Question

what is the area of this figure? 2 ft, 5 ft, 2 ft, 3 ft, 4 ft, 16 ft, 8 ft, 8 ft

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, looking at the base which is 8 m. Wait, maybe better to calculate the total height. Wait, the total height on the left is 16 m? Wait, maybe the figure is composed of three parts:
Wait, maybe the correct way is to split the figure into three rectangles:
1. Top rectangle: width = 2 m, height = 5 m.
2. Middle rectangle: width = 2 + 2 = 4 m? Wait, no, looking at the horizontal segments: 2 m, then 2 m, then 4 m (since 2 + 2 + 4 = 8 m, which is the base). And vertical segments: 5 m, 3 m, and 8 m (since 5 + 3 + 8 = 16 m, which is the left side).
So:
- Rectangle 1: width = 2 m, height = 5 m. Area = 2 * 5 = 10 m².
- Rectangle 2: width = 2 + 2 = 4 m? Wait, no, the middle part: after the first 2 m, there's another 2 m, so width = 2 + 2 = 4 m, height = 3 m. Area = 4 * 3 = 12 m².
- Rectangle 3: width = 8 m, height = 8 m. Area = 8 * 8 = 64 m².
Wait, but 2 + 2 + 4 = 8, which matches the base. And 5 + 3 + 8 = 16, which matches the left side.
Now sum the areas: 10 + 12 + 64 = 86? Wait, no, that can't be. Wait, maybe I made a mistake.
Wait, another approach: The total area can be calculated as the area of the large rectangle (8 m by 16 m) minus the areas of the missing parts. Wait, but the missing parts? Wait, no, the figure is a composite of three rectangles. Wait, let's check the dimensions again.
Wait, the base is 8 m. The left side is 16 m. The right side has a height of 8 m, then 3 m, then 5 m (8 + 3 + 5 = 16). The horizontal segments: from the right, 4 m, then 2 m, then 2 m (4 + 2 + 2 = 8).
So:
- Rightmost rectangle: width = 4 m, height = 8 m. Area = 4 * 8 = 32 m².
- Middle rectangle: width = 2 m, height = 8 + 3 = 11 m. Area = 2 * 11 = 22 m².
- Leftmost rectangle: width = 2 m, height = 8 + 3 + 5 = 16 m? No, that's not right. Wait, I think I messed up.
Wait, let's do it properly. Let's split the figure vertically:
- First vertical strip: width = 2 m, height = 16 m. Area = 2 * 16 = 32 m².
- Second vertical strip: width = 2 m, height = 16 - 5 = 11 m (since the top 5 m is only in the first strip). Area = 2 * 11 = 22 m².
- Third vertical strip: width = 4 m, height = 16 - 5 - 3 = 8 m (since 5 + 3 = 8, 16 - 8 = 8). Area = 4 * 8 = 32 m².
Now sum: 32 + 22 + 32 = 86? No, that's not right. Wait, maybe the correct split is:
Wait, the figure can be divided into three rectangles:
1. Top: 2m (width) * 5m (height) = 10
2. Middle: (2+2)m (width) * 3m (height) = 4*3=12
3. Bottom: 8m (width) * 8m (height)=64
Total: 10+12+64=86? Wait, but 2+2+4=8 (width), 5+3+8=16 (height). But 8*16=128. Wait, that's a problem. So maybe my split is wrong.
Wait, another way: The figure is a large rectangle of 8m (width) and 16m (height) minus two rectangles? Wait, no, the figure is green, so maybe the missing parts are not there. Wait, looking at the diagram:
- The top part: 2m wide, 5m tall.
- Then, to the right of that, a 2m wide, (5+3)=8m tall? No, the middle part: 2m wide, 3m tall, and then the bottom part: 8m wide, 8m tall. Wait, no, let's look at the horizontal lengths:
From left to right:
- First segment: 2m (width), height: 5m (top) + 3m (middle) + 8m (bottom) = 16m? No, that can't be. Wait, the vertical lengths:
From bottom to top:
- Bottom: 8m tall, 8m wide.
- Middle: 3m tall, (8 - 4)=4m wide? No, I'm getting confused.
Wait, let's use the formula for composite figures: sum of areas of individual rectangles.
Let's list the rectangles:
1. Rectangle 1: width = 2 m, height = 5 m. Area = 2 * 5 = 10 m².
2. Rectangle 2: width = 2 m (adjacent to rectangle 1) + 2 m? Wait, no, the next horizontal segment is 2 m, so width = 2 + 2 = 4 m, height = 3 m. Area = 4 * 3 = 12 m².
3. Rectangle 3: width = 8 m, height = 8 m. Area = 8 * 8 = 64 m².
Now sum: 10 + 12 + 64 = 86 m²? Wait, but 2 + 2 + 4 = 8 (width), and 5 + 3 + 8 = 16 (height). But 8 * 16 = 128, which is larger than 86. So maybe there's a mistake.
Wait, another approach: The total height is 16 m, total width is 8 m. The missing parts:
- To the right of the first 2 m, there's a rectangle missing? No, the figure is green, so it's the area we need. Wait, maybe the correct dimensions are:
- Top rectangle: 2m (width) * 5m (height) = 10
- Middle rectangle: (2+2)m (width) * 3m (height) = 4*3=12
- Bottom rectangle: 8m (width) * 8m (height)=64
Total: 10+12+64=86. But let's check the vertical height: 5+3+8=16, which matches the left side. Horizontal width: 2+2+4=8, which matches the base. So that works.
Wait, but maybe I made a mistake in the middle rectangle's width. Let's see: the first rectangle is 2m wide. Then, next to it, there's another 2m, so total width for the middle rectangle is 2+2=4m, height 3m. Then the bottom rectangle is 8m wide, height 8m. So yes, 2+2+4=8, 5+3+8=16. So the areas are 10, 12, 64. Sum is 86. Wait, but 8*16=128, so 128 - (missing areas). Wait, what's the area of the missing parts?
Wait, the total area of the rectangle 8m by 16m is 128 m². If our calculated area is 86, then the missing area would be 128 - 86 = 42. But that doesn't help. Wait, maybe my initial split is wrong.
Wait, let's look at the diagram again. The right side has a height of 8m, then 3m, then 5m (8+3+5=16). The bottom base is 8m. The horizontal segments: 4m, 2m, 2m (4+2+2=8). So:
- Rectangle 1: width=4m, height=8m. Area=4*8=32
- Rectangle 2: width=2m, height=8+3=11m. Area=2*11=22
- Rectangle 3: width=2m, height=16m. Area=2*16=32
Sum: 32+22+32=86. Same as before. So that must be the correct area.
Wait, but let's verify:
- 2m (left) * 16m (height) = 32
- 2m (middle) * (16 - 5) = 2*11=22 (since the top 5m is only in the left 2m)
- 4m (right) * 8m (height) = 32
32+22+32=86. Yes, that makes sense.

Step2: Calculate each area and sum

- Area of rectangle 1: \( 2 \times 5 = 10 \) m²
- Area of rectangle 2: \( (2 + 2) \times 3 = 4 \times 3 = 12 \) m² (Wait, no, earlier I did 2*11, but maybe this is another way. Wait, maybe the middle rectangle is width=2+2=4m, height=3m, and the bottom is 8m width, 8m height. Then 10+12+64=86. Yes, that's consistent.
So total area = 10 + 12 + 64 = 86 m²? Wait, no, 10+12=22, 22+64=86. Yes.
Wait, but let's check with another method. The total area of the large rectangle (8m x 16m) is 128 m². Now, are there any missing areas? Let's see:
- To the right of the first 2m, there's a rectangle of width=6m (8-2) and height=5m. Area=6*5=30
- To the right of the first 4m (2+2), there's a rectangle of width=4m (8-4) and height=3m. Area=4*3=12
So missing areas: 30 + 12 = 42
Total area of large rectangle: 8*16=128
Subtract missing areas: 128 - 42 = 86. Yes, that matches. So the area is 86 m².

Answer:

The area of the figure is \(\boxed{86}\) square meters.