Question

what is the area of this figure
2 m
5 m
2 m
3 m
4 m
16 m
8 m
8 m

Explanation:

Step1: Analyze the figure's structure

The figure can be divided into three rectangles. Let's identify their dimensions:
- First rectangle (top): width = 2 m, height = 5 m.
- Second rectangle (middle): width = 2 + 2 = 4 m? Wait, no. Wait, looking at the diagram, maybe better to split into three parts: bottom, middle, top. Wait, the bottom part: width 8 m, height 8 m? Wait, no, the total height is 16 m? Wait, maybe the figure is composed of three rectangles:

Wait, let's re-express:

1. Bottom rectangle: width = 8 m, height = 8 m (since the bottom part's height is 8 m, as per the diagram: 8 m height, 8 m width? Wait, no, the middle part: height 3 m, width: 8 - 4 = 4? Wait, maybe I misread. Let's try again.

Wait, the figure has three sections:

- Top section: width 2 m, height 5 m.
- Middle section: width (2 + 2) = 4 m? Wait, no, the middle section's width: 2 (top) + 2 (middle) = 4? Wait, the middle section's height is 3 m, and its width is 2 + 2 = 4? Then the bottom section: width 8 m, height 16 - 5 - 3 = 8 m? Wait, 5 + 3 + 8 = 16, which matches the total height of 16 m. And the bottom width is 8 m.

Wait, maybe:

- Top rectangle: \( \text{width} = 2 \, \text{m}, \text{height} = 5 \, \text{m} \)
- Middle rectangle: \( \text{width} = 2 + 2 = 4 \, \text{m}? \) No, wait, the middle part's width: 2 (top) + 2 (middle) = 4? Wait, the middle section's height is 3 m, and its width is \( 2 + 2 = 4 \, \text{m} \) (since the top is 2, middle adds 2, so 2 + 2 = 4). Then the bottom section: \( \text{width} = 8 \, \text{m}, \text{height} = 8 \, \text{m} \) (since total height is 16, 16 - 5 - 3 = 8).

Wait, alternatively, maybe the figure is:

1. Bottom rectangle: width 8 m, height 8 m (area \( 8 \times 8 \))
2. Middle rectangle: width (8 - 4) = 4 m? Wait, no, the middle part's width: 8 - 4 = 4? Wait, the middle section's height is 3 m, and its width is 8 - 4 = 4? Then the top section: width 2 m, height 5 m.

Wait, maybe a better approach: split the figure into three rectangles:

1. Top: \( 2 \times 5 \)
2. Middle: \( (2 + 2) \times 3 = 4 \times 3 \) (since the middle section's width is 2 (top) + 2 (middle) = 4)
3. Bottom: \( 8 \times 8 \) (since bottom width is 8, height is 16 - 5 - 3 = 8)

Wait, let's check the total width: bottom is 8, middle is 4? No, that can't be. Wait, the bottom width is 8, so the middle and top must fit into that. Wait, maybe the top is 2 m wide, middle is (2 + 2) = 4 m wide, and bottom is 8 m wide. So total width: 8 m (bottom) = 2 (top) + 2 (middle) + 4 (bottom)? No, that doesn't make sense. Wait, maybe the figure is:

- Bottom rectangle: width 8 m, height 8 m (area \( 8 \times 8 \))
- Middle rectangle: width (8 - 4) = 4 m? Wait, 8 - 4 = 4, so middle width 4 m, height 3 m (area \( 4 \times 3 \))
- Top rectangle: width 2 m, height 5 m (area \( 2 \times 5 \))

Wait, but 4 + 2 = 6, which is less than 8. No, that's not right. Wait, maybe the correct split is:

1. Bottom part: width 8 m, height 8 m (area \( 8 \times 8 \))
2. Middle part: width (8 - 4) = 4 m? No, 8 - 4 = 4, height 3 m (area \( 4 \times 3 \))
3. Top part: width 2 m, height 5 m (area \( 2 \times 5 \))

Wait, but 4 + 2 = 6, and 8 - 6 = 2, which is missing. Oh, maybe I made a mistake. Let's look at the diagram again. The bottom width is 8 m, the middle part's width is 8 - 4 = 4 m? Wait, the middle part has a 4 m indent? No, the diagram shows:

- Top: 2 m wide, 5 m tall.
- Middle: 2 m (top) + 2 m (middle) = 4 m wide, 3 m tall.
- Bottom: 8 m wide, 16 - 5 - 3 = 8 m tall.

Yes, that makes sense. So:

- Top area: \( 2 \times 5 = 10 \, \text{m}^2 \)
- Middle area: \( (2 + 2) \times 3 = 4 \times 3 = 12 \, \text{m}^2 \) (since the middle section's width is 2 (top) + 2 (middle) = 4)
- Bottom area: \( 8 \times 8 = 64 \, \text{m}^2 \) (since 5 + 3 + 8 = 16, total height 16 m)

Wait, but 2 + 2 + 4 = 8? No, 2 (top) + 2 (middle) + 4 (bottom) = 8? Wait, no, the bottom width is 8 m, so the middle section's width is 8 - 4 = 4? No, I'm confused. Let's try another approach. Let's use the total area minus the missing parts, but maybe it's easier to split into three rectangles:

1. Leftmost rectangle: width 2 m, height 16 m? No, that's not right. Wait, the figure is a composite of three rectangles:

- Rectangle 1: width = 2 m, height = 5 m (top left)
- Rectangle 2: width = 2 m, height = 5 + 3 = 8 m (middle left)
- Rectangle 3: width = 8 - 2 - 2 = 4 m, height = 16 m? No, that's not matching. Wait, the bottom part's height is 8 m, as per the diagram (8 m height, 8 m width).

Wait, maybe the correct dimensions are:

- Bottom rectangle: 8 m (width) × 8 m (height) = 64 m²
- Middle rectangle: (8 - 4) m? Wait, 8 - 4 = 4 m width, 3 m height: 4 × 3 = 12 m²
- Top rectangle: 2 m width, 5 m height: 2 × 5 = 10 m²

Now, sum them up: 64 + 12 + 10 = 86? Wait, no, 64 + 12 is 76, +10 is 86. Wait, but let's check the widths: 2 (top) + 4 (middle) + 2 (bottom)? No, 2 + 4 + 2 = 8, which matches the bottom width of 8 m. Yes! So:

- Top rectangle: width 2 m, height 5 m: area = 2×5 = 10
- Middle rectangle: width 4 m (8 - 2 - 2), height 3 m (since middle height is 3 m): area = 4×3 = 12
- Bottom rectangle: width 8 m, height 8 m (16 - 5 - 3 = 8): area = 8×8 = 64

Now, total area = 10 + 12 + 64 = 86? Wait, no, 10 + 12 is 22, +64 is 86. Wait, but let's verify the heights:

- Top height: 5 m
- Middle height: 3 m
- Bottom height: 8 m
- Total height: 5 + 3 + 8 = 16 m (matches)

Widths:

- Top width: 2 m
- Middle width: 4 m (8 - 2 - 2)
- Bottom width: 8 m (matches)

Yes, that works. So:

Step2: Calculate each rectangle's area

- Top rectangle: \( A_1 = 2 \times 5 = 10 \, \text{m}^2 \)
- Middle rectangle: \( A_2 = 4 \times 3 = 12 \, \text{m}^2 \)
- Bottom rectangle: \( A_3 = 8 \times 8 = 64 \, \text{m}^2 \)

Step3: Sum the areas

Total area \( A = A_1 + A_2 + A_3 = 10 + 12 + 64 = 86 \, \text{m}^2 \)? Wait, no, 10 + 12 is 22, 22 + 64 is 86. Wait, but let's check again. Wait, maybe the middle rectangle's width is 2 + 2 = 4 m (since the top is 2 m, middle is 2 m to the right of top, so total middle width 2 + 2 = 4 m), and its height is 3 m. Then the bottom rectangle's width is 8 m, height is 8 m (16 - 5 - 3 = 8). Then:

- Top: 2×5=10
- Middle: (2+2)×3=4×3=12
- Bottom: 8×8=64
- Total: 10+12+64=86.

Wait, but maybe I made a mistake in the middle width. Let's check the bottom width: 8 m. The top width is 2 m, middle width is 2 + 2 = 4 m, so bottom width is 8 - 2 - 2 = 4 m? No, that's not. Wait, no, the bottom rectangle's width is 8 m, so the middle rectangle's width is 8 - 4 = 4 m? I'm getting confused. Let's try another way. Let's look at the horizontal dimensions:

- The rightmost part: width 4 m, height 8 m (since the bottom height is 8 m, and the right part is 4 m wide, 8 m tall)
- The middle part: width 2 m, height 8 + 3 = 11 m? No, this is not working. Wait, maybe the figure is:

Total width: 8 m, total height: 16 m.

Split into three rectangles:

1. Top: 2 m (width) × 5 m (height)
2. Middle: (2 + 2) m (width) × 3 m (height) → 4 m × 3 m
3. Bottom: 8 m (width) × (16 - 5 - 3) m → 8 m × 8 m

Yes, that makes sense. So:

- Top area: 2×5=10
- Middle area: 4×3=12
- Bottom area: 8×8=64
- Total area: 10+12+64=86.

Wait, but let's check with another split. Suppose we have:

- Rectangle 1: 2m (w) × 16m (h) → no, that's too big.
- Rectangle 2: 2m (w) × (16 - 5)m = 11m (h) → 2×11=22
- Rectangle 3: (8 - 2 - 2)=4m (w) × 16m (h) → 4×16=64
- Total: 2×5 (top) + 2×(16-5) (middle) + 4×16 (bottom)? No, that's not. Wait, I think my initial split is correct. So the total area is 86 m²? Wait, no, maybe I messed up the middle height. Wait, the middle section's height is 3 m, as per the diagram: 3 m height. So 5 (top) + 3 (middle) + 8 (bottom) = 16 (total height). Yes. And the bottom width is 8 m, middle width is 2 + 2 = 4 m, top width is 2 m. 2 + 4 + 2 = 8, which matches the bottom width. So:

Top: 2×5=10

Middle: 4×3=12

Bottom: 8×8=64

Total: 10+12+64=86.

Wait, but let's check with actual numbers. If the bottom is 8x8=64, middle 4x3=12, top 2x5=10, sum is 86. Alternatively, maybe the middle width is 8 - 4 = 4 m (since the right part is 4 m wide, 8 m tall), so:

- Right rectangle: 4m × 8m = 32
- Middle rectangle: (8 - 4)m = 4m? No, this is too confusing. Wait, maybe the correct answer is 86. But let's re-express:

Wait, another approach:

The figure can be divided into three rectangles:

1. Top: width = 2 m, height = 5 m → area = 2×5 = 10
2. Middle: width = 2 m, height = 5 + 3 = 8 m → area = 2×8 = 16
3. Bottom: width = 8 - 2 - 2 = 4 m, height = 16 m → area = 4×16 = 64

Total area: 10 + 16 + 64 = 90. No, that's different. I think I made a mistake in the middle height. Wait, the middle section's height is 3 m, so the middle rectangle's height is 3 m, and its width is 2 + 2 = 4 m (since top is 2, middle is 2, so total 4). Then the bottom rectangle's height is 16 - 5 - 3 = 8 m, width 8 m. So:

Top: 2×5=10

Middle: 4×3=12

Bottom: 8×8=64

Total: 10+12+64=86.

Yes, that seems correct.

Answer:

\boxed{86} (in square meters)