Question

what is the area of this figure?
3 mm
7 mm
4 mm
8 mm
2 mm
3 mm
9 mm
18 mm
square millimeters
submit

Explanation:

Step1: Divide the figure into three rectangles.

Let's divide the composite figure into three rectangles:
- Rectangle 1: Base = 18 mm, Height = 3 mm.
- Rectangle 2: Base = 7 + 8 = 15 mm? Wait, no. Wait, let's re - examine. Wait, the middle rectangle: the height is 4 mm, and the base: let's see, the total length is 18 mm. The first rectangle (bottom) is 18 mm long and 3 mm tall. Then the middle rectangle: the height is 4 mm, and the base length: let's see, the rightmost rectangle is 3 mm wide (from the top part) and 9 - 3 - 4=2? Wait, no, maybe a better way.

Alternative division:
- Bottom rectangle: length = 18 mm, height = 3 mm. Area \( A_1=18\times3 = 54\) square mm.
- Middle rectangle: length = 18 - 3=15? No, wait, looking at the horizontal segments. The middle part: the horizontal length for the middle rectangle (height 4 mm) is 7 + 8=15? Wait, no, the left part of the middle rectangle is 7 mm, the middle part is 8 mm, and the right part is 3 mm. Wait, maybe:

First rectangle (bottom): width = 18 mm, height = 3 mm. Area \( A_1 = 18\times3\)
Second rectangle (middle): width = 7 + 8=15 mm? No, wait, the height of the middle rectangle is 4 mm, and the width: the total width is 18 mm, but the rightmost rectangle (top) has width 3 mm. Wait, maybe:

Bottom rectangle: 18 mm (length) × 3 mm (height) = 54

Middle rectangle: (7 + 8) mm? Wait, no, the middle rectangle's length: from the left, after the bottom rectangle, the middle part has a horizontal length of 7 + 8 = 15? No, wait, the middle rectangle's height is 4 mm, and the length: let's see, the bottom rectangle is 18 mm long. The middle rectangle is on top of the bottom rectangle, but shifted? Wait, no, the figure can be divided into three rectangles:

1. Bottom rectangle: length = 18 mm, height = 3 mm. Area \( A_1=18\times3 = 54\)
2. Middle rectangle: length = (18 - 3) mm? No, wait, the middle rectangle (height 4 mm) has a length of 7+8 = 15 mm? Wait, 7 mm (left part) + 8 mm (middle part) = 15 mm, and height 4 mm. Area \( A_2=15\times4=60\)
3. Top rectangle: length = 3 mm, height = 2 mm (since 9 - 3 - 4=2? Wait, 9 mm is the total height. The bottom rectangle is 3 mm, middle is 4 mm, so top is 9 - 3 - 4 = 2 mm? Wait, no, the top rectangle's height is 2 mm (given in the figure: 2 mm) and width 3 mm. Area \( A_3 = 3\times2=6\)? No, wait, the top rectangle's height is 9 - 3 - 4=2? Wait, the total height is 9 mm. Bottom: 3 mm, middle: 4 mm, so top: 9 - 3 - 4 = 2 mm. And the width of the top rectangle is 3 mm. So \( A_3=3\times2 = 6\)? Wait, no, the top rectangle's height is given as 2 mm in the figure. Wait, the figure shows: the top part has a height of 2 mm (the vertical segment labeled 2 mm) and width 3 mm. Then the middle part has height 4 mm, and the bottom part has height 3 mm. And the total height is 3 + 4+ 2=9 mm, which matches.

Wait, maybe a better division:

- Rectangle 1: bottom, width = 18 mm, height = 3 mm. Area \( A_1=18\times3 = 54\)
- Rectangle 2: middle, width = 18 - 3=15 mm? No, wait, the middle rectangle: the horizontal length is 7 + 8=15 mm (7 mm on the left, 8 mm in the middle), and height = 4 mm. Area \( A_2=15\times4=60\)
- Rectangle 3: top, width = 3 mm, height = 2 mm. Area \( A_3=3\times2 = 6\)

Wait, no, that can't be right. Wait, let's check the total width. 7 + 8+ 3=18 mm, which matches the total length of 18 mm. Yes! Because 7 (left middle) + 8 (middle middle) + 3 (right top) = 18 mm. And the heights: 3 (bottom) + 4 (middle) + 2 (top) = 9 mm, which matches the total height of 9 mm.

So now, calculate each area:

Step2: Calculate area of each rectangle.

- Area of bottom rectangle (\( A_1\)): \( A_1=18\times3 = 54\)
- Area of middle rectangle (\( A_2\)): The width of the middle rectangle is 7 + 8 = 15 mm, height is 4 mm. So \( A_2=15\times4=60\)
- Area of top rectangle (\( A_3\)): The width is 3 mm, height is 2 mm. So \( A_3=3\times2 = 6\)

Step3: Sum the areas.

Total area \( A = A_1+A_2+A_3=54 + 60+6=120\)? Wait, that can't be. Wait, no, I think I made a mistake in the middle rectangle's width. Wait, the middle rectangle: the left part is 7 mm, the middle part is 8 mm, and the right part is 3 mm. But the middle rectangle (height 4 mm) should have a width of 7 + 8=15 mm? Wait, no, the bottom rectangle has width 18 mm. The middle rectangle is on top of the bottom rectangle, but the rightmost 3 mm is part of the top rectangle. Wait, maybe another division:

Alternative division:

- Rectangle 1: bottom, width = 18 mm, height = 3 mm. Area \( A_1 = 18\times3=54\)
- Rectangle 2: middle, width = 18 - 3=15 mm? No, wait, the middle rectangle (height 4 mm) has a width of 7 + 8=15 mm (because the rightmost 3 mm is for the top rectangle). So \( A_2=15\times4 = 60\)
- Rectangle 3: top, width = 3 mm, height = 2 mm. Area \( A_3=3\times2=6\)

Wait, but 54 + 60+6 = 120. But let's check with another method. Let's use the formula for the area of a composite figure by subtracting the missing parts, but in this case, it's easier to add.

Wait, another way:

The total height is 9 mm, total width is 18 mm. But there are no missing parts, we are adding. Wait, let's check the dimensions again.

Wait, the bottom rectangle: height 3 mm, width 18 mm. Correct.

Middle rectangle: height 4 mm, width: the horizontal length from the left (after the bottom rectangle) is 7 mm, then 8 mm, so 7 + 8 = 15 mm. Correct, because 15 + 3=18 mm.

Top rectangle: height 2 mm, width 3 mm. Correct, because 3 + 4+ 2=9 mm.

So \( A_1 = 18\times3=54\), \( A_2=15\times4 = 60\), \( A_3=3\times2=6\). Sum: 54+60 + 6=120. Wait, but let's check with another approach.

Wait, maybe I messed up the middle rectangle's width. Let's look at the horizontal segments:

- The bottom rectangle: spans the entire 18 mm, height 3 mm.

- The middle rectangle: above the bottom rectangle, but the rightmost 3 mm is not part of the middle rectangle (it's part of the top rectangle). So the middle rectangle's width is 18 - 3=15 mm, height 4 mm. So \( A_2=15\times4 = 60\)

- The top rectangle: width 3 mm, height 2 mm (since 9 - 3 - 4=2). So \( A_3=3\times2=6\)

Total area: 54+60 + 6=120. Wait, but let's check the vertical segments. The total height is 9 mm. 3 (bottom) + 4 (middle) + 2 (top)=9. Correct.

Wait, but let's calculate using another division. Let's divide into three rectangles with widths 7, 8, 3 and heights 3 + 4, 4, 2? No, that's more complicated.

Wait, another way:

- Left - bottom rectangle: width = 7 mm, height = 3 mm. Area \( A_{lb}=7\times3 = 21\)
- Middle - bottom rectangle: width = 8 mm, height = 3 mm. Area \( A_{mb}=8\times3=24\)
- Right - bottom rectangle: width = 3 mm, height = 3 mm. Area \( A_{rb}=3\times3 = 9\)
- Middle - middle rectangle: width = 7 + 8=15 mm? No, wait, the middle rectangle (height 4 mm) would be on top of the bottom rectangles. Wait, this is getting confusing. Let's go back to the first division.

First division:

\( A_1=18\times3 = 54\)

\( A_2=(7 + 8)\times4=15\times4 = 60\)

\( A_3=3\times2=6\)

Total area \( A=54 + 60+6=120\)? Wait, no, 18×3=54, 15×4=60, 3×2=6. 54+60=114, 114 + 6=120.

Wait, but let's check the total height. 3 (bottom) + 4 (middle) + 2 (top)=9, which matches. And the total width: 7 + 8+ 3=18, which matches. So the area should be 120? Wait, no, wait, the middle rectangle's height is 4 mm, and the width: 7 + 8=15 mm (because the right 3 mm is for the top rectangle). And the top rectangle's height is 2 mm, width 3 mm. And the bottom rectangle's width is 18 mm, height 3 mm. So yes, the sum is 54+60 + 6=120.

Wait, but let's check with a different division. Let's take the rightmost rectangle: width 3 mm, height 9 mm. Area \( A_r=3\times9 = 27\)

Then the middle rectangle: width 8 mm, height 9 - 3=6 mm? No, 9 - 3=6? Wait, 3 (bottom) + 4 (middle) + 2 (top)=9. If we take the middle rectangle as width 8 mm, height 4 + 2=6 mm? No, this is getting messy.

Wait, going back to the first division:

Bottom rectangle: 18×3=54

Middle rectangle: (7 + 8)×4=15×4=60 (because 7 mm is the left part of the middle section, 8 mm is the middle part of the middle section, and together they make 15 mm, with height 4 mm)

Top rectangle: 3×2=6 (width 3 mm, height 2 mm)

Total: 54+60 + 6=120.

Yes, that seems correct.

Answer:

120