Question

what is the area of this figure? 16 m 2 m 5 m 8 m 4 m 4 m 2 m 7 m square meters

Explanation:

Step1: Analyze the figure structure

The figure can be divided into three rectangles. Let's identify their dimensions.
- Rectangle 1: Width = 7 m, Height = 8 m. Area formula: $A = w \times h$. So area $A_1 = 7 \times 8 = 56$ m².
- Rectangle 2: Width = 4 m, Height = (16 - 7 - 5) = 4 m? Wait, no. Wait, let's re - check. Wait, the top part: the total length is 16 m. The bottom left rectangle is 7 m wide. The middle part: the vertical side is 8 - 2 = 6? No, maybe better to split as:
Wait, another approach: The main large rectangle (if we consider the outer bounds) but with some adjustments. Wait, actually, let's split the figure into three parts:
1. The bottom left rectangle: 7m (width) × 8m (height) = 56 m².
2. The middle rectangle: (16 - 7 - 5)m = 4m (width) × (8 - 2)m? No, wait the vertical length: the height of the middle part. Wait, the right - most part: 5m (width) × 2m (height) = 10 m².
The middle part: between the bottom left and the right - most. The width is 16 - 7 - 5 = 4m, and the height is 8 - 2 = 6m? Wait, no. Wait the height of the middle rectangle: the total height is 8m, and the top right rectangle has height 2m, the bottom right has height 2m? Wait, no, looking at the figure:
Wait, the figure has a height of 8m. The top right rectangle is 2m tall (height) and 5m wide. The bottom right has a rectangle that is 4m tall and 5m wide? No, the white space is 4m tall and 5m wide. So the area of the figure is the area of the big rectangle (16m × 8m) minus the area of the white rectangle (5m × 4m) plus the area of the small rectangle at the bottom (4m × 2m)? Wait, no, let's do it correctly.
Alternative split:
- Rectangle 1: 7m (width) × 8m (height) = 56 m².
- Rectangle 2: (16 - 7)m = 9m? No, wait 16 - 7 = 9, but then subtract 5 (the width of the right - most) gives 4. Wait, 16 - 7 - 5 = 4m (width) and height is 8 - 2 = 6m? No, 8 - 2 = 6? Wait the top right rectangle is 2m tall, so the middle rectangle (between bottom left and top right) has height 8 - 2 = 6m? And width 16 - 7 - 5 = 4m. So area of rectangle 2: 4×6 = 24 m².
- Rectangle 3: 5m (width) × 2m (height) = 10 m².
Now sum them up: 56+24 + 10=90? Wait, no, that's not right. Wait, let's use the method of subtracting the white area from the big rectangle.
The big rectangle (if we consider the outer dimensions) would be 16m (length) × 8m (height) = 128 m². But there is a white rectangle of 5m (width) × 4m (height) = 20 m². But also, is there a small rectangle added? Wait, no, the figure is not the big rectangle minus the white rectangle. Wait, looking at the figure again:
The figure can be divided into three rectangles:
1. Bottom left: 7m × 8m = 56 m².
2. Middle: (16 - 7 - 5)m = 4m, and height: 8 - 2 = 6m? No, 8 - 2 = 6? Wait the top right rectangle is 2m tall, so the middle rectangle (the one above the bottom left and to the left of the white space) has height 8 - 2 = 6m? And width 4m. So area 4×6 = 24 m².
3. Top right: 5m × 2m = 10 m².
Now total area: 56+24 + 10 = 90? Wait, but let's check another way.
Wait, the white rectangle is 5m (width) × 4m (height) = 20 m². The big rectangle (16m × 8m) = 128 m². But then we have a small rectangle at the bottom (4m × 2m) that is part of the figure? No, maybe my initial split is wrong.
Wait, let's do it step by step:
1. The left - most rectangle: width = 7m, height = 8m. Area $A_1=7\times8 = 56$ m².
2. The middle rectangle: width = 16 - 7 - 5=4m, height = 8 - 2 = 6m? Wait, 8 - 2 = 6? The top right rectangle has height 2m, so the middle rectangle (between left and top right) has height 8 - 2 = 6m. Area $A_2 = 4\times6=24$ m².
3. The top right rectangle: width = 5m, height = 2m. Area $A_3=5\times2 = 10$ m².
Now sum all areas: $A = A_1+A_2+A_3=56 + 24+10=90$ m². Wait, but let's check with the big rectangle minus white area. The big rectangle is 16×8 = 128. The white area is 5×4 = 20. But 128-20 = 108, which is wrong. So my split is wrong.
Wait, another approach: The figure is composed of:
- A rectangle of 7m (width) × 8m (height) = 56 m².
- A rectangle of (16 - 7)m = 9m (width) × 2m (height) = 18 m².
- A rectangle of (16 - 7 - 5)m = 4m (width) × (8 - 2)m = 4×6 = 24 m². Wait, 9m width: 16 - 7 = 9, but 9 - 5 = 4. Wait, no, 16 - 7 = 9, which is the width from the left - most 7m to the right end. Then, the top part is 2m tall, so area 9×2 = 18. Then the middle part (below the top 2m) is 9 - 5 = 4m wide and 8 - 2 = 6m tall, area 4×6 = 24. Then the left - most is 7×8 = 56. Wait 56+18 + 24=98? No, that's not right.
Wait, I think I made a mistake in the height. Let's look at the figure again. The total height is 8m. The white rectangle is 4m tall and 5m wide. The top right rectangle is 2m tall and 5m wide. The bottom left rectangle is 7m wide and 8m tall. The middle rectangle (between bottom left and white rectangle) has width 16 - 7 - 5 = 4m and height 8 - 2 = 6m? No, 8 - 2 = 6? Wait the top right is 2m, so the middle rectangle (the one that is above the bottom and to the left of the white) has height 8 - 2 = 6m. And the bottom right rectangle (below the white) has height 8 - 4 = 4m? No, the white is 4m tall. So the bottom right rectangle is 5m wide and (8 - 4)m = 4m tall? No, the white is 4m tall, so the area of the figure is:
Area of left rectangle: 7×8 = 56.
Area of middle rectangle (above left, between left and white): (16 - 7 - 5)×(8 - 2)=4×6 = 24.
Area of top right: 5×2 = 10.
Area of bottom right: 5×(8 - 4)=5×4 = 20? Wait, no, the white is 4m tall, so the bottom right is part of the figure? Wait, no, the white is a hole. So the figure is:
Left rectangle (7×8) + middle rectangle (4×6) + top right (5×2) + bottom right (5×2)? No, this is getting confusing. Let's use the correct method of adding all the parts of the figure (excluding the white space).
The white space is a rectangle with width 5m and height 4m, area = 5×4 = 20 m².
The area of the big rectangle (if we consider the outer length 16m and outer height 8m) is 16×8 = 128 m². But wait, no, because the top right has a height of 2m, and the bottom right has a height of 2m? Wait, no, the figure's total height is 8m, and the white space is 4m tall. So the area of the figure is the area of the big rectangle (16×8) minus the area of the white rectangle (5×4) plus the area of the small rectangle at the bottom (4×2)? No, I think I need to re - split.
Wait, let's look at the coordinates:
- The bottom left corner is (0,0).
- The bottom right of the left rectangle is (7,8).
- The top right of the top rectangle is (16,2).
- The bottom right of the middle rectangle is (11,8) (since 16 - 5 = 11).
- The white rectangle is from (11,2) to (16,6) (since height 4m, so from y = 2 to y = 6).
So the figure consists of:
1. Rectangle 1: (0,0) to (7,8): area 7×8 = 56.
2. Rectangle 2: (7,2) to (11,8): width = 11 - 7 = 4, height = 8 - 2 = 6, area 4×6 = 24.
3. Rectangle 3: (11,0) to (16,2): width = 16 - 11 = 5, height = 2 - 0 = 2, area 5×2 = 10.
Now sum these areas: 56+24 + 10 = 90. Wait, but let's check with another method. The area of the figure can also be calculated as:
Area = (7×8)+((16 - 7)×2)+((16 - 7 - 5)×(8 - 2))
= 56+(9×2)+(4×6)
= 56 + 18+24
= 98? No, 9×2 = 18, 4×6 = 24, 56+18 = 74, 74+24 = 98. Wait, now I'm confused. There must be a mistake in the coordinate system.
Wait, let's use the formula for the area of a composite figure by adding all the rectangular parts:
- Part 1: 7m (width) × 8m (height) = 56 m².
- Part 2: (16 - 7)m = 9m (width) × 2m (height) = 18 m².
- Part 3: (16 - 7 - 5)m = 4m (width) × (8 - 2)m = 4×6 = 24 m².
Now sum: 56+18 + 24 = 98 m². Wait, where was the mistake before? Ah, the height of the middle part: when we take the part between the left rectangle and the top rectangle, the height is 8 - 2 = 6m, and the width is 16 - 7 - 5 = 4m. The top part (part 2) has width 16 - 7 = 9m and height 2m. And the left part is 7×8. Now 56+18 = 74, 74+24 = 98.
Wait, let's check the white area. The white area is 5m (width) × 4m (height) = 20 m². The area of the big rectangle (16×8) is 128 m². But 128 - 20 = 108, which is not 98. So my split is wrong.
Wait, another way: The figure has three rectangles:
1. Bottom rectangle: 7m (width) × 8m (height) = 56.
2. Middle rectangle: (16 - 7)m = 9m (width) × 2m (height) = 18.
3. Top - middle rectangle: (9 - 5)m = 4m (width) × (8 - 2)m = 4×6 = 24.
Now 56+18 + 24 = 98.
Wait, let's calculate the area of each part:
- 7×8 = 56.
- 9×2 = 18 (this is the top part above the middle rectangle).
- 4×6 = 24 (this is the middle part between the bottom and the top right).
- 5×2 = 10 (this is the top right part). Wait, no, 9 - 5 = 4, so 4×6 = 24, and 5×2 = 10, then 56+18+24 + 10=108? No, 56+18 = 74, 74+24 = 98, 98+10 = 108. Ah! Here's the mistake. I forgot the top right rectangle. So:
1. Bottom left: 7×8 = 56.
2. Middle top: (16 - 7)×2 = 9×2 = 18.
3. Middle middle: (16 - 7 - 5)×(8 - 2)=4×6 = 24.
4. Top right: 5×2 = 10.
Now sum: 56+18+24 + 10 = 108.
Wait, now let's check with the big rectangle minus white area. The big rectangle is 16×8 = 128. The white area is 5×4 = 20. 128 - 20 = 108. Yes! That's correct. Because the white rectangle is 5m wide and 4m tall, so area 20. The big rectangle (16m long, 8m tall) has area 128. Subtract the white area (20) gives 108. So where was the previous mistake? I forgot that the middle top rectangle (16 - 7)×2 and the middle middle (16 - 7 - 5)×(8 - 2) and the top right 5×2 and the bottom left 7×8: 7×8=56, (16 - 7)×2=18, (16 - 7 - 5)×(8 - 2)=24, 5×2=10. 56+18=74, 74+24=98, 98+10=108. And 16×8 - 5×4=128 - 20 = 108. So that's correct.

Step2: Calculate the area of the big rectangle

The length of the big rectangle is 16m and the height is 8m. Using the formula for the area of a rectangle $A = l\times h$, we get $A_{big}=16\times8 = 128$ m².

Step3: Calculate the area of the white rectangle

The white rectangle has a width of 5m and a height of 4m. Using the formula $A = w\times h$, we get $A_{white}=5\times4 = 20$ m².

Step4: Calculate the area of the figure

The area of the figure is the area of the big rectangle minus the area of the white rectangle. So $A = A_{big}-A_{white}=128 - 20 = 108$ m².

Answer:

108