Compound shapes worksheet for calculating area and perimeter of geometric figures.

Let’s solve each problem step by step. We’ll find the area of each rectangle (if split), add them for total area, and then add all outer sides for perimeter.

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Problem 1)

Shape is an L-shape made of two rectangles:
- Rectangle A (top): width = 4 cm, height = 3 cm → Area = 4 × 3 = 12 cm²
- Rectangle B (bottom): width = 8 cm, height = 3 cm → Area = 8 × 3 = 24 cm²
→ Total area = 12 + 24 = 36 cm²

Perimeter: Walk around the outside.
Top: 4 cm
Right side top: 3 cm
Middle horizontal right: 4 cm
Right side bottom: 3 cm
Bottom: 8 cm
Left side: 6 cm
Wait — let’s list all outer edges carefully:

Starting from top-left corner going clockwise:
- Top: 4 cm
- Right down: 3 cm
- Right across: 4 cm
- Down: 3 cm
- Bottom: 8 cm
- Left up: 6 cm

But wait — that double-counts or misses? Let’s trace properly.

Actually, better to count all outer segments:

From diagram:
Top edge: 4 cm
Then down 3 cm (right side of A)
Then right 4 cm (top of B’s extension)
Then down 3 cm (right side of B)
Then left 8 cm (bottom)
Then up 6 cm (left side) — but this includes both A and B heights.

Wait — actually, the full left side is 6 cm, which is correct.

So perimeter = 4 + 3 + 4 + 3 + 8 + 6 = let’s add:
4+3=7; 7+4=11; 11+3=14; 14+8=22; 22+6=28 → 28 cm

✔ Check: Another way — if it were a full rectangle 8x6, perimeter would be 2*(8+6)=28, but we have a notch? Wait no — in this case, the shape is like a big rectangle with a piece missing on top right? Actually no — looking again, it's composed of two rectangles stacked such that the total outline is still rectangular? Wait no.

Wait — actually, the shape has a “step” on the right. So the perimeter should include the inner step.

Let me label points mentally:

Start at top-left:
→ right 4 cm (top of A)
↓ down 3 cm (right of A)
→ right 4 cm (this is the top of the lower part extending right)
↓ down 3 cm (right side of B)
← left 8 cm (bottom)
↑ up 6 cm (left side) — back to start.

Yes, that’s 6 sides: 4, 3, 4, 3, 8, 6 → sum = 28 cm. Correct.

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Problem 2)

L-shape, rotated.

Split into:
- Rectangle A (top horizontal): width = 7 cm, height = 1 cm → Area = 7 × 1 = 7 cm²
- Rectangle B (vertical stem): width = 4 cm, height = 6 cm → Area = 4 × 6 = 24 cm²

Total area = 7 + 24 = 31 cm²

Perimeter: Trace outer edges.

Start at top-left:
→ right 7 cm (top)
↓ down 1 cm (right end of top bar)
← left 3 cm (this is the indentation — wait, no, let’s look at dimensions.

Actually, from diagram:

Top: 7 cm
Then down 1 cm (on right)
Then left 3 cm? No — after going down 1 cm, you go left along the inside? No — perimeter is only outer.

Better: List all outer sides.

From top-left:
→ 7 cm (top)
↓ 1 cm (right side of top bar)
← 3 cm? Wait — no, after descending 1 cm, the next segment is going left? But that would be internal.

Actually, looking at the shape: It’s like a backwards L.

Dimensions given:
Top horizontal: 7 cm long, 1 cm high.
Vertical part: 6 cm tall, 4 cm wide, attached below the left part of the top bar.

So the vertical bar starts under the first 4 cm of the top bar? Because total top is 7 cm, and the vertical bar is 4 cm wide, so there’s a 3 cm overhang on the right.

So outer path:

Start at top-left:
→ right 7 cm (full top)
↓ down 1 cm (right end)
← left 3 cm (along the bottom of the top bar’s overhang — this is outer!)
↓ down 6 cm (right side of vertical bar)
← left 4 cm (bottom)
↑ up 7 cm? Wait no — left side is 1 cm (top) + 6 cm (bottom) = 7 cm? But we already went down 1 and then 6, so going up the left side should be 7 cm.

Wait — let’s list segments:

1. Top: 7 cm
2. Right drop: 1 cm
3. Left along bottom of top bar: 3 cm (because 7 - 4 = 3)
4. Down the right side of vertical bar: 6 cm
5. Bottom: 4 cm
6. Up the left side: 1 + 6 = 7 cm

Sum: 7 + 1 + 3 + 6 + 4 + 7 = let’s compute:
7+1=8; 8+3=11; 11+6=17; 17+4=21; 21+7=28 → 28 cm

Check: Alternatively, imagine bounding box: width 7, height 7 (1+6). Perimeter of bounding box is 2*(7+7)=28, and since the shape fills the corners without indentations affecting outer perimeter? Actually yes — because the "notch" is filled by the vertical bar, so outer perimeter equals bounding box. So 28 cm is correct.

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Problem 3)

Another L-shape.

Split into:
- Rectangle A (left vertical): width = 2 cm, height = 10 cm → Area = 2 × 10 = 20 cm²
- Rectangle B (bottom horizontal): width = 9 cm, height = 3 cm → Area = 9 × 3 = 27 cm²

But wait — they overlap? The bottom rectangle extends under the left one? Actually, the left rectangle is 2 cm wide, and the bottom is 9 cm wide, starting from the same left edge? Looking at diagram:

The vertical part is 2 cm wide, 10 cm tall.
The horizontal part is 9 cm long, 3 cm tall, attached to the bottom of the vertical part, extending right.

But the horizontal part’s left 2 cm is under the vertical part? So if we just add 20 + 27, we’re double-counting the overlapping 2x3 area?

No — actually, in compound shapes like this, when they say “rectangle A” and “rectangle B”, they usually mean non-overlapping parts. Looking at the labels in problem 1, they split without overlap.

In problem 3, likely:

Rectangle A: the vertical strip: 2 cm x 10 cm
Rectangle B: the horizontal base excluding the part under A? Or including?

Wait — the diagram shows:

Total height on left: 10 cm
Width of left column: 2 cm
Then from the bottom, a horizontal part going right 9 cm, height 3 cm.

But the 9 cm includes the 2 cm under the vertical part? Probably not — because if you look, the horizontal part starts at the bottom of the vertical part and goes right 9 cm, meaning the total width is 2 + 7 = 9? Wait, the diagram says “7 cm” for the horizontal extension beyond the vertical part.

Looking back at image description:

For problem 3:
Left side: 10 cm total height
Top of left column: 2 cm wide
Then from the bottom of that, a horizontal part: labeled “7 cm” for the extension, and total bottom is 9 cm, so yes — the horizontal rectangle is 9 cm long, but the part under the vertical is included? Actually, no — typically in these problems, when they say “rectangle A” and “B”, they are adjacent without overlap.

In the diagram, it’s drawn as:

- Vertical rectangle: 2 cm wide, 10 cm high
- Horizontal rectangle: attached to the bottom-right, 7 cm long (since 9 - 2 = 7), and 3 cm high.

But the label says “9 cm” for the bottom, and “7 cm” for the horizontal part extending right.

Also, the vertical part has a segment labeled “7 cm” on its right side — that must be the part above the horizontal rectangle.

So:

Rectangle A: the top part of the vertical column? Or the whole vertical?

Actually, re-examining standard approach: In such L-shapes, we can split into two rectangles that don’t overlap.

Option 1:
- Rectangle A: left vertical: 2 cm x 10 cm
- Rectangle B: bottom horizontal: 7 cm x 3 cm (since it extends 7 cm right from the vertical part)

But then the bottom total width is 2 + 7 = 9 cm, which matches.

And the height of the vertical part above the horizontal is 10 - 3 = 7 cm, which is labeled.

So:

Area A = 2 × 10 = 20 cm²? But that includes the part where B is attached? No — if B is only the extension, then A should be the entire vertical, and B is the horizontal extension, but then the corner is counted twice? No, because B is only the part sticking out.

Actually, better to define:

Rectangle A: the vertical rectangle including the part down to the bottom: 2 cm x 10 cm
Rectangle B: the horizontal rectangle to the right: 7 cm x 3 cm

They share no area — because B starts at the right edge of A.

Yes, that makes sense.

So:

Area A = 2 × 10 = 20 cm²
Area B = 7 × 3 = 21 cm²
Total area = 20 + 21 = 41 cm²

Perimeter: Trace outer edges.

Start at top-left:
→ right 2 cm (top of A)
↓ down 10 cm (right side of A — but wait, at the bottom, it connects to B)

Actually, after going down 10 cm on the right of A, we are at the bottom-right of A. Then we go right along the top of B? No — B is below? I think I have orientation wrong.

Looking at typical L-shape for problem 3: It’s like a capital L, with long leg on left, short leg on bottom.

So:

- Left side: 10 cm up
- Top: 2 cm right
- Then down 7 cm (to the level where the bottom bar starts) — this is the inner corner
- Then right 7 cm (the bottom bar)
- Then down 3 cm? No.

Standard tracing:

Start at top-left corner:
→ move right 2 cm (top edge)
↓ move down 7 cm (this is the right side of the upper part of the vertical bar — until the bottom bar begins)
→ move right 7 cm (top edge of the bottom bar)
↓ move down 3 cm (right end of bottom bar)
← move left 9 cm (bottom edge — total width 9 cm)
↑ move up 10 cm (left edge) — back to start.

Segments:
1. Right: 2 cm
2. Down: 7 cm
3. Right: 7 cm
4. Down: 3 cm
5. Left: 9 cm
6. Up: 10 cm

Sum: 2+7=9; 9+7=16; 16+3=19; 19+9=28; 28+10=38 → 38 cm

Check: Bounding box would be 9 cm wide, 10 cm high, perimeter 2*(9+10)=38 cm. And since the shape is L-shaped filling the corner, the outer perimeter equals the bounding box perimeter. Yes, correct.

So total area 41 cm², perimeter 38 cm.

But earlier I said Area A = 2x10=20, Area B=7x3=21, total 41. Good.

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Problem 4)

T-shape.

Can split into:
- Top rectangle: 12 cm wide, 4 cm high → Area = 12 × 4 = 48 cm²
- Bottom rectangle (stem): 2 cm wide, 9 cm high → Area = 2 × 9 = 18 cm²

Total area = 48 + 18 = 66 cm²

Perimeter: Trace outer edges.

Start at top-left:
→ right 12 cm (top)
↓ down 4 cm (right side of top bar)
← left 5 cm (along the bottom of the top bar’s right overhang — since stem is centered? Diagram shows 5 cm on each side, so yes)
↓ down 9 cm (right side of stem)
← left 2 cm (bottom of stem)
↑ up 9 cm (left side of stem)
→ right 5 cm (along the bottom of the top bar’s left overhang)
↑ up 4 cm (left side of top bar) — back to start.

List segments:
1. Top: 12 cm
2. Right down: 4 cm
3. Left along bottom of top bar: 5 cm
4. Down stem right: 9 cm
5. Bottom stem: 2 cm
6. Up stem left: 9 cm
7. Right along bottom of top bar: 5 cm
8. Up left side: 4 cm

Sum: 12 + 4 + 5 + 9 + 2 + 9 + 5 + 4

Calculate step by step:
12+4=16
16+5=21
21+9=30
30+2=32
32+9=41
41+5=46
46+4=50 → 50 cm

Check: Another way — the shape has symmetry. Total perimeter should account for all outsides. Yes, 50 cm seems correct.

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Problem 5)

U-shape or frame.

Outer rectangle: 15 cm wide, 11 cm high
Inner cutout: 7 cm wide, 6 cm high, centered? From diagram, the cutout is in the middle bottom.

To find area: subtract inner rectangle from outer.

Outer area = 15 × 11 = 165 cm²
Inner cutout area = 7 × 6 = 42 cm²
Total area = 165 - 42 = 123 cm²

Perimeter: This is trickier. The perimeter includes the outer boundary plus the inner boundary of the cutout.

Outer perimeter: 2*(15 + 11) = 52 cm
But because there’s a hole, we add the inner perimeter.

The cutout is a rectangle 7 cm by 6 cm, so its perimeter is 2*(7+6)=26 cm, but since it’s open to the bottom? No — in this shape, the cutout is like a notch at the bottom, so it’s not a closed hole; it’s indented.

Looking at diagram: It’s a rectangle with a rectangular bite taken out of the bottom center.

So the perimeter will be: outer perimeter minus the bottom side where the bite is, plus the three sides of the bite.

Specifically:

Original outer perimeter: top 15, left 11, right 11, bottom 15 → total 52 cm.

But the bottom is interrupted: instead of a straight 15 cm bottom, we have:

- Left bottom segment: 4 cm
- Then up 6 cm (left side of cutout)
- Then right 7 cm (top of cutout)
- Then down 6 cm (right side of cutout)
- Then right 4 cm (right bottom segment)

So compared to original bottom of 15 cm, we removed the middle 7 cm and added: up 6 + right 7 + down 6 = 19 cm extra? Let’s calculate total perimeter directly.

Trace the path:

Start at top-left:
→ right 15 cm (top)
↓ down 11 cm (right side)
← left 4 cm (bottom right segment)
↑ up 6 cm (right side of cutout)
← left 7 cm (top of cutout)
↓ down 6 cm (left side of cutout)
← left 4 cm (bottom left segment)
↑ up 11 cm (left side) — back to start.

Segments:
1. Top: 15
2. Right down: 11
3. Bottom right: 4
4. Up cutout right: 6
5. Left across cutout top: 7
6. Down cutout left: 6
7. Bottom left: 4
8. Left up: 11

Sum: 15+11=26; 26+4=30; 30+6=36; 36+7=43; 43+6=49; 49+4=53; 53+11=64 → 64 cm

Check: Original outer perimeter 52 cm. We replaced the bottom 15 cm with: 4 + 6 + 7 + 6 + 4 = 27 cm. So difference: 27 - 15 = 12 cm extra. 52 + 12 = 64 cm. Correct.

Area: 15*11 - 7*6 = 165 - 42 = 123 cm². Good.

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Problem 6)

Irregular shape, like a zigzag.

We can split into rectangles.

Looking at dimensions:

It can be divided into three rectangles:

Option:
- Left small rectangle: 2 cm wide, 3 cm high → Area = 2×3=6
- Middle rectangle: 5 cm wide, 7 cm high? Wait, let's see.

From diagram:

Start from bottom-left:
- Bottom: 5 cm
- Up 3 cm
- Right 2 cm
- Up 6 cm
- Right 10 cm
- Down 2 cm
- Left 13 cm? Wait, top is 13 cm.

Better to use coordinates or decompose.

Notice that the shape can be seen as a large rectangle minus some parts, but perhaps easier to split vertically.

Split into:

Rectangle 1 (leftmost): width 2 cm, height 3 cm (bottom part)
Rectangle 2 (middle): width 5 cm, height 7 cm (from y=3 to y=10) — but wait, the middle part goes up 6 cm from the step, so from bottom, it's 3 + 6 = 9 cm high? Let's define.

Actually, from the bottom:

- From x=0 to x=2: height 3 cm (rectangle A)
- From x=2 to x=7: height 3 + 6 = 9 cm? But the diagram shows from the step, it goes up 6 cm, so total height at x=2 to x=7 is 3 (bottom) + 6 = 9 cm? But then at x=7 to x=17? Top is 13 cm.

Top width is 13 cm, and from the right, it drops down 2 cm.

Perhaps split horizontally.

Another way: The shape consists of:

- A bottom rectangle: 5 cm wide, 3 cm high → Area = 15 cm²
- A middle rectangle: 5 cm wide, 6 cm high (stacked on top of the left part of the bottom) → Area = 30 cm²
- A top rectangle: 13 cm wide, 2 cm high → Area = 26 cm²

But do they overlap? The middle rectangle is from x=0 to x=5, y=3 to y=9. The top rectangle is from x=0 to x=13, y=9 to y=11? But the diagram shows the top is at height 2 cm from the top of the middle? Let's check heights.

From diagram:

- Bottom: 3 cm high, 5 cm wide
- Above the left 2 cm of the bottom, there is a 6 cm high section (so total height 3+6=9 cm for that column)
- Then on top of that, a 2 cm high section spanning 13 cm wide.

But the 13 cm wide top includes the part over the middle and right.

Specifically:

- The left part: from x=0 to x=2: height 3 + 6 + 2 = 11 cm? No.

Let's read the labels:

From bottom-left:
- Move right 5 cm (bottom)
- Up 3 cm
- Right 2 cm (this is a step)
- Up 6 cm
- Right 10 cm
- Down 2 cm
- Left 13 cm (top) — but 5+2+10=17, and top is 13, inconsistency? Perhaps the 13 cm is the top width.

Actually, the top horizontal is labeled 13 cm, and it's at the very top.

The right side has a drop of 2 cm, then left 10 cm, then down 7 cm? Labels are:

On the right: after the top 13 cm, down 2 cm, then left 10 cm, then down 7 cm, then left 5 cm, then up 3 cm, then right 2 cm, then up 6 cm, then left 13 cm? Messy.

Better to use the fact that we can calculate area by dividing into non-overlapping rectangles.

Let me define:

Rectangle A: the bottom-left part: width 5 cm, height 3 cm → Area = 15 cm²
Rectangle B: the part above A on the left: width 2 cm, height 6 cm → Area = 12 cm² (since from the step, it goes up 6 cm)
Rectangle C: the top part: width 13 cm, height 2 cm → Area = 26 cm²

Now, do these overlap? Rectangle A is from y=0 to y=3, x=0 to x=5
Rectangle B is from y=3 to y=9, x=0 to x=2
Rectangle C is from y=9 to y=11, x=0 to x=13

No overlap. Total area = 15 + 12 + 26 = 53 cm²

Check if this covers the whole shape.

The shape also has a part from x=2 to x=5 at y=3 to y=9? In my division, between x=2 to x=5, from y=3 to y=9, is that covered? In rectangle A, only up to y=3. Rectangle B is only x=0 to 2. So the region x=2 to 5, y=3 to 9 is not covered! Oh no.

Mistake.

From the diagram, after going up 3 cm from bottom-left, then right 2 cm, then up 6 cm — so at x=2 to x=5, from y=3 to y=9, there is a rectangle of width 3 cm (5-2=3), height 6 cm.

I missed that.

So correct decomposition:

- Rectangle 1: bottom: x=0 to 5, y=0 to 3 → 5×3=15
- Rectangle 2: left-middle: x=0 to 2, y=3 to 9 → 2×6=12
- Rectangle 3: middle: x=2 to 5, y=3 to 9 → 3×6=18
- Rectangle 4: top: x=0 to 13, y=9 to 11 → 13×2=26

But now, rectangle 2 and 3 together make x=0 to 5, y=3 to 9, which is 5×6=30, same as 12+18.

Then total area = 15 (bottom) + 30 (middle layer) + 26 (top) = 71 cm²

But is the top layer covering x=0 to 13, while the middle is only x=0 to 5? Yes, and the diagram shows the top extends further right.

Now, what about the right part? After the top, it goes down 2 cm, then left 10 cm, then down 7 cm.

From the top-right, down 2 cm brings us to y=9 (since top is at y=11, down 2 to y=9), then left 10 cm to x=3 (since 13-10=3), then down 7 cm to y=2, then left 5 cm to x=-2? That can't be.

I think I need to interpret the labels differently.

Let me list the vertices based on the path.

Assume start at bottom-left corner (0,0).

Move right 5 cm → (5,0)
Up 3 cm → (5,3)
Right 2 cm → (7,3)
Up 6 cm → (7,9)
Right 10 cm → (17,9)
Down 2 cm → (17,7)
Left 13 cm → (4,7) [since 17-13=4]
Then down? The diagram doesn't show, but probably down to connect.

From (4,7), if we go down, but the bottom is at y=0, and we have a segment labeled 7 cm down? The label "7 cm" is on the left side of the right part.

Perhaps from (4,7) down 7 cm to (4,0), then left to (0,0)? But then the bottom from x=0 to x=4 is already covered, but we have a bottom of 5 cm from (0,0) to (5,0), so conflict.

This is confusing. Let's look for another approach.

Notice that the shape can be seen as a large rectangle minus two smaller rectangles.

Or use the grid method.

From the dimensions given, the total width at the top is 13 cm, at the bottom is 5 cm, but with steps.

Another idea: calculate the area by adding the areas of the parts as per the labels.

From the diagram, it's composed of:

- A rectangle on the bottom-left: 5 cm by 3 cm
- A rectangle on the top-left: 2 cm by 6 cm (above the first 2 cm of the bottom)
- A rectangle on the top: 13 cm by 2 cm
- And a rectangle on the right-middle: but wait, after the top, it goes down 2 cm, then left 10 cm, then down 7 cm.

Perhaps the "7 cm" down is from y=9 to y=2, and "3 cm" up is from y=0 to y=3, etc.

Let's assume the following coordinates:

Set bottom-left as (0,0).

- Go right 5 cm to (5,0)
- Up 3 cm to (5,3)
- Right 2 cm to (7,3)
- Up 6 cm to (7,9)
- Right 10 cm to (17,9)
- Down 2 cm to (17,7)
- Left 13 cm to (4,7) [17-13=4]
- Down 7 cm to (4,0) [7-7=0]
- Left 4 cm to (0,0) [4-4=0] — but the bottom was from 0 to 5, so from (4,0) to (0,0) is 4 cm, but we have a segment from (0,0) to (5,0), so this would overlap or miss.

From (4,0) to (0,0) is 4 cm, but the initial bottom is 5 cm from (0,0) to (5,0), so if we go from (4,0) to (0,0), we are missing from (4,0) to (5,0).

This suggests that the point (4,0) is not correct.

Perhaps the "left 13 cm" from (17,7) is to (4,7), then down 7 cm to (4,0), then left to (0,0), but then the bottom from (0,0) to (5,0) is not fully used; only from (0,0) to (4,0) is used in this path, but the shape has a bottom of 5 cm, so maybe the bottom is from (0,0) to (5,0), and the left side from (0,0) to (0,3), etc.

I think there's a mistake in my interpretation.

Let me try to calculate the area using the shoelace formula or by dividing into known parts.

From the diagram, the shape can be divided into three rectangles:

1. The bottom rectangle: 5 cm wide, 3 cm high → area 15 cm²
2. The middle rectangle: 5 cm wide, 6 cm high, but shifted? No.

Notice that the total height on the left is 3 + 6 = 9 cm for the first 2 cm width, then from x=2 to x=5, height 3 cm (only the bottom), but that doesn't match.

Another way: the shape has a "base" of 5 cm by 3 cm.

Then on top of the left 2 cm of the base, there is a 2 cm by 6 cm rectangle.

Then on top of that, a 13 cm by 2 cm rectangle.

Additionally, on the right, from x=5 to x=17, but the top is only 13 cm, so perhaps from x=4 to x=17 or something.

Let's calculate the area as the area under the curve.

From x=0 to x=2: height = 3 + 6 + 2 = 11 cm? No, because the top 2 cm is only from y=9 to y=11, and the 6 cm is from y=3 to y=9, and 3 cm from y=0 to y=3, so for x=0 to 2, height is 11 cm.

For x=2 to x=5: height = 3 cm (only the bottom) + 6 cm (middle) = 9 cm? But the top 2 cm may not cover it.

From the top being 13 cm wide, and it's at the top, so for x=0 to 13, there is the top 2 cm high.

Then below that, from y=0 to y=9, the shape varies.

Perhaps:

- For x=0 to 2: height 11 cm (3+6+2)
- For x=2 to 5: height 9 cm (3+6) — but the top 2 cm is only up to y=11, and if the middle is up to y=9, then for x=2 to 5, from y=0 to y=9, height 9 cm.
- For x=5 to 13: height 2 cm (only the top)
- For x=13 to 17: but the top is only 13 cm, so perhaps not.

The right part: after the top 13 cm, it goes down 2 cm, then left 10 cm, then down 7 cm.

So from x=13 to x=17, there is a part.

Let's define the right part.

From the top-right (x=13, y=11), down 2 cm to (13,9), then left 10 cm to (3,9), then down 7 cm to (3,2), then left to (0,2)? But the bottom is at y=0.

This is complicated.

Perhaps the "7 cm" down is from y=9 to y=2, and "3 cm" up is from y=0 to y=3, so the bottom is from y=0 to y=3 for x=0 to 5, and from y=2 to y=3 for x=3 to 5 or something.

I recall that in such problems, the area can be calculated by considering the bounding box and subtracting, but let's try a different decomposition.

Let me split the shape into:

- Rectangle A: the very bottom: 5 cm by 3 cm = 15 cm²
- Rectangle B: the part above A on the left: 2 cm by 6 cm = 12 cm² (from y=3 to y=9, x=0 to 2)
- Rectangle C: the part to the right of B at the same level: 3 cm by 6 cm = 18 cm² (x=2 to 5, y=3 to 9) — but is this present? In the diagram, after going up 3 cm at x=5, then right 2 cm to x=7, then up 6 cm, so at x=2 to 5, from y=3 to y=9, it should be filled, yes.
- Rectangle D: the top: 13 cm by 2 cm = 26 cm² (y=9 to 11, x=0 to 13)
- Rectangle E: the right part: from x=5 to x=17, but the top is only to x=13, and after that, from x=13 to x=17, there is a part from y=7 to y=9 or something.

From the path: after reaching (7,9), go right 10 cm to (17,9), then down 2 cm to (17,7), then left 13 cm to (4,7), then down 7 cm to (4,0), then left to (0,0).

So from (4,0) to (0,0) is 4 cm, but the bottom is from (0,0) to (5,0), so from (4,0) to (5,0) is 1 cm not covered in this path, but in the shape, it is covered by the bottom rectangle.

So the shape includes:

- From x=0 to 5, y=0 to 3: rectangle 5x3=15
- From x=0 to 2, y=3 to 9: 2x6=12
- From x=2 to 5, y=3 to 9: 3x6=18
- From x=0 to 13, y=9 to 11: 13x2=26
- From x=4 to 17, y=7 to 9: but from (4,7) to (17,7) is 13 cm, and up to y=9, so 13x2=26, but this overlaps with the top rectangle.

This is messy.

Notice that from (4,7) to (17,7) to (17,9) to (4,9) would be a rectangle, but (4,9) is not a vertex; the top is at y=11 for x=0 to 13.

Perhaps the region from x=4 to 13, y=7 to 9 is already included in the top rectangle? No, the top rectangle is y=9 to 11.

Let's calculate the area using the path and shoelace formula.

List the vertices in order, say clockwise.

Start at (0,0)
-> (5,0) // right 5 cm
-> (5,3) // up 3 cm
-> (7,3) // right 2 cm
-> (7,9) // up 6 cm
-> (17,9) // right 10 cm
-> (17,7) // down 2 cm
-> (4,7) // left 13 cm (17-13=4)
-> (4,0) // down 7 cm (7-7=0)
-> (0,0) // left 4 cm (4-4=0) — but this closes to (0,0), and the bottom from (0,0) to (5,0) is already included, but from (4,0) to (0,0) is 4 cm, while the initial bottom is 5 cm, so the point (5,0) is not connected directly to (4,0); there's a gap.

From (4,0) to (0,0) is fine, but the shape has a bottom from (0,0) to (5,0), so when we go from (4,0) to (0,0), we are missing the segment from (4,0) to (5,0), but in the path, we have (5,0) to (5,3), so the region between x=4 to 5, y=0 to 3 is included in the first rectangle.

In the polygon defined by the vertices: (0,0), (5,0), (5,3), (7,3), (7,9), (17,9), (17,7), (4,7), (4,0), and back to (0,0).

This polygon includes the area, and we can use shoelace formula.

List the coordinates in order:

1. (0,0)
2. (5,0)
3. (5,3)
4. (7,3)
5. (7,9)
6. (17,9)
7. (17,7)
8. (4,7)
9. (4,0)
10. (0,0) // close

Shoelace formula:

Sum1 = sum of x_i * y_{i+1}
Sum2 = sum of y_i * x_{i+1}
Area = |Sum1 - Sum2| / 2

Compute:

i=1: x1=0,y1=0; x2=5,y2=0 -> 0*0 = 0
i=2: x2=5,y2=0; x3=5,y3=3 -> 5*3 = 15
i=3: x3=5,y3=3; x4=7,y4=3 -> 5*3 = 15
i=4: x4=7,y4=3; x5=7,y5=9 -> 7*9 = 63
i=5: x5=7,y5=9; x6=17,y6=9 -> 7*9 = 63
i=6: x6=17,y6=9; x7=17,y7=7 -> 17*7 = 119
i=7: x7=17,y7=7; x8=4,y8=7 -> 17*7 = 119
i=8: x8=4,y8=7; x9=4,y9=0 -> 4*0 = 0
i=9: x9=4,y9=0; x10=0,y10=0 -> 4*0 = 0
Sum1 = 0+15+15+63+63+119+119+0+0 = let's calculate: 15+15=30; 30+63=93; 93+63=156; 156+119=275; 275+119=394; 394+0+0=394

Sum2 = y_i * x_{i+1}

i=1: y1=0,x2=5 -> 0*5=0
i=2: y2=0,x3=5 -> 0*5=0
i=3: y3=3,x4=7 -> 3*7=21
i=4: y4=3,x5=7 -> 3*7=21
i=5: y5=9,x6=17 -> 9*17=153
i=6: y6=9,x7=17 -> 9*17=153
i=7: y7=7,x8=4 -> 7*4=28
i=8: y8=7,x9=4 -> 7*4=28
i=9: y9=0,x10=0 -> 0*0=0
Sum2 = 0+0+21+21+153+153+28+28+0 = 21+21=42; 42+153=195; 195+153=348; 348+28=376; 376+28=404; 404+0=404

Area = |394 - 404| / 2 = | -10 | / 2 = 10 / 2 = 5 cm²? That can't be right; too small.

I must have the order wrong. Perhaps counter-clockwise.

Or I missed that from (4,0) to (0,0) is correct, but the point (5,0) is included, so the polygon is self-intersecting or something.

Perhaps the last point should be (0,0), but the path from (4,0) to (0,0) is fine, but let's list the vertices again.

From the description, after (4,0), it should go to (0,0), but the bottom from (0,0) to (5,0) is already covered by the first edge, so in the polygon, the edge from (4,0) to (0,0) and from (0,0) to (5,0) are both there, so the region between x=0 to 4, y=0 is covered twice? No, in polygon, it's the boundary.

Perhaps the correct sequence is:

Start at (0,0)
-> (5,0) // bottom
-> (5,3) // up
-> (7,3) // right
-> (7,9) // up
-> (17,9) // right
-> (17,7) // down
-> (4,7) // left
-> (4,0) // down
-> (0,0) // left

But then the edge from (4,0) to (0,0) and from (0,0) to (5,0) means that the point (0,0) is visited twice, but in shoelace, it's ok as long as we close.

But in my calculation, Sum1=394, Sum2=404, difference 10, area 5, which is impossible.

I think I have a mistake in the coordinates.

When we go from (17,7) left 13 cm to (4,7), that's correct.

Then down 7 cm to (4,0), correct.

Then left to (0,0), so from (4,0) to (0,0), distance 4 cm.

But the initial bottom is from (0,0) to (5,0), so the polygon has vertices at (0,0), (5,0), (5,3), ..., (4,0), (0,0).

So the edge from (4,0) to (0,0) and from (0,0) to (5,0) are both present, so the line from x=0 to x=4 on y=0 is traversed twice: once from (0,0) to (5,0) , and once from (4,0) to (0,0), which is backwards, so in shoelace, it might cancel or something.

To avoid confusion, let's calculate the area as the sum of rectangles without overlap.

From the shape, it can be divided into:

- Rectangle 1: x=0 to 5, y=0 to 3: 5*3 = 15
- Rectangle 2: x=0 to 2, y=3 to 9: 2*6 = 12
- Rectangle 3: x=2 to 5, y=3 to 9: 3*6 = 18
- Rectangle 4: x=0 to 13, y=9 to 11: 13*2 = 26
- Rectangle 5: x=4 to 17, y=7 to 9: but this overlaps with rectangle 4 for x=4 to 13, y=9 to 11? No, y=7 to 9 is below y=9 to 11.

Rectangle 5: x=4 to 17, y=7 to 9: width 13 cm, height 2 cm, area 26 cm²

But is this part of the shape? From the path, yes, from (4,7) to (17,7) to (17,9) to (4,9), but (4,9) is not a vertex; in the path, from (17,9) to (17,7) to (4,7), so the top of this rectangle is at y=9, but in the shape, at x=4 to 13, y=9 is the bottom of the top rectangle, so for x=4 to 13, y=9 to 11 is rectangle 4, and y=7 to 9 is rectangle 5, so no overlap.

For x=13 to 17, y=7 to 9 is only rectangle 5.

So total area = rect1 + rect2 + rect3 + rect4 + rect5 = 15 + 12 + 18 + 26 + 26 = let's add: 15+12=27; 27+18=45; 45+26=71; 71+26=97 cm²

But is rect5 necessary? In the shape, from (4,7) to (17,7) to (17,9) to (4,9), but (4,9) is not connected; in the path, after (17,9) we go to (17,7), then to (4,7), then to (4,0), so the region between y=7 to 9, x=4 to 17 is included, and it is not covered by other rectangles, so yes.

Now, for x=0 to 4, y=7 to 9: is it covered? In rect2 and rect3, we have up to y=9 for x=0 to 5, so for x=0 to 4, y=7 to 9 is covered by rect2 and rect3 (since rect2 is x=0-2, rect3 x=2-5, so x=0-5, y=3-9).

Rect