 32 - 1 = 31 cm? But we calculated 36, contradiction.

I see the mistake: when you cut out a rectangle from the bottom, if it's not at the corner, you remove one segment and add three segments.

In this case, cutting out a 5x2 rectangle from the bottom center of the 11x5 rectangle.

Originally, the bottom side is 11 cm.

After cutting out a 5 cm wide rectangle from the center, the bottom is now in two parts: left 3 cm and right 3 cm, so total bottom length 6 cm, whereas originally 11 cm, so you lose 5 cm from the bottom.

But you add the two vertical sides of the cut, each 2 cm, and the top of the cut is not added because it's internal or already there.

In the new shape, the perimeter includes:

- The top: 11 cm
- The two sides: each 5 cm, so 10 cm
- The bottom: two segments of 3 cm each, so 6 cm
- Plus the two vertical sides of the cut: each 2 cm, so 4 cm

Total: 11 + 10 + 6 + 4 = 31 cm.

But earlier I calculated 36 cm, so where did I go wrong in tracing?

In my tracing, I had:

From (0,5) to (11,5): 11

(11,5) to (11,0): 5 — but this is incorrect because from (11,5) to (11,2) is 3 cm (through top bar), then from (11,2) to (11,0) is 2 cm (through leg), so total 5 cm, ok.

Then (11,0) to (8,0): 3 cm

Then (8,0) to (8,2): 2 cm — this is up the left side of the right leg, which is correct for the inner side.

Then (8,2) to (3,2): 5 cm — this is left along y=2, which is the bottom of the top bar, and this is part of the perimeter? In the U-shape, this line y=2 from x=3 to x=8 is the top of the gap, and it is exposed, so yes, it is part of the perimeter.

Then (3,2) to (3,0): 2 cm — down the right side of the left leg.

Then (3,0) to (0,0): 3 cm

Then (0,0) to (0,5): 5 cm

Sum: 11+5+3+2+5+2+3+5 = let's add: 11+5=16, 16+3=19, 19+2=21, 21+5=26, 26+2=28, 28+3=31, 31+5=36 cm.

But according to the other method, it should be 31 cm. Contradiction.

I see the error: when I go from (8,2) to (3,2), that is 5 cm, but in the shape, at y=2, from x=3 to x=8, this is the line between the top bar and the gap, and for the perimeter, this line is internal if the top bar is solid, but in this case, since the gap is below, this line is the bottom of the top bar, and it is exposed to the gap, so it should be part of the perimeter.

However, in the standard U-shape, this line is indeed part of the perimeter.

But why the discrepancy with the other calculation?

In the other calculation, I said outer rectangle perimeter 32 cm, remove bottom 5 cm, add two vertical 2 cm each, so 32 -5 +4 = 31 cm.

But in this, I forgot that when you cut out the rectangle, you also need to account for the fact that the bottom is now split, but in the removal, when you remove the 5 cm from the bottom, you are left with two segments of 3 cm each, so the bottom contribution is 6 cm instead of 11 cm, so net loss of 5 cm from bottom, but you add the two vertical sides of 2 cm each, so net change -5 +4 = -1 cm, so 32 -1 = 31 cm.

But in my tracing, I have the bottom as 3+3=6 cm, and the two vertical sides of the cut as 2+2=4 cm, and the top 11 cm, sides 5+5=10 cm, total 11+10+6+4=31 cm. Oh! I miscalculated the sum earlier.

In my list:

1. 11 (top)
2. 5 (right side down) — but this 5 cm includes from y=5 to y=0 at x=11, which is correct for the outer right side.
3. 3 (bottom right leg) — from (11,0) to (8,0)
4. 2 (up to (8,2)) — this is the inner right side of the cut
5. 5 (left to (3,2)) — this is the top of the cut, or bottom of top bar
6. 2 (down to (3,0)) — inner left side of the cut
7. 3 (left to (0,0)) — bottom left leg
8. 5 (up to (0,5)) — left side

Now sum: 11 + 5 = 16
16 + 3 = 19
19 + 2 = 21
21 + 5 = 26
26 + 2 = 28
28 + 3 = 31
31 + 5 = 36? 28 + 3 = 31, then 31 + 5 = 36, but it should be 31.

Item 8 is from (0,0) to (0,5): 5 cm, but in item 2, I already have the right side from (11,5) to (11,0): 5 cm, and item 8 is left side from (0,0) to (0,5): 5 cm, so that's correct for the two vertical sides.

But 11 (top) + 5 (right) + 3 (bottom right) + 2 (up inner right) + 5 (across top of cut) + 2 (down inner left) + 3 (bottom left) + 5 (left) = let's list the lengths: 11,5,3,2,5,2,3,5

Add them: start with 11+5=16
16+3=19
19+2=21
21+5=26
26+2=28
28+3=31
31+5=36

But 36 is too big. I see the mistake: when I go from (0,0) to (0,5), that is 5 cm, but in the left side, from y=0 to y=5, but at x=0, from y=0 to y=2 is the left leg, and from y=2 to y=5 is the left part of the top bar, so it is continuous, so 5 cm is correct.

But according to the other method, it should be 31 cm.

Perhaps the segment from (8,2) to (3,2) is not part of the outer perimeter? But in a U-shape, it is.

Let's calculate the perimeter as the sum of all outer edges.

The shape has:

- Top edge: 11 cm
- Right edge: from y=0 to y=5 at x=11: 5 cm
- Bottom edge: from x=8 to x=11 at y=0: 3 cm (bottom of right leg)
- Then from (8,0) to (8,2): 2 cm (left side of right leg)
- Then from (8,2) to (3,2): 5 cm ( this is the line at y=2 from x=3 to x=8, which is the bottom of the top bar, and since the gap is below, this edge is exposed, so yes)
- Then from (3,2) to (3,0): 2 cm (right side of left leg)
- Then from (3,0) to (0,0): 3 cm (bottom of left leg)
- Then from (0,0) to (0,5): 5 cm (left edge)

Same as before.

But 11+5+3+2+5+2+3+5 = 36 cm.

Now, the area is 45 cm², as calculated.

For the perimeter, perhaps it is 36 cm, and my other method is wrong.

Why is the other method wrong? When I said outer rectangle perimeter 32 cm, and cut out a 5x2 rectangle from the bottom center, then the new perimeter is original minus the bottom segment of 5 cm (which is removed), plus the two vertical sides of 2 cm each, and plus the top of the cut, but the top of the cut is already part of the original rectangle's interior, so when you cut, you expose it, so you add it.

In general, when you cut out a rectangle from the interior, you add the perimeter of the cut-out rectangle, but since it's on the boundary, it's different.

For a cut-out from the edge, if you cut a rectangle from the bottom edge, not at corner, then you remove the segment of length w (width of cut) from the bottom, and add the two vertical sides of length h (height of cut), and the top of the cut is new, but in this case, the top of the cut is at y=2, which was previously internal, so you add it.

So, original bottom: 11 cm

After cut, bottom is split into two parts: left 3 cm, right 3 cm, so total bottom length 6 cm, so you lose 5 cm from bottom.

You add two vertical sides: each 2 cm, so +4 cm.

You add the top of the cut: 5 cm ( from x=3 to x=8 at y=2).

So net change: -5 +4 +5 = +4 cm.

Original perimeter 32 cm, so new perimeter 32 +4 = 36 cm. Yes! I forgot to add the top of the cut.

So perimeter = 36 cm.

Area = 45 cm², as before.

So Shape 4: Perimeter = 36 cm, Area = 45 cm²

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Shape 5: Bottom left — T-shape or something

Dimensions:

Left side: 8 cm

Bottom: 12 cm

Top has "2cm" and "3cm" on the sides, and "4cm" on the right side.

Labels: "2cm" on top-left, "3cm" on top-right, "4cm" on the right side, "8cm" on left, "12cm" on bottom.

So, likely, the shape has a base of 12 cm wide, 8 cm tall on left, but on right, only 4 cm tall, and on top, there are extensions.

From the description, it might be a rectangle with two rectangles on top.

Assume the main body is 12 cm wide, and height varies.

The left side is 8 cm, right side is 4 cm, so probably the bottom is at y=0, left side up to y=8, right side up to y=4.

Then on top, there are two protrusions: on left, a 2 cm wide extension, on right, a 3 cm wide extension.

But the top is not specified.

Perhaps the "2cm" and "3cm" are the widths of the top parts, and the height is additional.

But the left side is 8 cm, which may include the top part.

Assume that the main rectangle is 12 cm wide, and height min(8,4) = 4 cm, but then the left side is taller.

So, the shape can be divided as:

- Bottom rectangle: 12 cm wide, 4 cm tall (since right side is 4 cm)
- On top of that, on the left, a rectangle 2 cm wide, 4 cm tall (since 8-4=4 cm)
- On top of that, on the right, a rectangle 3 cm wide, ? tall, but the right side is only 4 cm, so if the bottom is 4 cm, and the right top part is on top, it would make the right side taller, but it's labeled 4 cm, so probably not.

Perhaps the 8 cm and 4 cm are the heights of the sides, and the top has the 2cm and 3cm as the overhangs.

Another interpretation: the shape is like a rectangle 12 cm wide, with a step on the right.

From left, height 8 cm, then at some point, it steps down to 4 cm on the right.

The "2cm" and "3cm" might be the widths of the top segments.

Assume that from left, for 2 cm width, height is 8 cm, then for the middle, height is h, then for 3 cm on right, height is 4 cm, but the bottom is 12 cm, so the middle width is 12 -2 -3 = 7 cm.

But what is the height of the middle part? Not specified.

Perhaps the top is flat, but the sides are different.

I think for this shape, it is a combination where the left part is taller.

Let's look for symmetry or standard.

Perhaps the "2cm" on top-left is the width of the left protrusion, "3cm" on top-right is the width of the right protrusion, and the main body is 12 cm wide, but the protrusions are on top, so the total width is still 12 cm, and the heights are given.

Assume that the main rectangle is 12 cm wide, 4 cm tall (since right side is 4 cm).

Then on top of the left part, a rectangle 2 cm wide, 4 cm tall (so total height on left 4+4=8 cm).

On top of the right part, a rectangle 3 cm wide, but if I add it, the right side would be 4 + h, but it's labeled 4 cm, so probably not.

Unless the 4 cm is the height of the main body, and the top parts are additional, but then the right side would be more than 4 cm.

Perhaps the 4 cm is the height of the right top part, but that doesn't make sense.

Another idea: the "4cm" on the right side is the height of the right vertical part, and the "8cm" on left is the height of the left vertical part, and the "2cm" and "3cm" are the widths of the top horizontal parts.

So, the shape has:

- Left vertical: 8 cm high, but width not specified.
- Right vertical: 4 cm high.
- Top horizontal: from left, 2 cm, then a gap, then 3 cm on right.

But the bottom is 12 cm, so the distance between the left and right verticals is 12 cm, but with top parts.

Perhaps it's like a bridge or something.

Let's assume that the bottom is 12 cm wide.

From left, a vertical rise of 8 cm, but only for a width of 2 cm? Then from there, it goes right at height 8 cm for some distance, then down to 4 cm, then right to the end.

But the "3cm" on top-right suggests that on the right, there is a 3 cm wide part at the top.

So, likely, the shape is:

- From x=0 to x=2, y=0 to y=8 (left tower)
- From x=2 to x=9, y=4 to y=8 (top bar, since 2+7=9, but 12-3=9, so from x=2 to x=9)
- From x=9 to x=12, y=0 to y=4 (right tower)

Then the bottom from x=0 to x=12 at y=0, but between x=2 to x=9, at y=0, is it filled? In this case, from x=2 to x=9, y=0 to y=4 is not included, so it's like a U-shape but with different heights.

So, the shape has:

- Left rectangle: 2x8
- Middle top rectangle: 7x4 (since 9-2=7, and height 8-4=4)
- Right rectangle: 3x4

But the middle top is from y=4 to y=8, x=2 to x=9.

Then the bottom is only under the left and right towers, so from x=0 to 2 and x=9 to 12 at y=0 to y=4 for left and right, but for left, it's already included in the 2x8, which includes y=0 to 8, so for x=0 to 2, y=0 to 8 is covered.

For x=9 to 12, y=0 to 4 is covered by the right rectangle.

For x=2 to 9, only y=4 to 8 is covered by the middle top.

So area = left: 2*8 = 16
middle: 7*4 = 28
right: 3*4 = 12
total 16+28+12 = 56 cm²

Bounding box 12x8 = 96 cm², but with cut-outs, but perhaps not necessary.

Perimeter: trace outer path.

Start at (0,0)

- Up to (0,8): 8 cm
- Right to (2,8): 2 cm
- Right to (9,8): 7 cm (along top)
- Down to (9,4): 4 cm
- Right to (12,4): 3 cm
- Down to (12,0): 4 cm
- Left to (0,0): 12 cm? But from (12,0) to (0,0) is 12 cm, but this would include the bottom, but between x=2 to x=9, at y=0, is not part of the shape, so we cannot go directly; we need to go along the bottom only where the shape is.

From (12,0) , move left to (9,0): 3 cm (bottom of right tower)
Then from (9,0) , but at x=9, y=0, to go to left, but the shape is not there until x=2, so we need to go up or something.

At (9,0), the shape has only up to y=4 for x>9, but for x<9, at y=0, only for x<2.

So from (9,0), we can only go up to (9,4), but we already did that.

After coming down to (12,0), we go left to (9,0): 3 cm
Then from (9,0) , since no shape below, we go up to (9,4): 4 cm — but we already have that segment from earlier.

This is messy.

From (12,0) left to (9,0): 3 cm
Then from (9,0) up to (9,4): 4 cm — but this is the same as the right side of the right tower, which we may have already traversed.

In the path, when we came from (9,4) down to (9,0)? No, in my earlier path, I had from (9,4) right to (12,4), then down to (12,0), so I haven't done the left side of the right tower yet.

Let's define the path carefully.

Start at (0,0)

1. Up to (0,8): 8 cm (left side)
2. Right to (2,8): 2 cm (top of left tower)
3. Right to (9,8): 7 cm (top of middle bar)
4. Down to (9,4): 4 cm (right side of middle bar)
5. Right to (12,4): 3 cm (top of right tower)
6. Down to (12,0): 4 cm (right side of right tower)
7. Left to (9,0): 3 cm (bottom of right tower)
8. Up to (9,4): 4 cm — but this is the same as step 4, and it's internal now.

Mistake: after step 7, at (9,0), we are at the bottom-left of the right tower, but to continue to the left, we need to go to the left tower, but there is no connection at y=0 between x=2 and x=9.

So from (9,0), we can only go up, but we already have the up path.

Actually, the shape is not connected at the bottom between left and right; it's only connected at the top.

So the bottom is separate for left and right.

So from (12,0) after step 6, we go left to (9,0): 3 cm (step 7)
Then from (9,0) , since no further left at y=0, we stop, but we need to close the path.

From (9,0) , we can go up to (9,4), but that is already done in step 4, and it's not outer anymore.

Perhaps start from a different point.

Start at (0,0)

- Up to (0,8): 8 cm
- Right to (2,8): 2 cm
- Right to (9,8): 7 cm
- Down to (9,4): 4 cm
- Right to (12,4): 3 cm
- Down to (12,0): 4 cm
- Left to (9,0): 3 cm
- Now from (9,0) , to go to the left part, but there is no direct path; the left part is at x=0 to 2, y=0 to 8, so from (9,0) , we need to go to (2,0) or something, but it's not connected.

I think I have a fundamental error in the shape interpretation.

Perhaps the "12cm" on bottom is the total width, and the shape is connected at the bottom.

In that case, for x=2 to x=9, y=0 to y=4 is part of the shape, but then the height on the right would be 4 cm, on left 8 cm, so at x=2, it steps down from 8 cm to 4 cm or something.

Assume that the shape has a constant bottom at y=0 from x=0 to x=12.

Then at x=0, height 8 cm, at x=12, height 4 cm, and the top is sloped or stepped.

With the "2cm" and "3cm" on top, likely it is stepped.

So, from x=0 to x=2, height 8 cm
From x=2 to x=9, height h
From x=9 to x=12, height 4 cm

But what is h? Not specified. Perhaps h=4 cm, so it steps down at x=2 from 8 cm to 4 cm, then at x=9, it is still 4 cm, but the "3cm" on top-right suggests that on the right, there is a 3 cm wide part at height 4 cm, which is consistent.

But then the top from x=2 to x=9 is at y=4 cm, so no issue.

So, the shape is:

- From x=0 to x=2, y=0 to y=8
- From x=2 to x=12, y=0 to y=4 (since at x=9 to 12, height 4 cm, and from x=2 to 9, also 4 cm)

Then the "3cm" on top-right might be redundant or for emphasis.

But the "2cm" on top-left is the width of the left part at full height.

So area = left part: 2*8 = 16
right part: 10*4 = 40 (since x=2 to 12 is 10 cm wide)
total 56 cm², same as before.

But in this case, from x=2 to x=12, y=0 to 4, so at x=9 to 12, it's included, and the "3cm" is just the width of the right end, but not necessary for calculation.

Perimeter: now it's a single polygon.

Start at (0,0)

- Up to (0,8): 8 cm
- Right to (2,8): 2 cm
- Down to (2,4): 4 cm (since at x=2, it steps down to y=4)
- Right to (12,4): 10 cm ( along y=4 from x=2 to x=12)
- Down to (12,0): 4 cm
- Left to (0,0): 12 cm

Sum: 8+2+4+10+4+12 = 40 cm

Calculate: 8+2=10, +4=14, +10=24, +4=28, +12=40 cm.

And area 2*8 + 10*4 = 16 + 40 = 56 cm².

The "3cm" on top-right might be to indicate that the right part is 3 cm wide, but in this case, from x=9 to 12 is 3 cm, but it's already included in the 10 cm.

So probably this is correct.

So Shape 5: Perimeter = 40 cm, Area = 56 cm²

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Shape 6: Bottom right — frame or annulus

Dimensions:

Outer: 9 cm by 8 cm? Labels: "9cm" on top, "8cm" on left, "6cm" on inner top, "3cm" on inner left.

So, outer rectangle 9 cm wide, 8 cm tall.

Inner rectangle 6 cm wide, 3 cm tall, centered or positioned.

Usually, for such frames, the inner rectangle is centered, but here the dimensions suggest it might not be.

The "6cm" on inner top, "3cm" on inner left, so likely the inner rectangle is 6 cm wide, 3 cm tall, and positioned with margins.

Outer width 9 cm, inner width 6 cm, so the side margins are (9-6)/2 = 1.5 cm each, but not specified.

Outer height 8 cm, inner height 3 cm, so top and bottom margins (8-3)/2 = 2.5 cm each.

But in the label, "3cm" on inner left might mean the height of the inner rectangle is 3 cm, and "6cm" on inner top means width 6 cm.

For area, area of frame = outer area - inner area = 9*8 - 6*3 = 72 - 18 = 54 cm².

For perimeter, it's the outer perimeter plus the inner perimeter, since it's a frame with a hole.

Outer perimeter: 2*(9+8) = 34 cm

Inner perimeter: 2*(6+3) = 18 cm

Total perimeter = 34 + 18 = 52 cm, because the hole adds its own perimeter.

Yes, for a shape with a hole, the perimeter includes both outer and inner boundaries.

So Shape 6: Perimeter = 52 cm, Area = 54 cm²

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Now, summarize all:

Shape 1 (top left): P=50 cm, A=118 cm²

Shape 2 (top right): P=50 cm, A=138 cm²

Shape 3 (middle left): P=26 cm, A=34 cm²

Shape 4 (middle right): P=36 cm, A=45 cm²

Shape 5 (bottom left): P=40 cm, A=56 cm²

Shape 6 (bottom right): P=52 cm, A=54 cm²

The task is to calculate area and perimeter for each compound shape, so I need to provide for each.

Since the user didn't specify which one, probably all.

But in the response, I should list them.

Perhaps box the answers for each.

Since the instruction is to provide final answer, and there are multiple, I'll list them.

Final Answer:

For the first shape (L-shape top left): Perimeter = 50 cm, Area = 118 cm²

For the second shape (L-shape top right): Perimeter = 50 cm, Area = 138 cm²

For the third shape (L-shape middle left): Perimeter = 26 cm, Area = 34 cm²

For the fourth shape (U-shape middle right): Perimeter = 36 cm, Area = 45 cm²

For the fifth shape (stepped shape bottom left): Perimeter = 40 cm, Area = 56 cm²

For the sixth shape (frame bottom right): Perimeter = 52 cm, Area = 54 cm²

To present neatly.

Since the problem says "solve the problem", and it's to calculate for each, I'll write the answers.

In the final answer section, perhaps list them.

Given the format, I'll put the final answer as a list.

But the instruction says "provide only the final correct answer", so perhaps for each shape.

Since it's multiple, I'll state them clearly.

Final Answer:
Shape 1: Perimeter = 50 cm, Area = 118 cm²
Shape 2: Perimeter = 50 cm, Area = 138 cm²
Shape 3: Perimeter = 26 cm, Area = 34 cm²
Shape 4: Perimeter = 36 cm, Area = 45 cm²
Shape 5: Perimeter = 40 cm, Area = 56 cm²
Shape 6: Perimeter = 52 cm, Area = 54 cm²

To match the order in the image: top row left to right, then middle row, then bottom row.

Yes.

I think that's it.