Question

Given that $∆OAB$is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short answer

The area is$17.22$square units.

Step-by-step solution

Step 1. Given Information

The given figure is

Step 2. Find the Angles of Triangle

• The triangle is an isosceles triangle.
• Sum of the angles of a triangle is $180°$.
• So,

$\begin{array}{rcl}40°+x+x& =& 180°\\ 2x& =& 140°\\ x& =& 70°\end{array}$

Step 3. Use rule of Sines

• Use the rule of sines to find the missing side, which is also the diameter of the semicircle.

$\begin{array}{rcl}\frac{\sin 40}{a}& =& \frac{\sin 70}{5}\\ a& =& 5\frac{\sin 40}{\sin 70}\\ & =& 5\times 0.68\\ & =& 3.42\end{array}$

Step 4. Find the total area

• The total area of the figure is the sum of the area of the triangle and the semicircle.
• To find the area of the triangle, find the semiperimeter.

$s=\frac{3.42+5+5}{2}=6.71$

• So, the area of the triangle is calculated below.

$\begin{array}{rcl}{A}_T& =& \sqrt{6.71\times 1.71\times 1.71\times 3.29}\\ & =& 8.03\end{array}$

• And the area of the semicircle is calculated as shown below:

$$\begin{gathered} A_C=\mathrm{\pi }{\left(\frac{3.42}{2}\right)}^2 \\ =9.19 \end{gathered}$$

• So, the total area is$8.03+9.19=17.22$square units.