What is the area of the shaded segment if the central angle is 50° and the radiu...

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nezuko89 @nezuko89

October 2022 1 6 Report

What is the area of the shaded segment if the central angle is 50° and the radius of the circle is 4cm?

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UK2019

Verified answer

✏️SEGMENT

===============================

Question: What is the area of the shaded segment if the central angle is 50° and the radius of the circle is 4cm?

Answer: $\Large \underline{ \rm \green{ \: \: 5.44 \: \: {cm}^{2} \: }}$

Solution: To find the area of the segment, we will subtract the area of the sector to the area of the isosceles triangle that is bounded by the two radii and a chord.

Find the Area of a Sector.

$\begin{aligned}& \bold{ \color{lightblue}Formula:} \\& \boxed{ A_{sec} = \frac{ \theta}{360 \degree} \cdot\pi {r}^{2} } \end{aligned}$

$\begin{aligned}{ A_{sec} = \frac{50 \degree}{360 \degree} \cdot\pi (4cm)^{2} } \end{aligned}$

$\begin{aligned}{ A_{sec} = \frac{5}{36} \cdot\pi (16cm^{2}) } \end{aligned}$

$\begin{aligned}{ A_{sec} = \frac{\pi(80 {cm}^{2} )}{36} } \end{aligned}$

$\begin{aligned}{ A_{sec} = \pi(2.22 {cm})^{2} } \end{aligned}$

- Let 3.14 be the approximate value of pi.

$\begin{aligned}{ A_{sec} ≈ (3.14)(2.22 {cm})^{2} } \end{aligned}$

$\begin{aligned}{ A_{sec} ≈ 6.97 {cm}^{2} } \end{aligned}$

$\large\sf$

Find the Area of a Triangle.

$\begin{aligned}& \bold{ \color{lightblue}Formula:} \\& \boxed{ A_{\Delta} = \frac{1}{2} \cdot {r}^{2} \sin( \theta) } \end{aligned}$

$\begin{aligned}{ A_{\Delta} = \frac{1}{2} \cdot (4cm^{2}) \sin(50 \degree) } \end{aligned}$

$\begin{aligned}{ A_{\Delta} = \frac{1}{2} \cdot (4cm^{2})(0.766) } \end{aligned}$

$\begin{aligned}{ A_{\Delta} = \frac{1}{2} \cdot (3.064cm^{2}) } \end{aligned}$

$\begin{aligned}{ A_{\Delta} = \frac{3.064cm^{2} }{2} } \end{aligned}$

$\begin{aligned}{ A_{\Delta} = 1.53 {cm}^{2} } \end{aligned}$

$\large\sf$

Find the Area of a Segment.

- Simply subtract the area of the sector and triangle.

$A_{seg} = A_{sec} - A_{\Delta}$

$A_{seg} = 6.97 {cm}^{2} - 1.53 {cm}^{2}$

$A_{seg} = 5.44 {cm}^{2}$

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Questions

Answers & Comments

Verified answer

✏️SEGMENT

===============================

Find the Area of a Sector.

Find the Area of a Triangle.

Find the Area of a Segment.

===============================

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