Math worksheet for calculating area and perimeter of composite figures.

Let’s solve each problem step by step. We’ll find the area and perimeter for each composite figure.

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Problem 1)

Figure is L-shaped. Dimensions:
- Top horizontal: 10 cm
- Left vertical top part: 3 cm
- Bottom right vertical: 7 cm (total height on right)
- Bottom horizontal right part: 5 cm
- Inner horizontal cutout: 4 cm

We can split this into two rectangles:

Option A: Split vertically
- Left rectangle: width = 10 - 5 = 5 cm, height = 3 cm → area = 5 × 3 = 15 cm²
- Right rectangle: width = 5 cm, height = 7 cm → area = 5 × 7 = 35 cm²
→ Total area = 15 + 35 = 50 cm²

Wait — let’s check with another split to verify.

Option B: Split horizontally
- Top rectangle: 10 cm wide × 3 cm high = 30 cm²
- Bottom rectangle: only the right part sticks down. The bottom part has width 5 cm and height = 7 - 3 = 4 cm → area = 5 × 4 = 20 cm²
→ Total area = 30 + 20 = 50 cm² ✔

Now perimeter:

Trace all outer edges:

Start at top-left corner, go clockwise:

- Top: 10 cm
- Right side: 7 cm
- Bottom right: 5 cm
- Up inner vertical: 4 cm (since total right height is 7, top part is 3, so drop is 4? Wait — actually, from bottom of right rectangle up to where it meets left part: that’s 7 - 3 = 4 cm upward? But in perimeter, we walk along the outside.

Actually, better to list all outer sides:

From diagram:

Top: 10 cm
Right: 7 cm
Bottom: 5 cm
Then up the inner step: 4 cm (this is vertical)
Then left along the inner horizontal: ? Let's see — the top part is 10 cm, bottom right is 5 cm, so the inner horizontal must be 10 - 5 = 5 cm? But labeled as 4 cm? Wait — look again.

In the diagram for #1:

It shows:

- Top edge: 10 cm
- Left edge (top part): 3 cm
- Then a horizontal segment going right: labeled 4 cm? Actually, looking at the drawing: after going down 3 cm on left, then goes right 4 cm, then down to meet the bottom.

Actually, standard way: label all outer segments.

Let me reconstruct coordinates mentally.

Assume bottom-left corner is origin.

But easier: use “add all outer sides”.

Perimeter = sum of all outer boundary lengths.

Looking at shape:

Starting from top-left:

→ Right 10 cm
↓ Down 7 cm (right side)
← Left 5 cm (bottom)
↑ Up 4 cm (inner vertical)
← Left 4 cm? Wait — no.

Actually, from the labels:

The figure has:

- Top: 10 cm
- Right: 7 cm
- Bottom: 5 cm
- Then an upward step: the vertical part between bottom and the lower part of left side — since left side total height should match right? Not necessarily.

Wait — left side has two parts: top 3 cm, then a gap, then... actually, the left side isn’t continuous.

Better approach: use the fact that for such L-shapes, perimeter equals the perimeter of the bounding box if you "push out" the indentations — but only if it’s rectilinear.

Actually, for any rectilinear polygon (all angles 90°), the perimeter is equal to twice the sum of the maximum width and maximum height — IF there are no holes or overlaps. But here, because of the notch, we have extra sides.

Standard method: trace the path.

Let’s assign points:

Start at top-left corner A.

A → B: right 10 cm (top)

B → C: down 7 cm (right side)

C → D: left 5 cm (bottom)

D → E: up 4 cm (this is the inner vertical — because from bottom, going up to where the horizontal cut starts; note that the left part above is 3 cm high, so from bottom to that level is 7 - 3 = 4 cm? Yes.)

E → F: left ? How far? From point E, we go left to meet the left side. The top was 10 cm, bottom right is 5 cm, so the horizontal distance from E to left side is 10 - 5 = 5 cm? But the diagram says “4cm” near that inner horizontal.

Look back at original image description:

In problem 1, it shows:

After going down 3 cm on left, then a horizontal segment labeled “4cm” going right, then down to connect to the bottom.

So actually:

Left side: from top, down 3 cm → then right 4 cm → then down to bottom.

And bottom right is 5 cm.

Total width at top: 10 cm.

So from left end to start of the 4cm segment: that’s the left rectangle width.

If top is 10 cm, and after going down 3 cm, we go right 4 cm, then down, and the bottom right part is 5 cm, then the remaining part on the bottom must be 10 - 4 - 5 = 1 cm? That doesn't make sense.

I think I misread.

Let me interpret the diagram correctly based on standard problems.

Typically in such diagrams:

For figure 1:

- The overall width is 10 cm (top)
- Overall height is 7 cm (right side)
- On the left, from top, down 3 cm, then right some amount, then down to bottom.
- The bottom has a segment of 5 cm on the right.
- The inner horizontal is labeled 4 cm — which is likely the length of the horizontal segment after dropping 3 cm on left.

So:

From top-left:

Down 3 cm → then right 4 cm → then down to bottom. Since total height is 7 cm, this last down segment is 7 - 3 = 4 cm.

Then from there, left to the left edge? No — after going down 4 cm, we are at the bottom level, and we need to go left to close the shape.

The bottom has a segment labeled 5 cm on the right — so from the right end, left 5 cm.

So the full bottom width: from left to right, it should be the same as top? Not necessarily, but in this case, let's calculate.

When we go from the inner corner (after down 3, right 4, down 4), we are now at a point that is 4 cm from the left? Let's set coordinates.

Set top-left as (0,7) — y increases downward for simplicity.

Point A: (0,7) — top-left

Go right to B: (10,7) — top-right

Go down to C: (10,0) — bottom-right (since height 7)

Go left to D: (5,0) — because bottom right segment is 5 cm, so from x=10 to x=5 at y=0

Now, from D (5,0), we go up to E: how far? The inner vertical is not directly given, but we know that on the left, from top, we went down 3 cm to a point, say F: (0,4) [since y=7-3=4]

Then from F (0,4) we go right to G: (4,4) — because labeled 4 cm

Then from G (4,4) we go down to H: (4,0) — to meet the bottom level

Then from H (4,0) to D (5,0)? That would be 1 cm, but not labeled.

This is messy.

Perhaps the "4cm" is the horizontal segment between the two verticals.

Another way: the figure can be seen as a large rectangle minus a small rectangle, but for perimeter, it's different.

Let's use the standard trick for rectilinear shapes: the perimeter is the same as the bounding box plus twice the depth of the indentation — but only if it's a single notch.

Bounding box: width 10 cm, height 7 cm → perimeter 2*(10+7)=34 cm

But because of the notch, we have two additional sides: the inner vertical and inner horizontal that are exposed.

In this case, the notch is on the bottom-left part.

From the diagram, when you go from left side down 3 cm, then right 4 cm, then down 4 cm to bottom, then left to the left edge? No.

Let's list all outer edges explicitly from the diagram description:

Segments:

1. Top: 10 cm (horizontal)

2. Right side: 7 cm (vertical down)

3. Bottom right: 5 cm (horizontal left)

4. Inner vertical up: this is the rise from bottom to the level of the horizontal cut. Since the left part has a drop of 3 cm from top, and total height is 7 cm, the distance from bottom to that level is 4 cm. So this segment is 4 cm up.

5. Inner horizontal left: this is the segment labeled "4cm" in the diagram? In the user's image, it says "4cm" on the horizontal part after the first drop.

In the text: "4cm" is written on the horizontal segment that is inside, between the left drop and the right part.

So after going up 4 cm from bottom, we go left 4 cm.

6. Then from there, we go up to the top-left? No, we are at a point that is 4 cm from the left? Let's see.

After step 5: we have gone left 4 cm from the inner corner.

Where is that inner corner? After step 3: we are at (5,0) if we assume bottom-right is (10,0), but let's define.

Assume the bottom-right corner is at (0,0) for ease.

Set coordinate system with origin at bottom-right.

So:

- Bottom-right: (0,0)

- Go left 5 cm to ( -5, 0) — this is the bottom right segment

- From (-5,0), go up 4 cm to (-5,4) — this is the inner vertical (since total height is 7, and top is at y=7, so from y=0 to y=4)

- From (-5,4), go left 4 cm to (-9,4) — this is the inner horizontal labeled "4cm"

- From (-9,4), go up 3 cm to (-9,7) — this is the left top part, labeled "3cm"

- From (-9,7), go right 10 cm to (1,7) — this is the top, labeled "10cm"

- From (1,7), go down 7 cm to (1,0) — but wait, this should connect to bottom-right (0,0)? No, (1,0) to (0,0) is 1 cm, but not accounted for.

Mistake.

If top is 10 cm from left to right, and left is at x= -9, then right should be at x= -9 +10 =1, so top-right is (1,7)

Bottom-right is (0,0), so from (1,7) down to (1,0), then left to (0,0)? But that would mean the right side is from (1,7) to (1,0), which is 7 cm, good, but then from (1,0) to (0,0) is 1 cm, which is not labeled, and the bottom right segment is labeled 5 cm, which would be from (0,0) to (-5,0), so total bottom from (-5,0) to (0,0) is 5 cm, but from (0,0) to (1,0) is additional 1 cm, so the full bottom is from (-5,0) to (1,0), which is 6 cm, but not consistent.

I think the issue is that the "5cm" on the bottom is the entire bottom right part, but in reality, for the shape to close, the bottom must extend to the left edge.

Perhaps the 5cm is the width of the bottom rectangle, and the left part is separate.

Let's calculate area first, which is easier, and for perimeter, use the fact that for such shapes, the perimeter can be found by adding all outer sides as per the labels.

From the diagram, the outer perimeter consists of:

- Top: 10 cm

- Right: 7 cm

- Bottom: 5 cm

- Then the inner vertical: 4 cm (up)

- Then the inner horizontal: 4 cm (left) — but this is not outer; it's inner, but in the boundary, it is part of the perimeter because it's exposed.

In composite figures, the perimeter includes all outer edges, including those of the notch.

So for figure 1, the perimeter path is:

Start at top-left:

1. Right 10 cm (top)

2. Down 7 cm (right side)

3. Left 5 cm (bottom right)

4. Up 4 cm (inner vertical) — this is the rise to the level of the horizontal cut

5. Left 4 cm (inner horizontal) — this is the segment labeled "4cm"

6. Up 3 cm (left side top part) — labeled "3cm"

7. Right ? to close to start. From the end of step 6, we are at the top-left, but after going left 4 cm in step 5, and up 3 cm in step 6, we are at a point that is 4 cm from the left edge? Let's see the net displacement.

After step 1: at (10,7) if start at (0,7)

Perhaps start at (0,7):

- Right to (10,7) : 10 cm

- Down to (10,0) : 7 cm

- Left to (5,0) : 5 cm (since bottom right is 5 cm)

- Up to (5,4) : 4 cm (because the horizontal cut is at y=4, since from top down 3 cm is y=4 if y=7 at top)

- Left to (1,4) : 4 cm (labeled "4cm")

- Up to (1,7) : 3 cm (labeled "3cm")

- Now from (1,7) to (0,7) : 1 cm left to close the shape.

Ah! So there is an additional 1 cm on the top-left.

But in the diagram, the top is labeled 10 cm, which is from (0,7) to (10,7), so from (1,7) to (0,7) is part of the top, but we already did the top as 10 cm from (0,7) to (10,7), so when we come back to (1,7), we need to go to (0,7), which is 1 cm, but that would mean the top is not fully covered.

This is confusing.

Perhaps the "10cm" is the total top width, and the left part is included.

Let's calculate the area using decomposition.

Split into two rectangles:

Rectangle 1: the top part: width 10 cm, height 3 cm → area = 30 cm²

Rectangle 2: the bottom right part: width 5 cm, height 4 cm (since 7-3=4) → area = 20 cm²

Total area = 30 + 20 = 50 cm²

For perimeter, the outer boundary:

- Top: 10 cm

- Right: 7 cm

- Bottom: 5 cm

- Left side: but the left side is not straight; it has a bite taken out.

From the bottom, on the left, we have a vertical segment from y=0 to y=4 at x=5? Let's think.

The shape has the following outer edges:

- From (0,7) to (10,7): 10 cm

- (10,7) to (10,0): 7 cm

- (10,0) to (5,0): 5 cm

- (5,0) to (5,4): 4 cm (up)

- (5,4) to (0,4): 5 cm? But labeled as 4 cm in the diagram? No, the diagram has "4cm" on the horizontal segment after the first drop.

In the user's image, for figure 1, it shows:

After the left 3cm down, then a horizontal segment labeled "4cm" going right, then down to the bottom.

So from the left edge, down 3 cm, then right 4 cm, then down 4 cm to bottom (since 7-3=4), then right to the right edge? But the bottom right is labeled 5 cm, so from the end of the down 4 cm, we go right 5 cm to the right edge.

So let's define:

Start at top-left (0,7)

- Down to (0,4): 3 cm

- Right to (4,4): 4 cm

- Down to (4,0): 4 cm

- Right to (9,0): 5 cm? 4 to 9 is 5 cm, so (9,0)

- Up to (9,7): 7 cm? But then from (9,7) to (0,7) is 9 cm, but top is labeled 10 cm.

Not matching.

If top is 10 cm, from (0,7) to (10,7)

Then from (10,7) down to (10,0): 7 cm

From (10,0) left to (5,0): 5 cm

From (5,0) up to (5,4): 4 cm

From (5,4) left to (0,4): 5 cm? But labeled as 4 cm.

I think there's a mistake in my interpretation.

Let me look for a standard solution or rethink.

Perhaps the "4cm" is the length of the horizontal segment that is the "step", and the left part is 10 - 4 - 5 = 1 cm, but that seems odd.

Another idea: the figure is composed of a 10x3 rectangle on top, and a 5x4 rectangle on the bottom right, but they overlap or something.

Area is definitely 10*3 + 5*4 = 30 + 20 = 50 cm², as before.

For perimeter, when you have two rectangles sharing a common part, but in this case, they are adjacent.

The shared edge is not part of the perimeter.

In this L-shape, the two rectangles share a edge of length min(3,4) or something.

Rectangle A: 10 cm x 3 cm (top)

Rectangle B: 5 cm x 4 cm (bottom right)

They are attached such that the bottom of A and top of B are at the same y-level? No.

Typically, the top rectangle sits on top of the bottom rectangle, but offset.

In this case, the bottom rectangle is under the right part of the top rectangle.

So the combined shape has:

- Width: max(10,5) = 10 cm

- Height: 3 + 4 = 7 cm

But with a notch on the bottom-left.

The perimeter can be calculated as:

Perimeter = 2 * (width + height) + 2 * depth of notch, but only if the notch is rectangular.

Here, the notch is on the bottom-left, with width w and depth d.

From the dimensions, the notch has width = 10 - 5 = 5 cm? But labeled as 4 cm for the horizontal.

Perhaps the 4 cm is the depth or something.

Let's calculate the perimeter by adding all sides as per the path.

Assume the following vertices in order:

Start at top-left: P1

P1 to P2: right 10 cm (top)

P2 to P3: down 7 cm (right side)

P3 to P4: left 5 cm (bottom right)

P4 to P5: up 4 cm (inner vertical) — this is the rise to the level of the horizontal cut

P5 to P6: left 4 cm (inner horizontal) — labeled "4cm"

P6 to P1: up 3 cm (left side) — labeled "3cm"

Now, from P6 to P1 is 3 cm up, but what is the horizontal distance? If P6 is at (x,y), P1 is at (0,7), and P6 is at (4,4) if we set P1 at (0,7), P2 at (10,7), P3 at (10,0), P4 at (5,0), P5 at (5,4), P6 at (1,4) because from (5,4) left 4 cm to (1,4), then from (1,4) to (0,7) is not direct; it should be to (0,4) then to (0,7), but that would be two segments.

From P6 (1,4) to P1 (0,7): this is not axis-aligned; it must be via (0,4) or (1,7).

So likely, from P6 (1,4) , we go left to (0,4): 1 cm, then up to (0,7): 3 cm.

But the 3 cm is already labeled as the left side, which is from (0,4) to (0,7).

So the segments are:

- P1(0,7) to P2(10,7): 10 cm

- P2(10,7) to P3(10,0): 7 cm

- P3(10,0) to P4(5,0): 5 cm

- P4(5,0) to P5(5,4): 4 cm

- P5(5,4) to P6(1,4): 4 cm (labeled)

- P6(1,4) to P7(0,4): 1 cm (not labeled, but necessary)

- P7(0,4) to P1(0,7): 3 cm (labeled)

So total perimeter = 10 + 7 + 5 + 4 + 4 + 1 + 3 = 34 cm

But 10+7=17, +5=22, +4=26, +4=30, +1=31, +3=34 cm.

And area is 50 cm², as before.

Is the 1 cm segment implied? In many such problems, the unlabeled segments are to be inferred.

Perhaps the "4cm" is meant to be the distance from the left, but let's check with other figures.

For figure 2, it might be clearer.

Let's do figure 2 first, as it might be simpler.

Problem 2)

Figure 2: L-shaped, dimensions:

- Top: 6 mm

- Left: 8 mm

- Bottom: 10 mm

- Right: 6 mm

So, similar L-shape.

Split into two rectangles:

Option: top rectangle: 6 mm x ? height.

Total height on left is 8 mm, on right is 6 mm, so the difference is 2 mm.

So, the top part has height h, bottom part has height 8-h, but on the right, the height is 6 mm, so probably the bottom rectangle has height 6 mm, and the top rectangle has height 8-6=2 mm.

Width: top is 6 mm, bottom is 10 mm, so the top rectangle is 6 mm wide, 2 mm high.

Bottom rectangle is 10 mm wide, 6 mm high, but they overlap on the right part.

Actually, the bottom rectangle extends full width 10 mm, height 6 mm.

The top rectangle is on the left, width 6 mm, height 2 mm, sitting on top of the bottom rectangle.

So area = area of bottom + area of top = 10*6 + 6*2 = 60 + 12 = 72 mm²

But is that correct? The top rectangle is only on the left, so yes, no overlap.

Total area = 72 mm²

Perimeter:

Outer boundary:

Start at top-left:

- Right 6 mm (top of top rectangle)

- Down 2 mm (right side of top rectangle) — but this is inner if there's something below, but in this case, below it is the bottom rectangle, so this down 2 mm is not outer; it's internal.

Mistake.

When we have the top rectangle on the left, and bottom rectangle full width, then the right side of the top rectangle is adjacent to the bottom rectangle, so it's not part of the perimeter.

So for perimeter, we need to trace the outer edge.

Vertices:

Set P1 at top-left (0,8) — y down.

P1 to P2: right 6 mm to (6,8) — top of top rectangle

P2 to P3: down to (6,6) — because the bottom rectangle starts at y=6? Total height is 8 mm, bottom rectangle height 6 mm, so from y=2 to y=8 for bottom? Let's define y=0 at bottom.

Set P1 at (0,8) top-left

P2 at (6,8) — top-right of top rectangle

P3 at (6,6) — down 2 mm (height of top rectangle)

P4 at (10,6) — right 4 mm? But bottom is 10 mm, so from x=6 to x=10 at y=6

P5 at (10,0) — down 6 mm (height of bottom rectangle)

P6 at (0,0) — left 10 mm (bottom)

P7 at (0,8) — up 8 mm (left side)

But from P6(0,0) to P7(0,8) is 8 mm, good.

From P3(6,6) to P4(10,6): this is the top of the bottom rectangle, but is it outer? Yes, because above it is air on the right, but on the left, above P3 is the top rectangle, so from P3 to P4 is exposed.

In this path, from P2(6,8) to P3(6,6): down 2 mm — this is the right side of the top rectangle, and since there is nothing to the right of it at that y-level, it is outer.

Then from P3(6,6) to P4(10,6): right 4 mm — this is the top of the bottom rectangle on the right part, which is exposed.

Then P4 to P5: down 6 mm

P5 to P6: left 10 mm

P6 to P7: up 8 mm

P7 to P1: but P7 is (0,8), P1 is (0,8), so done.

Segments:

- P1 to P2: 6 mm

- P2 to P3: 2 mm (down)

- P3 to P4: 4 mm (right) — since 10-6=4

- P4 to P5: 6 mm (down)

- P5 to P6: 10 mm (left)

- P6 to P7: 8 mm (up)

Sum: 6+2+4+6+10+8 = 36 mm

Area: top rectangle 6*2=12, bottom rectangle 10*6=60, total 72 mm²

But is the bottom rectangle height 6 mm? From y=0 to y=6, and top rectangle from y=6 to y=8, so yes.

In the diagram, the right side is labeled 6 mm, which is from y=0 to y=6, good.

Left side 8 mm, from y=0 to y=8, good.

Top 6 mm, good.

Bottom 10 mm, good.

So for figure 2, area = 72 mm², perimeter = 36 mm.

Now back to figure 1.

For figure 1, similarly, let's assume:

Total height 7 cm, total width 10 cm.

On the left, from top, down 3 cm, then right 4 cm, then down to bottom.

So the top rectangle is 10 cm wide, 3 cm high? But then the bottom part.

The bottom part has width 5 cm on the right, and the inner horizontal is 4 cm, so perhaps the left part of the bottom is missing.

From the path:

P1(0,7) to P2(10,7): 10 cm

P2(10,7) to P3(10,0): 7 cm

P3(10,0) to P4(5,0): 5 cm

P4(5,0) to P5(5,4): 4 cm (up) — since the horizontal cut is at y=4 (because from top down 3 cm is y=4 if y=7 at top)

P5(5,4) to P6(1,4): 4 cm (left) — labeled "4cm"

P6(1,4) to P7(0,4): 1 cm (left) — not labeled, but necessary to reach left edge

P7(0,4) to P1(0,7): 3 cm (up) — labeled "3cm"

So perimeter = 10+7+5+4+4+1+3 = 34 cm

Area: can be calculated as large rectangle minus the missing part.

Large rectangle 10x7 = 70 cm²

Missing part: a rectangle at bottom-left, from x=0 to x=1, y=0 to y=4? From P6(1,4) to P7(0,4) to (0,0) to (1,0), but (1,0) to (5,0) is part of the shape, so the missing part is from x=0 to x=1, y=0 to y=4, size 1x4 = 4 cm²

So area = 70 - 4 = 66 cm²? But earlier I had 50, contradiction.

With the decomposition: top rectangle 10x3 = 30

Bottom right rectangle 5x4 = 20, but they don't overlap, so 50, but according to this, it should be 66, so inconsistency.

I think the error is in the coordinate system.

If P1(0,7), P2(10,7), P3(10,0), P4(5,0), then from P4(5,0) to P5(5,4), then to P6(1,4), then to P7(0,4), then to P1(0,7).

The region enclosed: from x=0 to x=10, but with a bite from x=0 to x=1, y=0 to y=4 removed? No, because from (0,4) to (0,7) is there, and from (0,0) to (1,0) is not part of the boundary; in this path, the bottom is from (5,0) to (10,0), and from (0,4) to (0,7), but what about from (0,0) to (0,4)? It's not included; the shape does not include the area below y=4 for x<1.

So the shape consists of:

- Rectangle A: x=0 to 10, y=4 to 7 (height 3) — area 10*3 = 30

- Rectangle B: x=1 to 5, y=0 to 4 (width 4, height 4) — area 4*4 = 16

- Rectangle C: x=5 to 10, y=0 to 4 (width 5, height 4) — area 5*4 = 20

But rectangle B and C are both at y=0 to 4, so together x=1 to 10, y=0 to 4, area 9*4 = 36, plus rectangle A 30, total 66 cm².

But in the diagram, the bottom right is labeled 5 cm, which would be from x=5 to 10, good, and the inner horizontal is 4 cm, from x=1 to 5? 5-1=4, yes.

And the left drop is 3 cm, from y=7 to y=4, good.

So area = 30 + 36 = 66 cm²? But 30 for top, and for bottom, from x=1 to 10, y=0 to 4, which is 9*4=36, total 66.

But earlier I thought 50, which is wrong.

In the initial decomposition, I said top 10x3=30, bottom right 5x4=20, but that misses the part from x=1 to 5, y=0 to 4, which is also there.

In the shape, after going down 3 cm on left, then right 4 cm, then down 4 cm, then right 5 cm to the right edge, so the bottom part is from x=1 to 10, y=0 to 4, with a "notch" on the left, but in this case, from x=0 to 1, y=0 to 4 is not part of the shape; it's cut out.

So yes, area = area of top rectangle (10x3) + area of bottom rectangle (9x4) = 30 + 36 = 66 cm²

Bottom rectangle is from x=1 to 10, width 9 cm, height 4 cm, area 36.

Total 66 cm².

Perimeter as calculated: 10 (top) + 7 (right) + 5 (bottom right) + 4 (up) + 4 (left) + 1 (left to edge) + 3 (up left) = 34 cm

The 1 cm is from x=1 to x=0 at y=4, which is necessary.

So for figure 1: area 66 cm², perimeter 34 cm.

But let's verify with another method.

Use the formula for rectilinear polygon: sum of all horizontal projections times 2, etc., but perhaps accept this.

For figure 3 and 4, we can do similarly.

Perhaps the "4cm" in figure 1 is the width of the bottom left part, but in the diagram, it's labeled on the horizontal segment after the drop, which is from the left drop to the start of the bottom right, so if bottom right is 5 cm, and top is 10 cm, then the horizontal segment should be 10 - 5 = 5 cm, but it's labeled 4 cm, so perhaps it's 4 cm, and the remaining is 1 cm, as above.

So I'll go with that.

So for figure 1:
Area: 66 cm²
Perimeter: 34 cm

But let's double-check with the numbers.

Another way: the area can be calculated as the area of the bounding box minus the missing rectangle.

Bounding box: 10 cm x 7 cm = 70 cm²

Missing rectangle: at bottom-left, from x=0 to x=1, y=0 to y=4, size 1x4 = 4 cm²

So 70 - 4 = 66 cm², good.

Perimeter: for a rectangle with a rectangular notch, the perimeter is 2*(L+W) + 2*depth, but here the notch is on the corner, so it adds two sides.

Original rectangle perimeter 2*(10+7)=34 cm

When you cut out a rectangle from the corner, you remove two sides but add two new sides of the same length, so perimeter remains the same if you cut from the corner.

In this case, cutting out a 1x4 rectangle from the bottom-left corner: you remove the bottom side from x=0 to x=1 and the left side from y=0 to y=4, but add the new sides: the top of the cut (from x=0 to x=1 at y=4) and the right of the cut (from y=0 to y=4 at x=1).

So you remove 1 cm (bottom) + 4 cm (left) = 5 cm, and add 1 cm (new top) + 4 cm (new right) = 5 cm, so perimeter unchanged, still 34 cm.

Yes! So for figure 1, perimeter is 34 cm, area 66 cm².

Great.

Now figure 2 we did: area 72 mm², perimeter 36 mm.

Confirm figure 2.

Bounding box: width 10 mm, height 8 mm, area 80 mm²

Missing part: at top-right, a rectangle from x=6 to x=10, y=6 to y=8, size 4x2 = 8 mm²

So area = 80 - 8 = 72 mm², good.

Perimeter: original rectangle 2*(10+8)=36 mm

Cutting out a rectangle from the top-right corner: remove top side from x=6 to x=10 (4 mm) and right side from y=6 to y=8 (2 mm), add new sides: bottom of cut (x=6 to x=10 at y=6) and left of cut (y=6 to y=8 at x=6), so add 4 mm + 2 mm = 6 mm, remove 4+2=6 mm, so perimeter unchanged, 36 mm.

Perfect.

Now figure 3.

Problem 3)

Dimensions:
- Top left: 5 m
- Vertical drop: 6 m
- Horizontal right: 8 m
- Bottom right vertical: 3 m

So L-shape.

Likely, total width and height.

From the labels:

From top-left, down 6 m, then right 8 m, then down 3 m to bottom.

Top is 5 m.

So probably, the top part is 5 m wide, and the bottom part extends further.

Assume:

P1(0,h) top-left

P2(5,h) — right 5 m (top)

P3(5,h-6) — down 6 m

P4(5+8,h-6) = (13,h-6) — right 8 m

P5(13,h-6-3) = (13,h-9) — down 3 m

Then to close, from P5 to P6(0,h-9) — left 13 m? But not labeled.

From P5(13,h-9) to P6(0,h-9) — bottom, but length not given.

From P6(0,h-9) to P1(0,h) — left side, length h - (h-9) = 9 m, but not labeled.

In the diagram, the bottom right vertical is 3 m, and the horizontal is 8 m, top is 5 m, left drop is 6 m.

Probably, the left side total height is 6 + 3 = 9 m.

So h = 9 m.

So P1(0,9)

P2(5,9) — top 5 m

P3(5,3) — down 6 m (since 9-6=3)

P4(13,3) — right 8 m (5+8=13)

P5(13,0) — down 3 m (3-0=3)

P6(0,0) — left 13 m (bottom)

P7(0,9) — up 9 m (left side)

But from P6(0,0) to P7(0,9) is 9 m, good.

Segments:

- P1 to P2: 5 m

- P2 to P3: 6 m (down)

- P3 to P4: 8 m (right)

- P4 to P5: 3 m (down)

- P5 to P6: 13 m (left) — since from x=13 to x=0

- P6 to P7: 9 m (up)

Sum: 5+6+8+3+13+9 = 44 m

Area: can be split into two rectangles.

Rectangle A: left part, x=0 to 5, y=3 to 9 (height 6) — area 5*6 = 30 m²

Rectangle B: bottom part, x=0 to 13, y=0 to 3 (height 3) — area 13*3 = 39 m²

But they overlap on x=0 to 5, y=3 to 3? No, rectangle A is y=3 to 9, rectangle B is y=0 to 3, so no overlap, but at y=3, it's shared, but for area, ok.

Total area = 30 + 39 = 69 m²

Is that correct? The shape includes from x=0 to 5, y=0 to 9? No, in rectangle B, it's x=0 to 13, y=0 to 3, and rectangle A is x=0 to 5, y=3 to 9, so together, it's x=0 to 5, y=0 to 9 and x=5 to 13, y=0 to 3.

Yes, which matches the path.

Area = 5*9 + 8*3 = 45 + 24 = 69 m²? 5*9=45 for left column, but the left column is only up to y=9, but in y=0 to 3, it's included in both? No.

Better: the shape is union of:

- Rectangle 1: x=0 to 5, y=0 to 9 — area 45 m²

- Rectangle 2: x=5 to 13, y=0 to 3 — area 8*3=24 m²

But they overlap on x=5 to 5? No, at x=5, it's a line, so no area overlap.

Rectangle 1 is x=0 to 5, y=0 to 9

Rectangle 2 is x=5 to 13, y=0 to 3

So together, for