Question

A triangle is placed in a semicircle with a radius of , as shown below. Find the area of the shaded region.Use the value for , and do not round your answer. Be sure to include the correct unit in your answer.

Answer 1

Given the figure below:

The area of the shaded region is expressed as

`area\text{ of shaded region = area of semicircle - area of triangle}`

step 1: Evaluate the area of the semicircle.

The area of the semicircle is expressed as

`\begin{gathered} area\text{ of semicircle=}(1)/(2)*\pi r^2 \\ where\text{ r is the radius of the circle} \end{gathered}`

Thus, we have

`\begin{gathered} area\text{ of semicircle = }(1)/(2)*3.14*4cm*4cm \\ \Rightarrow area\text{ of semicircle =25.12 cm}^2 \end{gathered}`

step 2: Evaluate the area of the triangle.

The area of the triangle is expressed as

`\begin{gathered} area\text{ of triangle =}(1)/(2)* base* height \\ thus,\text{ we have} \\ area\text{ of triangle =}(1)/(2)*8cm*4cm \\ =16\text{ cm}^2 \end{gathered}`

step 3: Evaluate the area of the shaded region.

Recall that

`\begin{gathered} area\text{ of shaded reg}\imaginaryI\text{on = area of sem}\imaginaryI\text{c}\imaginaryI\text{rcle- area of tr}\imaginaryI\text{angle} \\ \end{gathered}`

Thus, we have

`\begin{gathered} area\text{ of shaded region = \lparen25.12 -16\rparen cm}^2 \\ =9.12\text{ cm}^2 \end{gathered}`

Hence, the area of the shaded region is

`9.12\text{ cm}^2`