Question

if $r = 18$ cm, what is the area of this sector? express your answer to the nearest tenth of a centimeter.
image of a semicircular sector with radius r
\bigcirc $686.3$ cm$^2$
\bigcirc $834.1$ cm$^2$
\bigcirc $424.2$ cm$^2$
\bigcirc $508.9$ cm$^2$

Explanation:

Step1: Identify the sector type

The sector is a semicircle (since the diagram shows a half - circle), so the central angle \(\theta = 180^{\circ}\) or \(\pi\) radians. The formula for the area of a sector is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\) (when \(\theta\) is in degrees) or \(A = \frac{1}{2}r^{2}\theta\) (when \(\theta\) is in radians). For a semicircle, \(\frac{\theta}{360^{\circ}}=\frac{1}{2}\), so the area of the sector (semicircle) is \(A=\frac{1}{2}\pi r^{2}\).

Step2: Substitute the value of \(r\)

Given \(r = 18\space cm\), substitute into the formula:
\(A=\frac{1}{2}\times\pi\times(18)^{2}\)
First, calculate \((18)^{2}=324\). Then, \(\frac{1}{2}\times\pi\times324 = 162\pi\).

Step3: Calculate the numerical value

Using \(\pi\approx3.14159\), we have \(162\times3.14159\approx162\times3.1416\).
\(162\times3.1416=(160 + 2)\times3.1416=160\times3.1416+2\times3.1416=502.656+6.2832 = 508.9392\space cm^{2}\)
Rounding to the nearest tenth, we get \(508.9\space cm^{2}\).

Answer:

\(508.9\space cm^{2}\)