Math worksheet for calculating the area of composite rectangles, featuring eight different shapes with given dimensions.

Let’s solve each problem one by one. We’ll break each composite shape into rectangles, find the area of each rectangle (length × width), and then add them up.

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Problem 1:
Shape is an L-shape.
We can split it into two rectangles:
- Top rectangle: 30 ft long × 15 ft high → Area = 30 × 15 = 450 sq ft
- Bottom left rectangle: 5 ft wide × (20 - 15) = 5 ft high → Area = 5 × 5 = 25 sq ft
Total Area = 450 + 25 = 475 sq ft

Wait — let me double-check. The total height on the left is 20 ft, and the top part is 15 ft tall, so the bottom vertical part is 5 ft tall. And its width is 5 ft. Yes, that’s correct.

✔ Final for #1: 475 sq ft

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Problem 2:
This is a big rectangle with a small rectangle cut out from the middle-left.
Big rectangle: 25 cm × 30 cm = 750 sq cm
Cut-out rectangle: 5 cm × 10 cm = 50 sq cm? Wait — look again.

Actually, the shape has a “notch” on the left side. The full height is 30 cm. The notch is 5 cm wide and goes down 10 cm from the top? But there’s also 10 cm below it. So the notch is 5 cm wide and 10 cm tall? Let me visualize:

The left side has two segments: 10 cm (top), then a 5 cm inward step, then another 10 cm down. So the missing piece is 5 cm wide and 10 cm tall? Actually, no — the entire left side is indented in the middle.

Better way: Think of it as three parts or subtract the hole.

Full outer box: 25 cm wide × 30 cm tall = 750 sq cm

But there’s a rectangular hole on the left: 5 cm wide and how tall? From the diagram, the indentation starts after 10 cm from top and ends before 10 cm from bottom — so the hole is 10 cm tall? Wait, the labels say:

Left side: 10 cm (top segment), then 5 cm (horizontal indent), then 10 cm (bottom segment). So the vertical gap where the rectangle is missing is between those two 10 cm sections — meaning the missing rectangle is 5 cm wide and (30 - 10 - 10) = 10 cm tall.

Yes! So missing area = 5 × 10 = 50 sq cm

So actual area = 750 - 50 = 700 sq cm

✔ Final for #2: 700 sq cm

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Problem 3:
T-shaped figure. Can be split into top rectangle and bottom stem.

Top rectangle: 36 m wide × ? How tall? Total height is 24 m, and the stem is 10 m wide at bottom. The stem’s height isn’t given directly.

Wait — better to split horizontally.

Option: Split into top bar and bottom rectangle.

Top bar: 36 m wide × (24 - ?) — we don’t know the height of the top bar.

Alternative: The bottom part is 10 m wide and extends downward. The total width is 36 m, so the overhang on each side is (36 - 10)/2 = 13 m? But the diagram doesn’t show symmetry.

Looking at the diagram: It says 14 m on the left side of the bottom part? Wait — label says “14 m” next to the left side of the lower rectangle? Actually, re-examining:

The shape has a top rectangle 36 m wide, and a bottom rectangle sticking down, which is 10 m wide. On the left side, from the bottom of the top rectangle to the bottom of the whole shape, it’s labeled 14 m? No — wait, the label “14 m” is on the left side of the entire figure? That can’t be because total height is 24 m.

I think I misread. Let me parse the labels:

- Top edge: 36 m
- Right side: 24 m (total height)
- Bottom of the protruding part: 10 m (width of the stem)
- Left side of the stem: 14 m — this must be the height of the stem? But 14 m + something = 24 m? Then the top part would be 10 m tall?

Actually, looking at standard T-shapes: usually the top bar has some height, and the stem has some height.

Assume: The top rectangle is 36 m wide and H1 tall, the stem is 10 m wide and H2 tall, and H1 + H2 = 24 m.

But we’re not given H1 or H2 directly. However, on the left side, it says “14 m” — that might be the height of the stem? Because if the stem is 14 m tall, then the top bar is 24 - 14 = 10 m tall.

That makes sense. Let’s go with that.

So:
- Top rectangle: 36 m × 10 m = 360 sq m
- Stem: 10 m × 14 m = 140 sq m
Total = 360 + 140 = 500 sq m

Is that right? Check: If stem is 14 m tall, and top is 10 m, total 24 m — matches. Widths match. Yes.

✔ Final for #3: 500 sq m

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Problem 4:
T-shape again, but upside-down? Or just a T.

Labels:
- Top bar: 12 ft wide, 6 ft tall? Wait — the 6 ft is on the right side of the top bar.
- Then the stem: 8 ft tall, and on the sides, it says 3 ft — probably the overhang on each side.

So top bar: 12 ft wide × 6 ft tall = 72 sq ft

Stem: width = total width minus overhangs? Overhangs are 3 ft on each side, so stem width = 12 - 3 - 3 = 6 ft? But the diagram shows the stem is centered, and height is 8 ft.

Wait — actually, the 3 ft is labeled on the left and right of the stem, meaning the horizontal distance from the edge of the top bar to the edge of the stem is 3 ft on each side. So yes, stem width = 12 - 3 - 3 = 6 ft.

Height of stem is 8 ft.

So stem area = 6 × 8 = 48 sq ft

Total area = 72 + 48 = 120 sq ft

✔ Final for #4: 120 sq ft

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Problem 5:
Zigzag shape — looks like three rectangles stacked with offsets.

From top to bottom:

Top rectangle: 14 yd long × 8 yd high? Wait — the 8 yd is on the right side of the top part.

Actually, let's trace:

- Top part: length 14 yd, height 8 yd? But then it steps down.

Better to divide into three horizontal rectangles:

1. Top rectangle: 14 yd × 8 yd = 112 sq yd
2. Middle rectangle: it’s offset. The horizontal part is 5 yd? Labels: after the top, it goes down 8 yd? No.

Look at the dimensions:

- Top: 14 yd (length), 8 yd (height) — but then it indents.
- Then a vertical drop of 8 yd? No, the label "8 yd" is on the right side of the top block.

Actually, the shape has:

- A top rectangle: 14 yd wide × 8 yd tall
- Then a middle section that sticks out to the left? The label "5 yd" is on the left extension.
- Then a bottom rectangle: 8 yd tall? And the right side has "5 yd" indentation.

Perhaps it's easier to see as three separate rectangles:

Rectangle A (top): 14 yd × 8 yd = 112

Rectangle B (middle left protrusion): 5 yd wide × 8 yd tall? But the height between top and bottom is not clear.

Wait — the total height: from top to bottom, there are two 8 yd segments? The diagram shows:

On the right side: top part 8 yd, then a gap, then bottom part 8 yd? And in the middle, there’s a connection.

Actually, looking closely: the shape consists of:

- A top horizontal rectangle: 14 yd long, 8 yd high
- A bottom horizontal rectangle: same size? 14 yd long, 8 yd high? But it’s shifted.
- And a vertical connector in the middle? But the labels show "5 yd" on the left and right of the middle part.

Another approach: the entire shape can be seen as a large rectangle minus two corners, but that might be messy.

Let me try adding the parts:

Part 1: Top rectangle — 14 yd × 8 yd = 112

Part 2: The middle vertical part — but it’s not vertical; it’s a horizontal bar connecting? No.

Notice that the left side has a 5 yd extension downward from the top, and the right side has a 5 yd extension upward from the bottom? This is confusing.

Let’s read the labels carefully:

- Top edge: 14 yd
- Right side of top block: 8 yd (so height of top block is 8 yd)
- Then, on the left, there’s a 5 yd horizontal segment going left, and then down 8 yd? The label "8 yd" is written vertically on the left side of the middle part.
- Then at the bottom, there’s a 5 yd horizontal segment going right, and the bottom block is 8 yd high.

Actually, I think the shape is made of three rectangles:

1. Top: 14 yd × 8 yd
2. Middle: a rectangle that is 5 yd wide (the overhang) and 8 yd tall (the vertical drop) — but positioned on the left
3. Bottom: 14 yd × 8 yd, but shifted right by 5 yd?

Wait, perhaps the total width is 14 + 5 = 19 yd? But that might not help.

Alternative method: use the fact that the shape is symmetric or calculate bounding box.

Bounding box: width = 14 + 5 = 19 yd? Height = 8 + 8 + 8 = 24 yd? No, that’s too big.

Let’s list all visible segments:

From top-left corner:
- Go right 14 yd (top)
- Down 8 yd (right side of top)
- Left 5 yd (this is the start of the middle part)
- Down 8 yd (left side of middle)
- Right 5 yd (bottom of middle)
- Down 8 yd? No, then it should connect to bottom.

I think I found a better way: the shape can be divided into three rectangles:

- Rectangle 1 (top): 14 yd × 8 yd = 112
- Rectangle 2 (middle left): 5 yd × 8 yd = 40 (this is the part that hangs down on the left)
- Rectangle 3 (bottom): 14 yd × 8 yd = 112, but wait, if the bottom is aligned to the right, then its left edge is 5 yd from the left of the top? So the bottom rectangle is from x=5 to x=19, but the top is from x=0 to x=14, so they overlap from x=5 to x=14.

This is getting complicated. Perhaps the intended division is:

The shape has a central vertical strip? No.

Let me search for a standard way. Notice that the total area can be calculated as the sum of the areas of the three horizontal bars, but they overlap? No, in this case, they are connected without overlap.

Actually, looking at the diagram again (mentally), it's like a "staircase" but mirrored.

Another idea: the shape is composed of:

- A large rectangle of 14 yd by (8+8+8)=24 yd, but with two rectangles cut out: one on the top-right and one on the bottom-left? Let's see.

If I imagine a big rectangle 14 yd wide and 24 yd tall, area = 336.

Then, on the top-right, there is a missing rectangle: 5 yd wide and 8 yd tall? Because the top bar only goes 14 yd, but if the big rectangle is 14 yd, then no.

Perhaps the big rectangle is wider.

Let's define coordinates.

Set origin at top-left.

- From (0,0) to (14,8): top rectangle
- From (0,8) to (5,16): middle left rectangle (since it goes down 8 yd and left 5 yd? But from (0,8) to (5,16) would be down and right, not left.

I think I have it:

After the top rectangle (0,0) to (14,8), the shape goes down to (0,16)? No.

From the description: after going down 8 yd on the right, it goes left 5 yd, so to (9,8), then down 8 yd to (9,16), then right 5 yd to (14,16), then down 8 yd to (14,24), then left 14 yd to (0,24), then up to (0,0)? That doesn't make sense.

Perhaps it's:

- Start at top-left (0,0)
- Right 14 yd to (14,0)
- Down 8 yd to (14,8)
- Left 5 yd to (9,8)
- Down 8 yd to (9,16)
- Right 5 yd to (14,16)
- Down 8 yd to (14,24)
- Left 14 yd to (0,24)
- Up 24 yd to (0,0) — but that would include extra area.

This is not working. Let me try a different strategy.

Notice that the shape has three identical rectangles of 14 yd by 8 yd, but arranged with offsets, and the overlapping parts are not there — actually, in this case, they are connected, so no overlap.

But when you place them, the middle one is shifted.

Let's calculate the area by adding the parts that are unique.

From the diagram, the top rectangle is 14x8.

Then, attached to its left side, there is a rectangle that is 5 yd wide and 8 yd tall (going down).

Then, attached to the bottom of that, there is a rectangle that is 14 yd wide and 8 yd tall, but shifted to the right by 5 yd, so it starts at x=5.

So the bottom rectangle is from x=5 to x=19, y=16 to y=24.

But then the total width is 19 yd, and height 24 yd.

Now, to find area:

- Top: 14*8 = 112
- Middle left: 5*8 = 40 (from x=0 to 5, y=8 to 16)
- Bottom: 14*8 = 112 (from x=5 to 19, y=16 to 24)

Do these overlap? The middle left is x=0-5, y=8-16; bottom is x=5-19, y=16-24; top is x=0-14, y=0-8. No overlap.

So total area = 112 + 40 + 112 = 264 sq yd

Is that correct? Let me verify with another method.

Imagine the bounding box: from x=0 to 19, y=0 to 24, area = 19*24 = 456

Now, what is missing? In the top-right, from x=14 to 19, y=0 to 8: that's 5*8=40 missing

In the bottom-left, from x=0 to 5, y=16 to 24: that's 5*8=40 missing

Also, in the middle, from x=5 to 14, y=8 to 16: is that filled? In our shape, from x=5 to 14, y=8 to 16 is empty? Let's see.

In our earlier division, we have:

- y=0-8: x=0-14 filled
- y=8-16: only x=0-5 filled (the middle left rectangle)
- y=16-24: x=5-19 filled

So in the region x=5-14, y=8-16, it is empty. That's a rectangle 9 yd wide (14-5=9) and 8 yd tall, area 72.

Also, in x=14-19, y=0-8: empty, 5*8=40

In x=0-5, y=16-24: empty, 5*8=40

So total missing area = 72 + 40 + 40 = 152

Bounding box 19*24=456, so area = 456 - 152 = 304? That doesn't match my previous 264.

I think I messed up the bounding box.

If the bottom rectangle goes to x=19, and top to x=14, then the rightmost point is x=19, leftmost x=0, so width 19.

Height from y=0 to y=24, so 24.

But in the shape, at y=0-8, it's only up to x=14, so from x=14 to 19 is empty.

At y=8-16, only x=0-5 is filled, so x=5-19 is empty.

At y=16-24, x=5-19 is filled, so x=0-5 is empty.

So the filled regions are:

- R1: [0,14] x [0,8]
- R2: [0,5] x [8,16]
- R3: [5,19] x [16,24]

Now, do R1 and R2 overlap? R1 is y=0-8, R2 is y=8-16, so they touch at y=8, but no area overlap.

R2 and R3: R2 is x=0-5, y=8-16; R3 is x=5-19, y=16-24; they touch at x=5 and y=16, no area overlap.

R1 and R3: R1 is y=0-8, R3 is y=16-24, no overlap.

So total area = area(R1) + area(R2) + area(R3) = (14*8) + (5*8) + (14*8) = 112 + 40 + 112 = 264 sq yd

But earlier when I did bounding box, I got 304, which is wrong because I miscalculated the missing areas.

Bounding box [0,19] x [0,24] = 456

Missing areas:

- Region A: [14,19] x [0,8] = 5*8 = 40
- Region B: [5,19] x [8,16] = 14*8 = 112? Wait, x from 5 to 19 is 14 yd, y from 8 to 16 is 8 yd, so 112
- Region C: [0,5] x [16,24] = 5*8 = 40

Total missing = 40 + 112 + 40 = 192

Then area = 456 - 192 = 264 sq yd. Yes! Matches.

So final answer for #5 is 264 sq yd.

✔ Final for #5: 264 sq yd

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Problem 6:
L-shaped or stepped shape.

Labels:
- Bottom: 28 m
- Right side: 18 m
- Top-right: 12 m
- Left side: 6 m
- And a 10 m on the top-left part.

Let me sketch mentally.

It looks like a rectangle with a bite taken out of the top-left.

Or, can be split into two rectangles.

One way:
- Bottom rectangle: 28 m wide × 6 m high (since left side is 6 m)
- Top rectangle: but it's not full width.

From the labels: the total height is 18 m, and the bottom part is 6 m high, so the top part is 12 m high? But the right side is 18 m, and there's a 12 m on the top-right.

Actually, the shape has:

- A bottom rectangle: 28 m × 6 m
- Above it, on the right, a rectangle that is 12 m wide and (18 - 6) = 12 m high? But 12 m is labeled on the top, and 18 m on the right.

Let's see the path:

Start at bottom-left: go right 28 m, up 6 m, then left? No.

From the diagram: it seems like from bottom-left, go right 28 m, up 18 m, but then there's a step.

The label "10 m" is on the top-left horizontal, and "6 m" on the left vertical.

Probably: the shape is composed of:

- A large rectangle on the right: 12 m wide × 18 m high? But 12 m is on the top, and 18 m on the right.

Another way: split vertically.

Left part: a rectangle 10 m wide? The "10 m" is labeled on the top-left horizontal segment.

Assume:

- The bottom part is 28 m wide and 6 m high.
- Above it, from x=0 to x=10, there is a rectangle of height (18 - 6) = 12 m? But the total height on the right is 18 m, and on the left, from bottom to top of the left part is 6 m + something.

Let's use the labels:

- The leftmost vertical side is 6 m (from bottom to the first step)
- Then, from there, it goes right 10 m (this is the top of the left part)
- Then up to the top, but the total height is 18 m, so from the 6 m level to top is 12 m.
- On the right, from bottom to top is 18 m, and the top-right horizontal is 12 m.

So, the shape can be divided into:

1. Bottom rectangle: 28 m × 6 m = 168 sq m
2. Top-left rectangle: 10 m × 12 m = 120 sq m (since from y=6 to y=18, height 12 m, and width 10 m)
3. Top-right rectangle: but if I add these, I have covered from x=0 to 10 for y=6-18, and x=0 to 28 for y=0-6.

But the top-right part from x=10 to 28, y=6 to 18 is not yet included? And it should be there because the right side is 18 m high.

In fact, the entire top part from y=6 to y=18 should be filled from x=0 to x=28? But no, because there's a step.

From the description, after going up 6 m on the left, it goes right 10 m, then up to the top, but the top is at 18 m, so from y=6 to y=18 is 12 m, and then it goes right to the end.

But the label "12 m" is on the top-right, which might be the width of the top part on the right.

Perhaps the top part is only 12 m wide on the right, but that doesn't make sense with the 28 m bottom.

Let's think differently.

The total width at the bottom is 28 m.

At the top, the width is less. The label "12 m" is on the top-right horizontal, and "10 m" on the top-left horizontal, but they are at different heights.

Actually, the shape has a "step" on the left side.

From bottom-left:
- Right 28 m (bottom)
- Up 18 m (right side)
- Left 12 m (top)
- Down ?
- Left 10 m? This is confusing.

Perhaps it's better to split into two rectangles:

- Rectangle A: the right part, which is 12 m wide and 18 m high? But 12 m is labeled on the top, and if it's the width, then yes.
- Rectangle B: the left part, which is (28 - 12) = 16 m wide? But the left side has a 6 m height and a 10 m top.

Let's calculate the area by subtraction.

Imagine a large rectangle 28 m wide × 18 m high = 504 sq m

Now, what is missing? In the top-left, there is a rectangle cut out.

From the labels, on the left, from the top, it goes down 6 m? No.

The left side has a vertical segment of 6 m from the bottom, then it steps right 10 m, then up to the top.

So, the missing part is in the top-left: a rectangle that is 10 m wide and (18 - 6) = 12 m high? But if the large rectangle is 28x18, and we remove a rectangle of 10x12 from the top-left, then the remaining shape would have:

- At y=0 to 6: full width 28 m
- At y=6 to 18: width from x=10 to 28, so 18 m wide

But in the diagram, the top-right is labeled 12 m, which would be the width at the top, but 28 - 10 = 18 m, not 12 m. Contradiction.

Unless the 12 m is not the width at the top.

Let's read the labels again:

- "28 m" at the bottom
- "18 m" on the right side (total height)
- "12 m" on the top-right horizontal segment
- "6 m" on the left-side vertical segment (from bottom to the step)
- "10 m" on the top-left horizontal segment (after the step)

So, from the bottom-left corner:
- Move right 28 m to bottom-right
- Move up 18 m to top-right
- Move left 12 m to a point
- Move down ?
- Move left 10 m to a point
- Move down 6 m to bottom-left? But that would close the shape.

Let's trace the perimeter:

Start at bottom-left (0,0)
- Right to (28,0) // bottom
- Up to (28,18) // right side
- Left to (16,18) // because 28 - 12 = 16? If "12 m" is the length moved left, then to x=28-12=16
- Down to (16,6) // because from y=18 to y=6 is 12 m down? But the label "6 m" is on the left, not here.
- Left to (6,6) // if "10 m" is moved left, from x=16 to x=6
- Down to (6,0) // but then to (0,0)? Not matching.

From (16,6) , if we go left 10 m to (6,6), then down to (6,0), then left to (0,0), but that would make the bottom from x=0 to 28, but from x=0 to 6 is additional.

This is messy.

Perhaps the "6 m" is the height from bottom to the step on the left, and "10 m" is the width of the step.

So, the shape has:

- A bottom rectangle: 28 m × 6 m
- Above it, a rectangle that is (28 - 10) = 18 m wide? But the top-right is labeled 12 m.

Another idea: the top part is 12 m wide, and it is on the right, so from x=16 to 28 (since 28-12=16), and height from y=6 to y=18, so 12 m high.

Then, the left part above the bottom is from x=0 to 16, but only up to y=6, and then from x=0 to 10, up to y=18? I'm confused.

Let's assume that the step is on the left: from the bottom, up 6 m, then right 10 m, then up to the top at y=18, so the height of the left column is 18 m, but the bottom 6 m is full width, and above that, only the right part is there.

Standard way for such shapes: split into two rectangles.

Rectangle 1: the bottom part: 28 m × 6 m = 168 sq m

Rectangle 2: the top part: which is from y=6 to y=18, and from x=10 to x=28? Why x=10? Because the "10 m" is the width of the left overhang or something.

From the label "10 m" on the top-left horizontal, and "6 m" on the left vertical, likely the left part has a width of 10 m for the top section.

So, the top section is 10 m wide and (18 - 6) = 12 m high, located on the left.

Then, the right section is from x=10 to x=28, y=0 to y=18, but that would overlap with the bottom.

Better: the shape consists of:

- A left rectangle: 10 m wide × 18 m high = 180 sq m
- A right rectangle: (28 - 10) = 18 m wide × 6 m high = 108 sq m (only the bottom part)

Then total area = 180 + 108 = 288 sq m

Check if this matches the labels.

In this case, the bottom width is 10 + 18 = 28 m, good.

The right side height is 6 m for the right rectangle, but the total height on the right should be 18 m, but in this division, the right rectangle is only 6 m high, so from y=6 to y=18 on the right is not filled, but in the diagram, the right side is 18 m high, so it should be filled.

So this is incorrect.

Perhaps the right rectangle is 18 m high, but then it would overlap.

Let's try:

- Rectangle A: the entire right part: 12 m wide × 18 m high = 216 sq m (since "12 m" is on the top-right, likely the width of the right section)
- Rectangle B: the left part: which is (28 - 12) = 16 m wide, but only 6 m high = 96 sq m

Then total = 216 + 96 = 312 sq m

Now, check the labels: the left side has "6 m" for the height of the left part, good.

The "10 m" label: where is it? In this case, the left part is 16 m wide, but the label says "10 m" on the top-left horizontal.

Perhaps the 10 m is the width of the top of the left part, but in this division, the left part is only 6 m high, so no top.

I think I need to accept that the "10 m" is the width of the step, and the left part above the bottom is 10 m wide.

So, let's define:

- Bottom rectangle: 28 m × 6 m = 168
- Top-left rectangle: 10 m × (18 - 6) = 10 × 12 = 120
- Top-right rectangle: but if I add these, I have from x=0 to 10 for y=6-18, and x=0 to 28 for y=0-6.

The region from x=10 to 28, y=6 to 18 is not included, but it should be, because the right side is 18 m high.

Unless the top-right is separate.

Perhaps the top part is only on the right: from x=16 to 28, y=6 to 18, with width 12 m (28-16=12), height 12 m.

Then, the left part above bottom is from x=0 to 16, but only up to y=6, and then from x=0 to 10, up to y=18? Still messy.

Let's calculate the area using the given numbers without splitting.

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

Large rectangle: 28 m × 18 m = 504 sq m

Missing rectangle: in the top-left, a rectangle of width 10 m and height (18 - 6) = 12 m? But then the missing area is 10*12 = 120, so area = 504 - 120 = 384 sq m

But is that correct? If we remove a 10x12 rectangle from the top-left, then the shape would have:

- At y=0 to 6: full 28 m
- At y=6 to 18: from x=10 to 28, so 18 m wide

But in the diagram, the top-right is labeled 12 m, which would be the width at the top, but 28 - 10 = 18 m, not 12 m. So unless the 12 m is not the width, but something else.

Perhaps the "12 m" is the height of the top part, but it's labeled on the top, so likely width.

Another possibility: the "12 m" is the width of the top-right section, and "10 m" is the width of the top-left section, but they are at the same height.

Assume that at the top, there are two parts: left 10 m, right 12 m, but 10+12=22 < 28, so there's a gap, which doesn't make sense.

Perhaps the 10 m and 12 m are not both at the top.

Let's look for a standard solution or rethink.

From the labels, the total height is 18 m.

The left side has a vertical segment of 6 m from the bottom.

Then, from there, it goes right 10 m (horizontal).

Then, from there, it goes up to the top, so the vertical distance from y=6 to y=18 is 12 m.

Then, from the top-left of that, it goes right to the end, but the end is at x=28, and if it went right 10 m from x=0, it's at x=10, then up to (10,18), then right to (28,18), so the top width is 18 m (28-10=18), but the label says "12 m" on the top-right, which might be a mistake or I misread.

Perhaps "12 m" is the length of the top-right horizontal, which is from x=16 to 28, so 12 m, and the left part is from x=0 to 16, but with a step.

Let's assume that the step is at x=10 on the left.

So, the shape is:

- From (0,0) to (28,0) to (28,18) to (16,18) to (16,6) to (10,6) to (10,0) to (0,0)? But then from (10,0) to (0,0) is not direct.

From (10,6) to (10,0) to (0,0), but then the bottom from x=0 to 10 is included, and from x=10 to 28 is also included, so bottom is full.

Then the area can be calculated as:

- Rectangle 1: x=0 to 28, y=0 to 6 = 28*6 = 168
- Rectangle 2: x=10 to 28, y=6 to 18 = 18*12 = 216? 28-10=18 m wide, 12 m high = 216
- But then the region x=0 to 10, y=6 to 18 is not included, which is correct for this shape.

In this case, the top-right horizontal is from x=16 to 28? No, in this division, from (16,18) to (28,18) is 12 m, yes! Because if the top-right is from x=16 to 28, that's 12 m, and the left part of the top is from x=10 to 16, but in our shape, from x=10 to 16, y=6 to 18 is included in rectangle 2, so the top is from x=10 to 28, which is 18 m, but the label "12 m" is only for the right part, perhaps indicating that from x=16 to 28 is 12 m, but why 16?

In this setup, if the top-right horizontal is 12 m, and it's at y=18, from x=a to a+12, and the total width is 28, and the left part has a step at x=10, then perhaps a=16, so from x=16 to 28 is 12 m, and from x=10 to 16 is 6 m, but then the width from x=10 to 28 is 18 m, as before.

But in the area calculation, if we have:

- Bottom: 28*6 = 168
- Top: from x=10 to 28, y=6 to 18 = 18*12 = 216
- Total = 168 + 216 = 384 sq m

And the "10 m" label is the horizontal segment from (0,6) to (10,6), which is 10 m, good.

The "6 m" is from (0,0) to (0,6), good.

The "12 m" is from (16,18) to (28,18)? But in this shape, from (10,18) to (28,18) is 18 m, not 12 m. So unless the 12 m is not the entire top, but only a part.

Perhaps the 12 m is the width of the right section, but in the diagram, it's labeled on the top-right, so likely the length of that segment.

Maybe the shape has a additional constraint.

Another idea: perhaps the "12 m" is the height of the top part, but it's labeled on the top, so probably not.

Let's calculate with the numbers given.

Suppose we split into:

- Left rectangle: 10 m wide × 18 m high = 180
- Right rectangle: 18 m wide × 6 m high = 108 (since 28-10=18)
- But then the right rectangle is only 6 m high, so from y=6 to 18 on the right is missing, but it should be there.

Unless the right rectangle is 18 m high, but then it would be 18*18=324, plus left 10*6=60, total 384, same as before.

If right rectangle is 18 m wide × 18 m high = 324, but then it overlaps with the left if left is 10 m wide.

No.

Perhaps the right part is from x=10 to 28, 18 m wide, 18 m high, area 324, and the left part is from x=0 to 10, 6 m high, area 60, total 384, and no overlap since different y-ranges for the left part.

In this case, the left part is only y=0 to 6, x=0 to 10.

The right part is y=0 to 18, x=10 to 28.

Then at x=10, y=0 to 6, it is included in both? No, if we define strictly, the right part starts at x=10, so if we include x=10 in both, there is overlap.

To avoid overlap, make the right part x>10, but usually in such problems, we consider the boundaries.

Typically, we can assign the line to one rectangle.

So area = area(left) + area(right) = (10*6) + (18*18) = 60 + 324 = 384 sq m

And the "12 m" label: on the top-right, from x=16 to 28 is 12 m, but in this shape, the top is from x=10 to 28, so from x=16 to 28 is 12 m, which is part of it, so perhaps the label is indicating that segment, but not the whole top.

Similarly, the "10 m" is the left horizontal at y=6.

So this seems consistent.

Total area = 384 sq m

✔ Final for #6: 384 sq m

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Problem 7:
F-shaped or E-shaped.

Labels:
- Left side: 8 mm
- Top: 6 mm
- Then on the right, various segments: 2 mm, 1 mm, 3 mm, etc.

Specifically:
- From top-left, right 6 mm
- Down 2 mm
- Left 3 mm (this is the first inward)
- Down 1 mm
- Right 3 mm
- Down 3 mm
- Left 6 mm? But the left side is 8 mm total.

Let's trace.

Start at top-left (0,0)
- Right 6 mm to (6,0)
- Down 2 mm to (6,2)
- Left 3 mm to (3,2)
- Down 1 mm to (3,3)
- Right 3 mm to (6,3)
- Down 3 mm to (6,6)
- Left 6 mm to (0,6)
- Up 2 mm to (0,8)? But the left side is 8 mm, so from (0,6) to (0,8) is 2 mm, but then to (0,0) is 8 mm, so from (0,6) to (0,0) is 6 mm, contradiction.

The left side is labeled 8 mm, so from bottom to top is 8 mm.

From the bottom: start at (0,0)
- Up 8 mm to (0,8)
- Right 6 mm to (6,8)
- Down 2 mm to (6,6)
- Left 3 mm to (3,6)
- Down 1 mm to (3,5)
- Right 3 mm to (6,5)
- Down 3 mm to (6,2)
- Left 6 mm to (0,2)
- Up 2 mm to (0,4)? Not closing.

From (6,2) left to (0,2), then up to (0,8), but then from (0,2) to (0,8) is 6 mm, but the left side is 8 mm, so from (0,0) to (0,8) is 8 mm, so if we have from (0,2) to (0,8), that's 6 mm, missing the bottom 2 mm.

Perhaps the bottom is from (0,0) to (6,0), but not labeled.

Let's list the vertices based on the labels.

From the diagram description:

- The overall height is 8 mm (left side)
- The top width is 6 mm
- Then, on the right side, from top: down 2 mm, then left 3 mm (so a notch), then down 1 mm, then right 3 mm (back to the edge), then down 3 mm, then left to the left side.

Also, on the left side, from bottom, up 3 mm? The label "3 mm" is on the left side near the bottom.

Specifically, the labels are:
- Left side: 8 mm (total)
- Near bottom on left: 3 mm (probably the height of the bottom part)
- Then above that, 1 mm? Let's see the sequence.

Typically for such shapes, it's divided into rectangles.

Let me try to split into three horizontal rectangles.

Bottom rectangle: from y=0 to y=3, x=0 to 6 mm? But the left side has a 3 mm label, so perhaps the bottom part is 3 mm high.

Then middle: from y=3 to y=4, but with a notch.

From the labels: after the bottom 3 mm, there is a 1 mm segment, then 3 mm, etc.

Assume:

- Bottom rectangle: 6 mm wide × 3 mm high = 18 sq mm
- Middle rectangle: but it has a bite.

From the path:

Start at (0,0)
- Right 6 mm to (6,0)
- Up 3 mm to (6,3) // but the label "3 mm" is on the left, not here.

Perhaps the 3 mm on the left is the height from bottom to the first horizontal.

Let's use the given numbers to define the parts.

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

Large rectangle: 6 mm wide × 8 mm high = 48 sq mm

Now, what is cut out? On the right side, there are two notches.

From the top: down 2 mm, then left 3 mm, so a rectangle of 3 mm wide × 2 mm high is cut out from the top-right? But then down 1 mm, then right 3 mm, so another cut.

Actually, the cut-outs are:

- One rectangle: 3 mm wide × 2 mm high, located at the top-right, but since it's indented, it's removed.

- Another rectangle: 3 mm wide × 1 mm high, located below that.

Let's see the positions.

From the top-right corner (6,8):
- Down 2 mm to (6,6)
- Left 3 mm to (3,6)
- Down 1 mm to (3,5)
- Right 3 mm to (6,5)
- Down 3 mm to (6,2)
- Left 6 mm to (0,2)
- Up 6 mm to (0,8)? But from (0,2) to (0,8) is 6 mm, but the left side is 8 mm, so from (0,0) to (0,2) is 2 mm, not labeled.

The label "3 mm" on the left side near bottom suggests that from y=0 to y=3 is the bottom part.

So perhaps from (0,0) to (0,3) is 3 mm, then to (0,8) is 5 mm, but not matching.

Let's calculate the area by adding the parts.

Divide into three parts:

1. Bottom rectangle: from y=0 to y=3, x=0 to 6 mm = 6*3 = 18 sq mm
2. Middle rectangle: from y=3 to y=4, but only from x=0 to 3 mm? Because at y=3, it goes right to x=6, but then down, so at y=3 to y=4, the width is from x=0 to 6, but with a notch? No.

From the sequence:

After the bottom 3 mm, at y=3, the shape goes right to x=6, then up? No.

From the standard interpretation of such diagrams:

- The bottom part is 6 mm wide and 3 mm high.
- Above it, from y=3 to y=4, the width is 6 mm, but then at y=4, it indents.
- Specifically, from y=3 to y=4, full width 6 mm.
- Then from y=4 to y=5, only from x=0 to 3 mm (because it indents 3 mm on the right)
- Then from y=5 to y=8, full width 6 mm again? But the top is 6 mm, and from y=5 to y=8 is 3 mm, but the label "2 mm" is on the top-right.

Let's use the labels as given in the diagram description:

- Left side: 8 mm
- Top: 6 mm
- On the right side, from top: down 2 mm, then the horizontal left 3 mm, then down 1 mm, then horizontal right 3 mm, then down 3 mm, then left to left side.
- On the left side, from bottom: up 3 mm (label "3 mm"), then the rest.

So, the height from bottom to the first horizontal on left is 3 mm.

Then from there to the next is 1 mm? The label "1 mm" is on the right side between the two horizontals.

So, let's define the y-coordinates.

Set y=0 at bottom.

- From y=0 to y=3: the shape is full width 6 mm (since no indent mentioned, and left side has 3 mm label)
- At y=3, it may change.

From the right side labels: from top (y=8) down 2 mm to y=6, then left 3 mm, so at y=6, it goes to x=3 (since 6-3=3)
- Then down 1 mm to y=5
- Then right 3 mm to x=6
- Then down 3 mm to y=2
- Then left to x=0

But at y=2, it goes left to x=0, so from y=2 to y=3, what happens?

Also, on the left side, from y=0 to y=3 is 3 mm, so at y=3, it is still at x=0.

So, the shape has:

- From y=0 to y=2: full width 6 mm? But at y=2, it goes left to x=0, so from y=2 to y=3, it is from x=0 to 6 mm.

Let's list the bounds.

For y from 0 to 2: the shape is from x=0 to 6 mm (full width)
For y from 2 to 3: still x=0 to 6 mm? But at y=2, it goes left to x=0, which is already the left, so no change.

From the path: at y=2, after coming down to (6,2), it goes left to (0,2), so at y=2, it is from x=0 to 6.

Then from (0,2) up to (0,3)? But the left side is continuous.

Perhaps from y=2 to y=3, it is full width.

Then at y=3, it may have a change, but no label.

From the right side, after down 3 mm to y=2, then left to x=0, then up to y=8, but with indents.

From (0,2) up to (0,8), but at y=6, it has a indent on the right.

So for y from 2 to 6: the shape is from x=0 to 6 mm, except between y=5 and y=6, it is only from x=0 to 3 mm? Let's see.

From the sequence:

- From (0,2) up to (0,6) // but at y=6, it goes right to (3,6)? No.

From earlier: after reaching (0,2), it goes up to (0,8), but at y=6, when it was coming down, it went left to (3,6), so at y=6, the right boundary is x=3 for some part.

Let's define the right boundary as a function of y.

From y=0 to y=2: right boundary x=6
From y=2 to y=5: right boundary x=6? But at y=5, it comes from (3,5) to (6,5), so at y=5, it is x=6.
From y=5 to y=6: right boundary x=3 (because from (3,5) to (3,6))
From y=6 to y=8: right boundary x=6 (because from (3,6) to (6,6) is not; from (6,6) to (6,8) is up, so at y=6 to 8, x=6)

Let's clarify the path from the beginning.

Start at (0,0)
- Right to (6,0) // bottom
- Up to (6,2) // right side, 2 mm up? But the label "3 mm" is on left, not here.
Perhaps the 2 mm is from the top.

Start at (0,8) // top-left
- Right to (6,8) // top
- Down to (6,6) // 2 mm down
- Left to (3,6) // 3 mm left
- Down to (3,5) // 1 mm down
- Right to (6,5) // 3 mm right
- Down to (6,2) // 3 mm down
- Left to (0,2) // 6 mm left
- Up to (0,8) // 6 mm up, but from y=2 to y=8 is 6 mm, but the left side is 8 mm, so from y=0 to y=2 is missing.

So to close, from (0,2) down to (0,0), then right to (6,0), but then from (6,0) to (6,2) is already there.

So the path is:
(0,0) -> (6,0) -> (6,2) -> (6,5) wait no.

From (6,2) to (0,2), then to (0,0), but (0,0) to (6,0) is already done.

So vertices: (0,0), (6,0), (6,2), (0,2), (0,8), (6,8), (6,6), (3,6), (3,5), (6,5), (6,2) — but (6,2) is repeated.

From (6,5) to (6,2), then to (0,2), then to (0,8), then to (6,8), then to (6,6), then to (3,6), then to (3,5), then back to (6,5) — this is not simple.

I think the correct path is:

Start at (0,0)
- Right to (6,0)
- Up to (6,2) // so far 2 mm up
- Left to (0,2) // but then to (0,8)
- Up to (0,8)
- Right to (6,8)
- Down to (6,6) // 2 mm down
- Left to (3,6) // 3 mm left
- Down to (3,5) // 1 mm down
- Right to (6,5) // 3 mm right
- Down to (6,2) // 3 mm down, but (6,2) is already visited.

So from (6,5) down to (6,2), which is the same as earlier, so the shape has a self-intersection or something.

Perhaps from (6,5) down to (6,2), and since (6,2) is already there, it's fine, but then the region between y=2 and y=5 on the right is filled, but with a bite between y=5 and y=6.

To calculate area, we can use the shoelace formula or divide into rectangles.

Let me divide into rectangles based on the y-levels.

The y-coordinates where changes happen: y=0,2,5,6,8

So intervals: y=0 to 2, y=2 to 5, y=5 to 6, y=6 to 8

For each interval, find the width.

- y=0 to 2: from x=0 to 6, width 6 mm
- y=2 to 5: from x=0 to 6, width 6 mm (since no indent in this range)
- y=5 to 6: from x=0 to 3, width 3 mm (because at y=5 to 6, the right boundary is x=3)
- y=6 to 8: from x=0 to 6, width 6 mm

Then area = sum of (width * height) for each interval.

- y0-2: 6 * 2 = 12
- y2-5: 6 * 3 = 18 (height 3 mm)
- y5-6: 3 * 1 = 3
- y6-8: 6 * 2 = 12
Total = 12 + 18 + 3 + 12 = 45 sq mm

Now, check if this matches the labels.

The left side is 8 mm, good.

Top is 6 mm, good.

On the right side, from y=8 to y=6: 2 mm down, then left 3 mm, so at y=6, it is at x=3, good.

Then down 1 mm to y=5, then right 3 mm to x=6, good.

Then down 3 mm to y=2, good.

On the left side, from y=0 to y=2: 2 mm, but the label "3 mm" is on the left side near bottom. In this case, from y=0 to y=2 is 2 mm, but the label says 3 mm, so discrepancy.

Perhaps the 3 mm is from y=0 to y=3.

In that case, adjust.

Suppose the bottom part is 3 mm high.

Then from y=0 to 3: width 6 mm
Then from y=3 to 5: width 6 mm? But at y=5, it has the indent.

From the right side, the down 3 mm is from y=5 to y=2, so if y=2 is not the bottom, perhaps y=0 to y=2 is 2 mm, but the label "3 mm" might be for a different part.

Looking back at the user's description: "3 mm" is labeled on the left side, and "1 mm" on the right side between the two horizontals, "2 mm" on the top-right vertical, "3 mm" on the bottom-left vertical? The user said: "3 mm" on the left side, and in the text: "3 mm" appears twice.

In the initial problem statement for #7: "6 mm" on top, "8 mm" on left, "2 mm" on the first right vertical, "3 mm" on the first horizontal left, "1 mm" on the second right vertical, "3 mm" on the second horizontal right, "3 mm" on the left side near bottom.

So likely, the "3 mm" on the left side near bottom is the height from bottom to the first horizontal on the left, but in our path, from (0,0) to (0,2) is 2 mm, not 3.

Perhaps the down 3 mm on the right is from y=5 to y=2, but y=2 is not the bottom; the bottom is at y=0, and from y=2 to y=0 is 2 mm, but the label says 3 mm for the left side bottom part.

To resolve, assume that the distance from y=0 to y=3 is 3 mm for the left side.

Then, in the right side, the down 3 mm is from y=5 to y=2, but y=2 to y=0 is 2 mm, so total from y=5 to y=0 is 5 mm, but the left side from y=0 to y=5 is 5 mm, while the total is 8 mm, so from y=5 to y=8 is 3 mm, but the top has 2 mm down from y=8 to y=6, etc.

Set y=0 at bottom.

- Left side: from y=0 to y=3: 3 mm (label)
- Then from y=3 to y=8: 5 mm, but not labeled.

On the right side:
- From y=8 down to y=6: 2 mm
- Then left 3 mm
- Down to y=5: 1 mm (so from y=6 to y=5)
- Right 3 mm
- Down to y=2: 3 mm (so from y=5 to y=2)
- Then left to x=0

But at y=2, it goes left to x=0, so from y=2 to y=3, what is the width? If from y=2 to y=3, it is from x=0 to 6, then the left side from y=2 to y=3 is 1 mm, but the label "3 mm" is for y=0 to y=3, so from y=0 to y=2 must be 2 mm, but 2+1=3, good.

So:
- y=0 to 2: width 6 mm (since no indent)
- y=2 to 3: width 6 mm (still full)
- y=3 to 5: width 6 mm? But at y=5, it has the indent starting.

From the path, at y=5, it is at x=6, then down to y=2, but y=2 is below, so for y=3 to 5, it should be full width.

Then at y=5 to 6: width 3 mm (as before)
y=6 to 8: width 6 mm

But y=3 to 5 is 2 mm high, width 6 mm.

So intervals:
- y0-2: 6*2 = 12
- y2-3: 6*1 = 6
- y3-5: 6*2 = 12 (height 2 mm)
- y5-6: 3*1 = 3
- y6-8: 6*2 = 12
Total = 12+6+12+3+12 = 45 sq mm same as before.

And the left side from y=0 to y=3 is 3 mm, which matches the label.

The "3 mm" on the bottom-left is satisfied.

So area = 45 sq mm

✔ Final for #7: 45 sq mm

---

Problem 8:
U-shaped or frame.

Labels:
- Overall width: 16 cm
- Overall height: 28 cm
- Bottom: left part 8 cm, right part 6 cm, so the gap in middle is 16 - 8 - 6 = 2 cm? But not labeled.
- Inside, there is a rectangle cut out: 18 cm high, and width? Not given, but from the context, the cut-out is in the middle.

The shape is like a rectangle with a rectangular hole in the middle, but not centered.

Specifically, the bottom has two parts: left 8 cm, right 6 cm, so the distance between them is 16 - 8 - 6 = 2 cm, but that might be the width of the cut-out at the bottom.

The cut-out is 18 cm high, and presumably 2 cm wide, but let's see.

The overall height is 28 cm, and the cut-out is 18 cm high, so it doesn't go to the top or bottom.

Typically, for such shapes, the cut-out is centered or positioned.

From the labels, the cut-out is 18 cm high, and the overall height is 28 cm, so the space above and below is (28-18)/2 = 5 cm on each side, but not specified.

The bottom has left 8 cm, right 6 cm, so the cut-out starts at x=8 cm from left, and has width w, then right part 6 cm, so 8 + w + 6 = 16, so w = 2 cm.

Yes! So the cut-out is 2 cm wide and 18 cm high.

Positioned at the bottom? The label "18 cm" is on the cut-out, and it's vertical, so height 18 cm.

Where is it located vertically? The overall height is 28 cm, and if the cut-out is 18 cm high, and assuming it is at the bottom, then from y=0 to y=18, but then the top would be from y=18 to y=28, 10 cm high.

But in the diagram, it might be centered or at the bottom.

The problem doesn't specify, but typically in such problems, the cut-out is flush with the bottom or top.

Given that the bottom has the two parts, likely the cut-out is at the bottom, so from y=0 to y=18.

Then the area is the area of the large rectangle minus the area of the cut-out.

Large rectangle: 16 cm × 28 cm = 448 sq cm

Cut-out: 2 cm × 18 cm = 36 sq cm

Area = 448 - 36 = 412 sq cm

But is the cut-out really 2 cm wide? From the bottom: left part 8 cm, then cut-out, then right part 6 cm, and 8+2+6=16, yes.

And height 18 cm, as labeled.

So yes.

To confirm, the shape has:
- Left rectangle: 8 cm wide × 28 cm high = 224
- Right rectangle: 6 cm wide × 28 cm high = 168
- Top middle: but if the cut-out is only 18 cm high, then above it, from y=18 to y=28, there is a rectangle of width 2 cm and height 10 cm, area 20
- But in this case, the left and right rectangles include the full height, so if I add left and right, I have 224 + 168 = 392, and then the top middle 2*10=20, total 412, same as before.

In the subtraction method, large rectangle 16*28=448, minus cut-out 2*18=36, get 412.

Good.

✔ Final for #8: 412 sq cm

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Now, compiling all answers:

1. 475 sq ft
2. 700 sq cm
3. 500 sq m
4. 120 sq ft
5. 264 sq yd
6. 384 sq m
7. 45 sq mm
8. 412 sq cm

Final Answer:
1. 475
2. 700
3. 500
4. 120
5. 264
6. 384
7. 45
8. 412