Area of Composite Shapes

Show your work for each problem.

1.
What is the first step when finding the area of a composite shape?
1. a)Draw a circle around the shape
2. b)Add all the sides together
3. c)Identify the basic shapes within the composite shape
4. d)Multiply length times width
Answer: Identify the basic shapes within the composite shape
The first step is to identify the basic shapes (rectangles, triangles, circles) that make up the composite shape. Only then can you apply the correct formulas.
2.
A rectangle with a circular hole cut out of it requires you to:
1. a)Add the circle area to the rectangle area
2. b)Multiply the areas together
3. c)Subtract the circle area from the rectangle area
4. d)Divide the rectangle area by the circle area
Answer: Subtract the circle area from the rectangle area
When there's a cutout or hole, you subtract its area. The hole removes material from the shape, so the remaining area is the rectangle area minus the circle area.
3.
An L-shaped figure is made of two rectangles: one is $6$ m by $4$ m and the other is $3$ m by $2$ m. What is the total area in m²?
Answer: 30
Rectangle 1: $6 \times 4 = 24$ m². Rectangle 2: $3 \times 2 = 6$ m². Total: $24 + 6 = 30$ m².
4.
A house-shaped figure has a square base with side $5$ m and a triangular roof with base $5$ m and height $2$ m. What is the total area in m²?
Answer: 30
Square: $5 \times 5 = 25$ m². Triangle: $\frac{1}{2} \times 5 \times 2 = 5$ m². Total: $25 + 5 = 30$ m².
5.
Find the area of a rectangle ( $8$ cm by $5$ cm) with a triangular extension (base $5$ cm, height $3$ cm).
Answer: 47.5
1. What is the area of the rectangle? ( $8 \times 5$ ) 40
2. What is the area of the triangle? ( $\frac{1}{2} \times 5 \times 3$ ) 7.5
3. What is the total area? 47.5
6.
A rectangle measures $10$ m by $6$ m with a circular pool of radius $2$ m in the center. Using $\pi \approx 3.14$ , what is the area of the lawn (excluding the pool)?
1. a) $72.56$ m²
2. b) $60$ m²
3. c) $12.56$ m²
4. d) $47.44$ m²
Answer: $47.44$ m²
Rectangle: $10 \times 6 = 60$ m². Circle: $\pi \times 2^2 = 3.14 \times 4 = 12.56$ m². Lawn: $60 - 12.56 = 47.44$ m².
7.
A metal sheet is $15$ cm by $10$ cm with a square hole of side $3$ cm cut out. What is the remaining area in cm²?
Answer: 141
Rectangle: $15 \times 10 = 150$ cm². Square hole: $3 \times 3 = 9$ cm². Remaining: $150 - 9 = 141$ cm².
8.
A wall is $6$ m wide and $4$ m tall with a triangular gable (base $6$ m, height $2$ m) on top. Find the total wall area.
Answer: 30
1. What is the area of the rectangular wall? 24
2. What is the area of the triangular gable? 6
3. What is the total wall area? 30
9.
An arrow-shaped sign has a rectangle ( $12$ cm by $8$ cm) with a triangle on one end (base $8$ cm, height $5$ cm). What is the total area in cm²?
Answer: 116
Rectangle: $12 \times 8 = 96$ cm². Triangle: $\frac{1}{2} \times 8 \times 5 = 20$ cm². Total: $96 + 20 = 116$ cm².
10.
A sign is a rectangle ( $14$ cm by $10$ cm) with TWO triangular points, one on each short end. Each triangle has base $10$ cm and height $4$ cm. Find the total area.
Answer: 180
1. What is the area of the rectangle? 140
2. What is the area of ONE triangle? 20
3. What is the area of BOTH triangles? 40
4. What is the total area? 180
11.
A square courtyard ( $20$ m side) has a rectangular garden ( $12$ m by $8$ m) and a circular fountain (radius $3$ m) inside. Using $\pi \approx 3.14$ , what is the remaining paved area?
1. a) $304$ m²
2. b) $275.74$ m²
3. c) $400$ m²
4. d) $371.74$ m²
Answer: $275.74$ m²
Square: $20 \times 20 = 400$ m². Garden: $12 \times 8 = 96$ m². Fountain: $3.14 \times 9 = 28.26$ m². Paved: $400 - 96 - 28.26 = 275.74$ m².
12.
A T-shaped platform consists of three rectangles: a horizontal bar ( $16$ m by $4$ m) and a vertical stem ( $4$ m by $10$ m). What is the total area in m²?
Answer: 104
Horizontal bar: $16 \times 4 = 64$ m². Vertical stem: $4 \times 10 = 40$ m². Total: $64 + 40 = 104$ m².
13.
A window consists of a rectangle ( $1.2$ m by $0.8$ m) with a semicircular top (diameter $0.8$ m). Using $\pi \approx 3.14$ , find the total glass area.
Answer: 1.21
1. What is the area of the rectangle? 0.96
2. What is the radius of the semicircle? (diameter ÷ 2) 0.4
3. What is the area of the semicircle? ( $\frac{\pi r^2}{2}$ ) 0.25
4. What is the total glass area? (rounded to 2 decimals) 1.21
14.
A hexagonal nut has a hexagonal outer boundary that can be approximated as a square ( $2$ cm side) with four corner triangles (each with legs $0.5$ cm). The circular hole in the center has radius $0.6$ cm. Using $\pi \approx 3.14$ , find the metal area to 2 decimal places.
Answer: 2.37
Square: $2 \times 2 = 4$ cm². Each corner triangle: $\frac{1}{2} \times 0.5 \times 0.5 = 0.125$ cm². Four triangles: $0.5$ cm². Circle: $3.14 \times 0.36 = 1.13$ cm². Metal: $4 - 0.5 - 1.13 = 2.37$ cm².
15.
A running track is a rectangle ( $100$ m by $60$ m) with semicircular ends (each with diameter $60$ m). Using $\pi \approx 3.14$ , find the total track area.
Answer: 8826
1. What is the area of the rectangular middle section? 6000
2. What is the radius of each semicircle? 30
3. What is the area of ONE semicircle? 1413
4. What is the total track area? (Two semicircles make a full circle) 8826