Question

finding composite area
the window shown is the shape of a semi - circle with a radius of 6 feet. the distance from f to e is 3 feet and the measure of arc ( widehat{ic}=45^{circ}). find the area of the glass in region bcih, rounded to the nearest square foot.
( \text{ft}^2)

Explanation:

Step1: Find the radius of the larger and smaller semi - circles

The radius of the larger semi - circle $R = 6$ feet. The radius of the smaller semi - circle $r=6 - 3=3$ feet.

Step2: Use the formula for the area of a sector

The formula for the area of a sector of a circle with radius $R$ and central angle $\theta$ (in degrees) is $A=\frac{\theta}{360}\times\pi R^{2}$.
For the larger semi - circle sector with $\theta = 45^{\circ}$ and $R = 6$ feet, $A_1=\frac{45}{360}\times\pi\times6^{2}=\frac{1}{8}\times\pi\times36=\frac{9\pi}{2}$ square feet.
For the smaller semi - circle sector with $\theta = 45^{\circ}$ and $r = 3$ feet, $A_2=\frac{45}{360}\times\pi\times3^{2}=\frac{1}{8}\times\pi\times9=\frac{9\pi}{8}$ square feet.

Step3: Calculate the area of the region $BCHI$

The area of the region $BCHI$ is $A = A_1 - A_2$.
$A=\frac{9\pi}{2}-\frac{9\pi}{8}=\frac{36\pi - 9\pi}{8}=\frac{27\pi}{8}\approx\frac{27\times3.14}{8}=\frac{84.78}{8}=10.5975\approx11$ square feet.

Answer:

$11$