Find the perimeter and total area of the composite shape shown below. All measurements are given in inches. Use pi 3.14

Question

in any formulas used.

An image shows a trapezoid and a semicircle connected together on the shorter base side of the trapezoid. The trapezoid has bases of 8 inches and 20 inches and side lengths of 10 inches and 10 inches with the height of trapezoid being 8 inches. The diameter of the semicircle is the same as the length of the shorter base side of the trapezoid of 8 inches.

Answer 1

To find the perimeter and total area of the composite shape, we will break the problem down into parts: the trapezoid and the semicircle.

Step 1: Identify the dimensions

• Trapezoid:

  • Longer base \( b_1 = 20 \) inches
  • Shorter base \( b_2 = 8 \) inches
  • Height \( h = 8 \) inches
  • Side lengths \( s_1 = 10 \) inches and \( s_2 = 10 \) inches (the non-parallel sides)

• Semicircle:

  • Diameter \( d = 8 \) inches (same as the shorter base of the trapezoid)
  • Radius \( r = \frac{d}{2} = \frac{8}{2} = 4 \) inches

Step 2: Calculate the area of the trapezoid

The area \( A_t \) of a trapezoid is given by the formula: \[ A_t = \frac{1}{2} \times (b_1 + b_2) \times h \]

Substituting the values: \[ A_t = \frac{1}{2} \times (20 + 8) \times 8 = \frac{1}{2} \times 28 \times 8 = 112 \text{ square inches} \]

Step 3: Calculate the area of the semicircle

The area \( A_s \) of a semicircle is given by the formula: \[ A_s = \frac{1}{2} \pi r^2 \]

Substituting the value for \( r \) and using \( \pi \approx 3.14 \): \[ A_s = \frac{1}{2} \times 3.14 \times (4^2) = \frac{1}{2} \times 3.14 \times 16 = \frac{1}{2} \times 50.24 = 25.12 \text{ square inches} \]

Step 4: Calculate the total area

To find the total area \( A \) of the composite shape: \[ A = A_t + A_s = 112 + 25.12 = 137.12 \text{ square inches} \]

Step 5: Calculate the perimeter

The perimeter \( P \) of the composite shape comprises:

• The lengths of the two bases of the trapezoid (20 inches and 8 inches)
• The lengths of the two non-parallel sides of the trapezoid (10 inches each)
• The curved part of the semicircle, which is half the circumference of a circle.

The formula for the circumference \( C \) of a full circle is: \[ C = \pi d \] The curved part of the semicircle is: \[ \text{Curved Part} = \frac{1}{2} \pi d = \frac{1}{2} \times 3.14 \times 8 = 12.56 \text{ inches} \]

Adding these up: \[ P = b_1 + b_2 + s_1 + s_2 + \text{Curved Part} \] Substituting the values: \[ P = 20 + 8 + 10 + 10 + 12.56 = 60.56 \text{ inches} \]

Final Results

• Total Area: \( 137.12 \) square inches
• Perimeter: \( 60.56 \) inches