Question 196·Medium·Geometry and Trigonometry

First, find the area of the entire circle using $A = \pi r^2$ with $r=8$. Keep your answer in terms of $\pi$.

Once you know the fraction $\dfrac{135}{360}$ and the full area of the circle, multiply them to get the area of the sector. Simplify the fraction before multiplying to make the arithmetic easier.

In Desmos, type (135/360)*pi*8^2 and press Enter to get a decimal value for the sector area. Then either divide that result by pi (type your result /pi) to see the coefficient of $\pi$, or separately evaluate each option (e.g., 18*pi, 24*pi, etc.) and compare the decimal values to see which answer choice matches your computed sector area.

A sector is a portion of a circle. Its area is given by

Here, the central angle is $135^\circ$ and the radius is $8$.

Compute what fraction of a full circle $135^\circ$ represents:

Simplify this fraction by dividing numerator and denominator by $45$:

So the sector is $\dfrac{3}{8}$ of the entire circle.

Use the circle area formula $A = \pi r^2$ with $r = 8$:

So the entire circle has area $64\pi$ square units.

The sector area is $\dfrac{3}{8}$ of $64\pi$:

First, multiply $\dfrac{3}{8}$ by $64$:

So the sector area is

square units, which corresponds to choice B) $24\pi$.