Math worksheet for calculating area and perimeter of compound shapes with six problems.

To solve the problem of finding the area and perimeter of each compound shape, we need to break down each shape into simpler geometric figures (e.g., rectangles, triangles, circles, etc.) and then calculate their areas and perimeters step by step. Below is a detailed explanation for each shape:

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1) Shape 1:

#### Description:
- The shape consists of a rectangle with a semicircle on top.
- Dimensions:
- Rectangle: Length = 22.2 in, Width = 11.5 in
- Semicircle: Diameter = 18.1 in (Radius = 9.05 in)

#### Steps:
1. Calculate the area of the rectangle:
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 22.2 \times 11.5 = 255.3 \, \text{in}^2
\]

2. Calculate the area of the semicircle:
\[
\text{Area}_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (9.05)^2 = \frac{1}{2} \pi (81.9025) \approx \frac{1}{2} \times 3.1416 \times 81.9025 \approx 128.57 \, \text{in}^2
\]

3. Total area:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{semicircle}} = 255.3 + 128.57 \approx 383.87 \, \text{in}^2
\]

4. Calculate the perimeter:
- Perimeter of the rectangle (excluding the top side): \(2 \times \text{Width} + \text{Length} = 2 \times 11.5 + 22.2 = 45.2 \, \text{in}\)
- Circumference of the semicircle: \(\pi \times \text{Diameter} / 2 = \pi \times 18.1 / 2 \approx 28.42 \, \text{in}\)
- Total perimeter: \(45.2 + 28.42 \approx 73.62 \, \text{in}\)

#### Final Answer:
\[
\boxed{383.87 \, \text{in}^2, 73.62 \, \text{in}}
\]

---

2) Shape 2:

#### Description:
- The shape consists of a rectangle with a semicircle on top.
- Dimensions:
- Rectangle: Length = 20.2 mm, Width = 3.1 mm
- Semicircle: Diameter = 20.2 mm (Radius = 10.1 mm)

#### Steps:
1. Calculate the area of the rectangle:
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 20.2 \times 3.1 = 62.62 \, \text{mm}^2
\]

2. Calculate the area of the semicircle:
\[
\text{Area}_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (10.1)^2 = \frac{1}{2} \pi (102.01) \approx \frac{1}{2} \times 3.1416 \times 102.01 \approx 160.28 \, \text{mm}^2
\]

3. Total area:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{semicircle}} = 62.62 + 160.28 \approx 222.90 \, \text{mm}^2
\]

4. Calculate the perimeter:
- Perimeter of the rectangle (excluding the top side): \(2 \times \text{Width} + \text{Length} = 2 \times 3.1 + 20.2 = 26.4 \, \text{mm}\)
- Circumference of the semicircle: \(\pi \times \text{Diameter} / 2 = \pi \times 20.2 / 2 \approx 31.74 \, \text{mm}\)
- Total perimeter: \(26.4 + 31.74 \approx 58.14 \, \text{mm}\)

#### Final Answer:
\[
\boxed{222.90 \, \text{mm}^2, 58.14 \, \text{mm}}
\]

---

3) Shape 3:

#### Description:
- The shape consists of a rectangle with a right triangle on top.
- Dimensions:
- Rectangle: Length = 12.1 cm, Width = 11.1 cm
- Triangle: Base = 12.1 cm, Height = 1.1 cm

#### Steps:
1. Calculate the area of the rectangle:
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 12.1 \times 11.1 = 134.31 \, \text{cm}^2
\]

2. Calculate the area of the triangle:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 12.1 \times 1.1 = 6.655 \, \text{cm}^2
\]

3. Total area:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{triangle}} = 134.31 + 6.655 \approx 140.97 \, \text{cm}^2
\]

4. Calculate the perimeter:
- Perimeter of the rectangle (excluding the top side): \(2 \times \text{Width} + \text{Length} = 2 \times 11.1 + 12.1 = 34.3 \, \text{cm}\)
- Hypotenuse of the triangle: \(\sqrt{\text{Base}^2 + \text{Height}^2} = \sqrt{12.1^2 + 1.1^2} = \sqrt{146.41 + 1.21} = \sqrt{147.62} \approx 12.15 \, \text{cm}\)
- Total perimeter: \(34.3 + 12.1 + 1.1 \approx 47.5 \, \text{cm}\)

#### Final Answer:
\[
\boxed{140.97 \, \text{cm}^2, 47.5 \, \text{cm}}
\]

---

4) Shape 4:

#### Description:
- The shape is a trapezoid with two parallel sides and two non-parallel sides.
- Dimensions:
- Top base = 3.1 m
- Bottom base = 20.0 m
- Height = 3.0 m
- Non-parallel sides = 4.1 m each

#### Steps:
1. Calculate the area of the trapezoid:
\[
\text{Area}_{\text{trapezoid}} = \frac{1}{2} \times (\text{Top base} + \text{Bottom base}) \times \text{Height} = \frac{1}{2} \times (3.1 + 20.0) \times 3.0 = \frac{1}{2} \times 23.1 \times 3.0 = 34.65 \, \text{m}^2
\]

2. Calculate the perimeter:
- Perimeter = Top base + Bottom base + Two non-parallel sides
\[
\text{Perimeter} = 3.1 + 20.0 + 4.1 + 4.1 = 29.3 \, \text{m}
\]

#### Final Answer:
\[
\boxed{34.65 \, \text{m}^2, 29.3 \, \text{m}}
\]

---

5) Shape 5:

#### Description:
- The shape is a pentagon composed of a rectangle and two right triangles.
- Dimensions:
- Rectangle: Length = 18.1 km, Width = 9.3 km
- Triangles: Base = 4.3 km, Height = 11.1 km (each)

#### Steps:
1. Calculate the area of the rectangle:
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 18.1 \times 9.3 = 168.33 \, \text{km}^2
\]

2. Calculate the area of one triangle:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4.3 \times 11.1 = 23.865 \, \text{km}^2
\]
- Since there are two triangles:
\[
\text{Total Area}_{\text{triangles}} = 2 \times 23.865 = 47.73 \, \text{km}^2
\]

3. Total area:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Total Area}_{\text{triangles}} = 168.33 + 47.73 = 216.06 \, \text{km}^2
\]

4. Calculate the perimeter:
- Perimeter = Two sides of the rectangle + Two bases of the triangles + One top side of the rectangle
\[
\text{Perimeter} = 18.1 + 9.3 + 9.3 + 4.3 + 4.3 + 18.1 = 53.4 \, \text{km}
\]

#### Final Answer:
\[
\boxed{216.06 \, \text{km}^2, 53.4 \, \text{km}}
\]

---

6) Shape 6:

#### Description:
- The shape consists of a rectangle with a semicircle on one side.
- Dimensions:
- Rectangle: Length = 21.2 yd, Width = 10.1 yd
- Semicircle: Diameter = 10.1 yd (Radius = 5.05 yd)

#### Steps:
1. Calculate the area of the rectangle:
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 21.2 \times 10.1 = 214.12 \, \text{yd}^2
\]

2. Calculate the area of the semicircle:
\[
\text{Area}_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (5.05)^2 = \frac{1}{2} \pi (25.5025) \approx \frac{1}{2} \times 3.1416 \times 25.5025 \approx 40.02 \, \text{yd}^2
\]

3. Total area:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{semicircle}} = 214.12 + 40.02 \approx 254.14 \, \text{yd}^2
\]

4. Calculate the perimeter:
- Perimeter of the rectangle (excluding the side with the semicircle): \(2 \times \text{Width} + \text{Length} = 2 \times 10.1 + 21.2 = 41.4 \, \text{yd}\)
- Circumference of the semicircle: \(\pi \times \text{Diameter} / 2 = \pi \times 10.1 / 2 \approx 15.83 \, \text{yd}\)
- Total perimeter: \(41.4 + 15.83 \approx 57.23 \, \text{yd}\)

#### Final Answer:
\[
\boxed{254.14 \, \text{yd}^2, 57.23 \, \text{yd}}
\]

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Final Answers for All Shapes:

1. \(\boxed{383.87 \, \text{in}^2, 73.62 \, \text{in}}\)
2. \(\boxed{222.90 \, \text{mm}^2, 58.14 \, \text{mm}}\)
3. \(\boxed{140.97 \, \text{cm}^2, 47.5 \, \text{cm}}\)
4. \(\boxed{34.65 \, \text{m}^2, 29.3 \, \text{m}}\)
5. \(\boxed{216.06 \, \text{km}^2, 53.4 \, \text{km}}\)
6. \(\boxed{254.14 \, \text{yd}^2, 57.23 \, \text{yd}}\)