Question

8 cm
16 cm
29 cm
22 cm
find the area and perimeter of the composite figure shown.
area:
perimeter:

Explanation:

Step1: Find the total width of the base

The base width is the sum of 8 cm and 22 cm, so \( 8 + 22 = 30 \) cm.

Step2: Calculate the area of the composite figure

We can divide the composite figure into two rectangles. The first rectangle has dimensions 29 cm (height) and 8 cm (width). The second rectangle has dimensions \( 29 - 16 = 13 \) cm (height) and 22 cm (width).
The area of the first rectangle is \( 29\times8 = 232 \) \( cm^2 \).
The area of the second rectangle is \( 13\times22 = 286 \) \( cm^2 \).
The total area is \( 232 + 286 = 518 \) \( cm^2 \).

Step3: Calculate the perimeter of the composite figure

For the perimeter, we can use the formula for the perimeter of a rectangle with length 30 cm (total width) and height 29 cm, but we need to check if there are any indentations. However, in this case, the horizontal and vertical segments that are indented are balanced. So the perimeter is \( 2\times(30 + 29) = 2\times59 = 118 \) cm. Wait, no, let's re - check.
Wait, actually, when we look at the figure, the top horizontal segments: the left part is 8 cm, then there is a 22 cm segment. The vertical segments: the left side is 29 cm, the right side is also 29 cm (since the lower rectangle's height is 13 cm and the upper left rectangle's height is 29 cm, and the indentation is 16 cm, but when calculating perimeter, we can use the "outer rectangle" method with some adjustments.
Wait, another way: sum all the outer sides.
Top: 8 + 22 = 30 cm.
Bottom: 30 cm.
Left: 29 cm.
Right: 29 cm.
Now the vertical sides in the middle: the upper indentation is 16 cm, and the lower part (the height of the lower rectangle) is 13 cm. But when calculating perimeter, the two vertical segments (the ones that are part of the indentation) are also included. Wait, no, let's list all the sides:
Starting from the top - left corner, going right: 8 cm, then down: 16 cm, then right: 22 cm, then down: 13 cm (since 29 - 16 = 13), then left: 30 cm (8 + 22), then up: 29 cm. Wait, no, that's not correct.
Wait, the correct way is to use the formula for the perimeter of a composite figure by considering the total length around. The horizontal length is 8+22 = 30 cm, and the vertical length is 29 cm. But we have to check the indentation. However, in this case, the horizontal and vertical "missing" parts are internal and do not affect the perimeter. Wait, actually, the perimeter is equal to the perimeter of a rectangle with length 30 cm and width 29 cm. So \( P=2\times(30 + 29)=118 \) cm. But wait, let's count the sides:
Top: 8 + 22 = 30
Right: 29
Bottom: 30
Left: 29
Wait, that's too simple. Wait, no, there are two vertical segments in the middle. Wait, no, the figure is like an L - shape but with the upper part on the left and the lower part spanning the width. Wait, maybe my initial division is wrong. Let's divide the figure into two rectangles:
Rectangle 1: height = 29 cm, width = 8 cm.
Rectangle 2: height = 29 - 16 = 13 cm, width = 22 cm.
Now, for the perimeter:
The left side of rectangle 1: 29 cm.
The right side of rectangle 2: 13 cm.
The bottom side of rectangle 2: 22 cm.
The top side of rectangle 1: 8 cm.
The top side of rectangle 2: 22 cm.
The right side of rectangle 1: 29 cm (but wait, no, the right side of rectangle 1 is adjacent to the left side of rectangle 2? No, actually, the figure is such that the upper rectangle (rectangle 1) is on the left, and the lower rectangle (rectangle 2) is below it, spanning from the left (under rectangle 1) to the right with width 22 cm.
Wait, maybe a better approach:
Perimeter = 2*(length + width) where length is the total horizontal length (8 + 22) and width is the total vertical length (29), but we have to add the two vertical segments that are part of the indentation? No, actually, in this case, the indentation is such that the horizontal and vertical segments that are indented are internal. Wait, no, let's draw the figure mentally:
- Left vertical side: 29 cm.
- Top horizontal side: 8 cm (right from the top - left corner), then down 16 cm, then right 22 cm, then down 13 cm (29 - 16), then left 30 cm (8 + 22), then up 29 cm.
Now, let's sum all these sides:
29 (left) + 8 (top - left horizontal) + 16 (down) + 22 (right - middle horizontal) + 13 (down) + 30 (bottom horizontal) + 29 (up - right vertical)
Wait, 29+8 = 37; 37+16 = 53; 53+22 = 75; 75+13 = 88; 88+30 = 118; 118+29 = 147. Wait, that's wrong. I see my mistake. The up - right vertical side is not 29 cm. The up - right vertical side should be equal to the left vertical side? No, let's start over.
The correct way to calculate the perimeter of a composite figure is to add the lengths of all the outer edges.
Let's identify all the outer edges:
1. Top - left to top - middle: 8 cm (horizontal, right)
2. Top - middle to middle - right: 16 cm (vertical, down)
3. Middle - right to bottom - right: 22 cm (horizontal, right)
4. Bottom - right to bottom - left: 13 cm (vertical, down) Wait, no, 29 - 16 = 13, but the bottom - left to top - left is 29 cm (vertical, up).
Wait, I think I messed up the direction. Let's go clockwise from the bottom - left corner:
- Bottom - left to bottom - right: 8 + 22 = 30 cm (horizontal, right)
- Bottom - right to top - right: 29 cm (vertical, up)
- Top - right to top - middle: 22 cm (horizontal, left)
- Top - middle to top - left: 16 cm (vertical, up)
- Top - left to bottom - left: 8 cm (vertical, down)? No, that's not right.
Wait, maybe the figure is composed of two rectangles:
Rectangle A: width = 8 cm, height = 29 cm.
Rectangle B: width = 22 cm, height = 29 - 16 = 13 cm.
Now, the perimeter of rectangle A is \( 2\times(8 + 29)=74 \) cm.
The perimeter of rectangle B is \( 2\times(22 + 13)=70 \) cm.
But when we combine them, the overlapping side (the side where rectangle A and rectangle B are joined) is 8 cm (the width of rectangle A? No, the overlapping side is the vertical side of length 13 cm? Wait, no, rectangle A is on the left, height 29, width 8. Rectangle B is on the right, below rectangle A, width 22, height 13. So the overlapping side is the vertical side of length 13 cm (the right side of rectangle A and the left side of rectangle B). So when we calculate the total perimeter, we have to subtract twice the length of the overlapping side (because it's internal now).
Perimeter of A: \( 2\times(8 + 29)=74 \)
Perimeter of B: \( 2\times(22 + 13)=70 \)
Overlapping side length: 13 cm.
Total perimeter = 74+70 - 2\times13=74 + 70-26 = 118 cm. Which matches the earlier "outer rectangle" method.
And the area: area of A + area of B = \( 8\times29+22\times13 = 232+286 = 518 \) \( cm^2 \)

Answer:

Area: \( 518 \) square centimeters
Perimeter: \( 118 \) centimeters