Math worksheet for calculating the area and perimeter of six compound shapes, each with labeled dimensions.

Here are the step-by-step solutions for finding the area and perimeter of each compound shape.

1) Rectangle with a Quarter Circle on Top

Perimeter:
The perimeter is the distance around the outside of the shape.
* Bottom side: $22.2$ in
* Left side: $11.5$ in
* Right side: $10.1$ in
* Top curve: This is $\frac{1}{4}$ of a circle's circumference. The radius ($r$) is $12.1$ in.
* Circumference formula: $C = 2 \times \pi \times r$
* Full circumference: $2 \times 3.14 \times 12.1 = 75.988$ in
* Quarter arc: $75.988 \div 4 \approx 19.00$ in
* Total Perimeter: $22.2 + 11.5 + 10.1 + 19.00 = \mathbf{62.8 \text{ in}}$

Area:
Split the shape into a rectangle and a quarter circle.
* Rectangle Area: Width $\times$ Height = $22.2 \times 11.5 = 255.3$ sq in
* Quarter Circle Area: $\frac{\pi \times r^2}{4}$
* $r^2 = 12.1 \times 12.1 = 146.41$
* Area = $(3.14 \times 146.41) \div 4 \approx 114.93$ sq in
* Total Area: $255.3 + 114.93 = \mathbf{370.23 \text{ sq in}}$

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2) Rectangle with a Semicircle on Top

Perimeter:
* Bottom side: $20.2$ mm
* Left side: $3.1$ mm
* Right side: $3.1$ mm
* Top curve: This is half a circle. Diameter is $20.2$ mm, so radius ($r$) is $10.1$ mm.
* Half Circumference: $(\pi \times d) \div 2$ or $\pi \times r$
* Arc: $3.14 \times 10.1 \approx 31.71$ mm
* Total Perimeter: $20.2 + 3.1 + 3.1 + 31.71 = \mathbf{58.11 \text{ mm}}$

Area:
* Rectangle Area: $20.2 \times 3.1 = 62.62$ sq mm
* Semicircle Area: $\frac{\pi \times r^2}{2}$
* $r^2 = 10.1 \times 10.1 = 102.01$
* Area = $(3.14 \times 102.01) \div 2 \approx 160.16$ sq mm
* Total Area: $62.62 + 160.16 = \mathbf{222.78 \text{ sq mm}}$

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3) Rectangle with a Triangle Attached

Perimeter:
Add all outer sides:
* Left side: $11.1$ cm
* Top side: $11.1$ cm
* Bottom side: $11.1$ cm
* Two slanted triangle sides: $12.1$ cm each
* Total Perimeter: $11.1 + 11.1 + 11.1 + 12.1 + 12.1 = \mathbf{57.5 \text{ cm}}$

Area:
* Rectangle Area: $11.1 \times 11.1 = 123.21$ sq cm
* Triangle Area: $\frac{1}{2} \times \text{base} \times \text{height}$
* Base (vertical dotted line) = $11.1$ cm
* Height (horizontal dotted line) = $13.4$ cm
* Area = $0.5 \times 11.1 \times 13.4 = 74.37$ sq cm
* Total Area: $123.21 + 74.37 = \mathbf{197.58 \text{ sq cm}}$

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4) Large Rectangle with Two Triangles Cut Out

Perimeter:
Add all outer boundary lines:
* Top: $20.1$ ft
* Left vertical: $4.1$ ft
* Left slant: $5.0$ ft
* Right vertical: $4.1$ ft
* Right slant: $5.0$ ft
* Bottom middle segment: We calculate this by subtracting the triangle bases from the total width.
* Total width = $20.1$ ft.
* Left triangle base = $3.1$ ft. Right triangle base = $3.1$ ft.
* Middle bottom = $20.1 - 3.1 - 3.1 = 13.9$ ft.
* Total Perimeter: $20.1 + 4.1 + 5.0 + 13.9 + 5.0 + 4.1 = \mathbf{52.2 \text{ ft}}$

Area:
Calculate the area of the full large rectangle and subtract the two empty triangles.
* Full Rectangle Area: Width $20.1$ ft $\times$ Height $4.1$ ft = $82.41$ sq ft
* Two Triangles Area:
* One triangle area = $0.5 \times \text{base } 3.1 \times \text{height } 4.1 = 6.355$ sq ft
* Two triangles = $6.355 \times 2 = 12.71$ sq ft
* Total Area: $82.41 - 12.71 = \mathbf{69.7 \text{ sq ft}}$

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5) Trapezoid with a Triangle Hanging Below

Perimeter:
Add all outer sides:
* Top: $18.1$ km
* Left slant: $11.1$ km
* Right slant: $11.1$ km
* Bottom left slant: $12.1$ km
* Bottom right slant: $12.1$ km
* Total Perimeter: $18.1 + 11.1 + 11.1 + 12.1 + 12.1 = \mathbf{64.5 \text{ km}}$

Area:
Split into a top trapezoid and a bottom triangle.
* Top Trapezoid Area: $\frac{(a + b) \times h}{2}$
* Top base ($a$) = $18.1$ km
* Bottom base ($b$) = $9.5$ km (from diagram)
* Height ($h$) = $4.3$ km
* Area = $\frac{(18.1 + 9.5) \times 4.3}{2} = \frac{27.6 \times 4.3}{2} = 59.34$ sq km
* Bottom Triangle Area: $\frac{1}{2} \times \text{base} \times \text{height}$
* Base = $9.5$ km
* Height = $12.1$ km
* Area = $0.5 \times 9.5 \times 12.1 = 57.475$ sq km
* Total Area: $59.34 + 57.475 = \mathbf{116.815 \text{ sq km}}$

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6) Triangle, Rectangle, and Quarter Circle Combined

Perimeter:
Add the outer boundaries:
* Left slant: $11.0$ yd
* Top: $21.2$ yd
* Right vertical: $9.6$ yd
* Bottom curve: This is $\frac{1}{4}$ of a circle. Radius ($r$) is $10.1$ yd.
* Arc length = $\frac{2 \times \pi \times 10.1}{4} \approx 15.86$ yd
* Bottom straight segment: $10.1$ yd
* Total Perimeter: $11.0 + 21.2 + 9.6 + 15.86 + 10.1 = \mathbf{67.76 \text{ yd}}$

Area:
Split into three parts: Triangle, Rectangle, Quarter Circle.
* Triangle Area: $\frac{1}{2} \times \text{base} \times \text{height}$
* Base = $10.0$ yd, Height = $11.1$ yd
* Area = $0.5 \times 10.0 \times 11.1 = 55.5$ sq yd
* Rectangle Area: Width $\times$ Height
* Width = $21.2$ yd, Height = $9.6$ yd
* Area = $21.2 \times 9.6 = 203.52$ sq yd
* Quarter Circle Area: $\frac{\pi \times r^2}{4}$
* Radius = $10.1$ yd
* Area = $\frac{3.14 \times 10.1^2}{4} = \frac{3.14 \times 102.01}{4} \approx 80.08$ sq yd
* Total Area: $55.5 + 203.52 + 80.08 = \mathbf{339.1 \text{ sq yd}}$

Final Answer:
1) Area: 370.23 sq in, Perimeter: 62.8 in
2) Area: 222.78 sq mm, Perimeter: 58.11 mm
3) Area: 197.58 sq cm, Perimeter: 57.5 cm
4) Area: 69.7 sq ft, Perimeter: 52.2 ft
5) Area: 116.82 sq km, Perimeter: 64.5 km
6) Area: 339.1 sq yd, Perimeter: 67.76 yd