Question

derive the area of a sector formula. move the answers to the boxes to complete the sentences. the ratio of the area of a circle to the radian measure of the circle being proportional to the area of the sector and the radian measure of the sector

we can write this using symbols as

rewrite the equation to isolate ( a ) to have

simplify the equation to get

Explanation:

Step1: Substitute circle area and radians of circle

The area of a circle is \(A_{circle}=\pi r^{2}\), and the radian measure of a full - circle is \(2\pi\). Let the area of the sector be \(A\) and the radian measure of the sector be \(\theta\). Substituting into \(\frac{\text{circle area}}{\text{radians of circle}}=\frac{\text{sector area}}{\text{radians of sector}}\), we get \(\frac{\pi r^{2}}{2\pi}=\frac{A}{\theta}\)

Step2: Cross - multiply

Cross - multiplying the equation \(\frac{\pi r^{2}}{2\pi}=\frac{A}{\theta}\) gives \(\pi r^{2}\theta = 2\pi A\)

Step3: Solve for \(A\)

Divide both sides of the equation \(\pi r^{2}\theta=2\pi A\) by \(2\pi\). We have \(A = \frac{1}{2}r^{2}\theta\)

Answer:

First box: \(\frac{\pi r^{2}}{2\pi}=\frac{A}{\theta}\)
Second box: \(\pi r^{2}\theta = 2\pi A\)
Third box: \(A=\frac{1}{2}r^{2}\theta\)