https://file.aiquickdraw.com/mathbot/file/1727404547115s5a3jcyl.jpg

The composite shape in the image consists of two distinct parts:

1. **A trapezoid** with the following dimensions:
   - Bottom base: 20 inches
   - Top base: 8 inches
   - Height: 8 inches
   - The slant height of each side of the trapezoid is 10 inches.

2. **A semicircle** placed on top of the trapezoid:
   - The diameter of the semicircle is 8 inches (since it matches the top base of the trapezoid).
   - Therefore, the radius of the semicircle is \( \frac{8}{2} = 4 \) inches.

### Step 1: **Perimeter Calculation**
The perimeter consists of:
- The perimeter of the trapezoid (two slanted sides and the bottom base).
- The perimeter of the semicircle (its curved part).

#### 1.1 **Trapezoid's perimeter**:
- Bottom base = 20 inches
- The two slanted sides = 10 inches each

The straight top edge is covered by the diameter of the semicircle, so we exclude it from the perimeter. The perimeter contribution from the trapezoid is:
\[
P_{\text{trapezoid}} = 20 + 10 + 10 = 40 \, \text{inches}
\]

#### 1.2 **Semicircle's perimeter**:
The curved part of the semicircle’s perimeter is half of the circumference of a full circle:
\[
C_{\text{semicircle}} = \frac{1}{2} \times 2 \pi r = \pi r = 3.14 \times 4 = 12.56 \, \text{inches}
\]

Thus, the total perimeter of the shape is:
\[
P_{\text{total}} = P_{\text{trapezoid}} + C_{\text{semicircle}} = 40 + 12.56 = 52.56 \, \text{inches}
\]

### Step 2: **Area Calculation**

The total area consists of the area of the trapezoid and the area of the semicircle.

#### 2.1 **Trapezoid's area**:
The area of a trapezoid is given by the formula:
\[
A_{\text{trapezoid}} = \frac{1}{2} \times (b_1 + b_2) \times h
\]
Where:
- \( b_1 = 20 \) inches (bottom base)
- \( b_2 = 8 \) inches (top base)
- \( h = 8 \) inches (height)

Substituting these values:
\[
A_{\text{trapezoid}} = \frac{1}{2} \times (20 + 8) \times 8 = \frac{1}{2} \times 28 \times 8 = 112 \, \text{square inches}
\]

#### 2.2 **Semicircle's area**:
The area of a semicircle is half the area of a full circle:
\[
A_{\text{semicircle}} = \frac{1}{2} \times \pi r^2 = \frac{1}{2} \times 3.14 \times 4^2 = \frac{1}{2} \times 3.14 \times 16 = 25.12 \, \text{square inches}
\]

#### 2.3 **Total area**:
The total area is the sum of the areas of the trapezoid and the semicircle:
\[
A_{\text{total}} = A_{\text{trapezoid}} + A_{\text{semicircle}} = 112 + 25.12 = 137.12 \, \text{square inches}
\]

### Final Results:
- **Perimeter**: 52.56 inches
- **Total area**: 137.12 square inches

Would you like further details or have any questions? Here are 5 related questions to expand your understanding:

1. How would the perimeter change if the radius of the semicircle doubled?
2. What is the formula for the perimeter of a full circle and how would it be applied here?
3. How do you calculate the area of a trapezoid if the bases are not parallel?
4. If the trapezoid had different slant heights, how would that affect the perimeter calculation?
5. What if the shape were a full circle instead of a semicircle—how would the area change?

**Tip**: When working with composite shapes, always calculate areas and perimeters for individual parts before combining them for accurate results.

Find the perimeter and total area of the composite shape consisting of a trapezoid and a semicircle. All measurements are given in inches. Use π = 3.14 in any formulas used.

This problem involves calculating the perimeter and area of a composite shape made of a trapezoid and a semicircle. Learn how to use key geometric formulas like the area of a trapezoid and semicircle, and the perimeter formula for composite shapes. Suitable for Grades 8-10.

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