A circle has a radius of 9 inches. What is the area, in square inches, of a sector of this… — GRE General Test

A circle has a radius of $9$ inches. What is the area, in square inches, of a sector of this circle formed by a central angle of $40^\circ$ ?

The area of the sector is

9\pi

square inches.

The area of a sector with radius

r

and central angle

\theta

is calculated as

\text{Area} = \pi r^2 \left(\frac{\theta}{360^\circ}\right)

. Substituting

r = 9

and

\theta = 40^\circ

yields

\pi (9^2) \left(\frac{40^\circ}{360^\circ}\right) = 81\pi \left(\frac{1}{9}\right) = 9\pi

square inches.

Calculate the area of the entire circle using the formula

A_{\text{circle}} = \pi r^2

A_{\text{circle}} = \pi (9^2) = 81\pi

square inches.

The full circle area provides the total measure from which the sector fraction is calculated.

Determine the fraction of the circle represented by the central angle of

40^\circ

\frac{40^\circ}{360^\circ} = \frac{1}{9}

A circle contains

360^\circ

in total, so the central angle over

360^\circ

gives the proportion of the circle covered by the sector.

Multiply the total area of the circle by the fraction of the sector.

\text{Sector Area} = 81\pi \times \frac{1}{9} = 9\pi

square inches.

Applying the fraction to the total area yields the specific sector area.

Sector Area of a Circle

Question