Question

Area of a Segment Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of $40°$.

Short answer

The area of the segment is 0.69 square inches.

Step-by-step solution

Step 1. Given information.

The radius of the circle is 5 inches.

The central angle is $40°$.

Step 2. Determine the area of the sector.

Area of the sector is:

$\begin{array}{rcl}{A}_1& =& \frac{\theta }{360°}\times \pi r^2\\ & =& \frac{40°}{360°}\times \pi \times (5{)}^2\\ & =& 8.7266...\\ & \approx & 8.727\end{array}$

Step 3. Determine the area of the triangle.

The area of a triangle is:

$\begin{array}{rcl}{A}_2& =& \frac{1}{2}ab\sin C\\ & =& \frac{1}{2}\left(5\right)\left(5\right)\sin (40°)\\ & =& 8.0348...\\ & \approx & 8.035\end{array}$

Step 4. Determine the area of the segment.

Subtract the area of the triangle from the area of the sector to obtain the area of the segment.

$\begin{array}{rcl}A& =& A_1-A_2\\ & =& 8.727-8.035\\ & =& 0.692\\ & \approx & 0.69\end{array}$

The area of the segment is 0.69 square inches.