Calculate Trapezoid Area with 7π Segment and Semicircle: Step-by-Step Solution

To solve the problem of finding the area of the trapezoid with a semicircle on its top base, we follow these steps:

• Step 1: Identify the semicircle's radius from the given circumference.
• Step 2: Find the upper base length, which is the diameter of the semicircle.
• Step 3: Calculate the trapezoid's area using the area formula.

Let's work through each step:

Step 1: The given length of the highlighted segment is $7\pi$, which is the half-circumference of a circle (since it's a semicircle). The formula for the circumference of a full circle is $2\pi r$, so for a semicircle, it is $\pi r$. Setting this equal to the length given:

$\pi r = 7\pi$

Canceling $\pi$ from both sides, we find:

$r = 7$

Step 2: The diameter of the semicircle is twice the radius, hence:

$\text{Diameter} = 2 \times 7 = 14 \, \text{cm}$

This diameter also serves as the length of the upper base of the trapezoid.

Step 3: We use the formula for the area of a trapezoid:

$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$

Substitute the known values ($\text{Base}_1 = 14$, $\text{Base}_2 = 18$, $\text{Height} = 7$):

$\text{Area} = \frac{1}{2} \times (14 + 18) \times 7$ $\text{Area} = \frac{1}{2} \times 32 \times 7$ $\text{Area} = 16 \times 7$ $\text{Area} = 112 \, \text{cm}^2$

Thus, the area of the trapezoid is 112 cm$^2$.