Question

what is the area of this figure?
2 mi
4 mi
4 mi
8 mi
18 mi
3 mi
6 mi
9 mi
square miles

Explanation:

Step1: Divide the figure into three rectangles

We can split the composite figure into three rectangles. Let's identify their dimensions:
- Rectangle 1: Width = 2 mi, Height = 18 mi (from top to bottom)
- Rectangle 2: Width = 4 mi, Height = 8 mi (middle section)
- Rectangle 3: Width = 9 mi, Height = 6 mi (bottom section) Wait, no, actually, let's re - examine the figure. A better way is to consider the horizontal and vertical divisions. Let's do it correctly.

Alternative division:
- The leftmost rectangle: width = 2 mi, height = 18 mi.
- The middle rectangle: width = 4 mi, height = 8 + 6=14 mi? No, wait, let's look at the vertical lengths. The total height is 18 mi. The bottom part is 6 mi, the middle part (between the bottom and the top indent) is 8 mi, and the top indent is 4 mi. The horizontal lengths: the bottom width is 9 mi, the middle width (after the first indent) is 4 mi, and the left width is 2 mi.

Wait, a more accurate division:
1. Rectangle 1: left - most, width = 2 mi, height = 18 mi. Area formula: $A = l\times w$, so $A_1=2\times18 = 36$ square miles.
2. Rectangle 2: middle, width = 4 mi, height = 8 + 6=14 mi? No, wait, the height of the middle rectangle: from the bottom (6 mi) up to the top of the 8 mi section. Wait, the 8 mi is above the 3 mi indent, and the 6 mi is below. Wait, maybe another approach: total width is 9 mi. The left part is 2 mi, the middle part (after the left) is 4 mi, and the right part (after the middle) is 9 - 2 - 4 = 3 mi.

Now, let's define three rectangles:
- Rectangle A: left, width = 2 mi, height = 18 mi. $A_A=2\times18 = 36$.
- Rectangle B: middle, width = 4 mi, height = 8 + 6=14 mi? No, the height of the middle rectangle: the total height is 18 mi. The bottom rectangle (right - most) has height 6 mi, width 9 mi? No, that's not right. Wait, let's use the method of subtracting the missing parts, but it's easier to add the areas of the three parts.

Wait, looking at the figure:
- The bottom rectangle: width = 9 mi, height = 6 mi. Area $A_3 = 9\times6=54$.
- The middle rectangle: width = 4 mi (since 9 - 3 - 2? Wait, no, the middle rectangle is above the bottom one, with height 8 mi, and width = 4 mi (because the right - most part has a width of 3 mi (9 - 2 - 4=3), and the middle part is 4 mi). So area $A_2=4\times(8 + 6)$? No, the height of the middle rectangle is 8 + 6=14? Wait, no, the 8 mi is above the 3 mi indent, and the 6 mi is below. Wait, the 8 mi is between the 3 mi indent and the 4 mi indent.

Wait, let's start over. Let's find the height of each part:
- Bottom part: height = 6 mi, width = 9 mi. Area $A_1 = 9\times6=54$.
- Middle part: above the bottom part, height = 8 mi, width = 9 - 3=6 mi? No, the 3 mi is the width of the right - most indent. Wait, the horizontal length: the bottom has width 9 mi. Then, there is a 3 mi width (on the right) that has height 6 mi, and above that, a 3 mi width that is indented by 3 mi? No, the figure has a left - most column (2 mi wide, 18 mi tall), a middle column (4 mi wide, from the bottom 6 mi up to 8 mi above the bottom, so total height 6 + 8=14 mi), and a right - most column (3 mi wide, height 6 + 8=14 mi? No, the right - most column has a height of 6 mi (bottom) and then 8 mi above, but there is a 4 mi tall indent at the top.

Wait, I think I made a mistake. Let's use the correct division:

1. Left rectangle: width = 2 mi, height = 18 mi. Area: $2\times18 = 36$.
2. Middle rectangle: width = 4 mi, height = 8 + 6=14 mi. Area: $4\times14 = 56$.
3. Right rectangle: width = 3 mi (since 9 - 2 - 4 = 3), height = 6 mi. Area: $3\times6 = 18$.

Now, sum the areas: $36+56 + 18=110$? Wait, no, that can't be. Wait, let's check the heights again. The middle rectangle's height: the total height is 18 mi. The bottom part is 6 mi, the middle part (above the bottom) is 8 mi, and the top part (indent) is 4 mi. So the middle rectangle (between the left and the right) has height 6 + 8=14 mi, and the right rectangle (the small one on the right) has height 6 mi? No, the right rectangle should have the same height as the middle rectangle? Wait, no, the right - most part: the bottom is 6 mi, then above it is 8 mi, and then there is a 4 mi tall indent at the top. Wait, the total height is 6+8 + 4=18 mi, which matches.

So the right rectangle: width = 3 mi (9 - 2 - 4), height = 6 mi (bottom) + 8 mi (middle) = 14 mi? Wait, no, the 4 mi is the indent (a missing part). So the right rectangle's height is 18 - 4=14 mi? Yes! Because the top 4 mi is an indent (a part that is not there). So:

- Left rectangle: width = 2 mi, height = 18 mi. Area: $2\times18 = 36$.
- Middle rectangle: width = 4 mi, height = 18 - 4=14 mi. Area: $4\times14 = 56$.
- Right rectangle: width = 3 mi (9 - 2 - 4), height = 18 - 4=14 mi? No, wait, the right rectangle's height: the bottom is 6 mi, then 8 mi, so total 14 mi, and the top 4 mi is missing? No, the indent is on the middle - top. Wait, I think the correct way is:

The figure can be divided into three rectangles:

1. Bottom rectangle: width = 9 mi, height = 6 mi. Area: $9\times6 = 54$.
2. Middle rectangle: width = 4 + 2=6 mi? No, wait, the middle rectangle (above the bottom) has height 8 mi, width = 9 - 3=6 mi (since the right - most 3 mi is not part of the middle rectangle? No, this is getting confusing. Let's use the initial correct division:

Left rectangle: 2 mi (width) * 18 mi (height) = 36.

Middle rectangle: 4 mi (width) * (8 + 6) mi (height) = 4*14 = 56.

Right rectangle: 3 mi (width) * 6 mi (height) = 18.

Wait, but 36+56 + 18=110. But let's check with another method. Let's calculate the area as the area of the big rectangle minus the area of the missing parts.

The big rectangle (if there were no indents) would have width 9 mi and height 18 mi. Area = 9*18 = 162.

Now, the missing parts:

- The top - right indent: width = 9 - 2 - 4=3 mi, height = 4 mi. Area of missing part 1: 3*4 = 12.
- The middle - right indent: width = 3 mi, height = 8 mi? No, wait, no. Wait, the figure has two indents? No, looking at the figure: there is a 4 mi tall indent (top) with width 9 - 2 - 4=3 mi, and a 3 mi wide indent (middle) with height? No, the given lengths: 4 mi (top indent height), 8 mi (middle height), 3 mi (middle indent width), 6 mi (bottom height).

Wait, I think my first method was wrong. Let's do it properly:

1. Bottom rectangle: width = 9 mi, height = 6 mi. Area: $A_1=9\times6 = 54$.
2. Middle rectangle: above the bottom, height = 8 mi, width = 9 - 3=6 mi (since the right - most 3 mi is not part of the middle rectangle? No, the 3 mi is the width of the right - most column in the bottom. Wait, the middle rectangle (the one with height 8 mi) has width = 4 + 2=6 mi (left 2 mi and middle 4 mi). So area $A_2=6\times8 = 48$.
3. Top rectangle: height = 4 mi, width = 2 mi. Area $A_3=2\times4 = 8$.

Now, sum them up: $54 + 48+8=110$. Yes, same result.

So the area is 110 square miles.

Step1: Calculate area of left rectangle

Width = 2 mi, Height = 18 mi. Area formula: $A = width\times height$. So $A_1 = 2\times18=36$.

Step2: Calculate area of middle rectangle

Width = 4 mi, Height = 8 + 6 = 14 mi. Area formula: $A = width\times height$. So $A_2=4\times14 = 56$.

Step3: Calculate area of right rectangle

Width = 9 - 2 - 4=3 mi, Height = 6 mi. Area formula: $A = width\times height$. So $A_3=3\times6 = 18$.

Step4: Sum the areas

Total area $A = A_1+A_2+A_3=36 + 56+18=110$.

Answer:

110