Question

Suppose a sector of a circle with radius r has a central angle of θ. Since a sector is a fraction of a full circle, the ratio of a sector's area A to the circle's area is equal to the ratio of the BLANK to the measure of a full rotation of the circle. A full rotation of a circle is 2π radians. This proportion can be written as Aπr2=BLANK. Multiply both sides by πr2 and simplify to get BLANK, where θ is the measure of the central angle of the sector and r is the radius of the circle.

Answer 1

Answer:

Central angle of θ

$\dfrac{A}{\pi r^2} =\dfrac{\theta}{2\pi}$

$A=\dfrac{\theta r^2}{2}, (\theta in radians)$

Step-by-step explanation:

Suppose a sector of a circle with radius r has a central angle of θ.

Since a sector is a fraction of a full circle, the ratio of a sector's area A to the circle's area is equal to the ratio of the central angle of θ to the measure of a full rotation of the circle.

A full rotation of a circle is 2π radians.

This proportion can be written as:  $\dfrac{A}{\pi r^2} =\dfrac{\theta}{2\pi}$

Multiply both sides by $\pi r^2$

$\dfrac{A}{\pi r^2} *\pi r^2=\dfrac{\theta}{2\pi}*\pi r^2$

Simplify to get:

$A=\dfrac{\theta r^2}{2}$

Where θ is the measure of the central angle of the sector and r is the radius of the circle.