Question

If two sectors have an area of 12π square units, and one sector is from a circle with radius 6 units, while the other is from a circle with a radius of 4 units, how do the central angles for these sectors compare?

Answer 1

4 : 9

Explanation:

Given:

Two sectors, each has an area of 12pi, but with radii r1=6 and r2=4 units.

Find ratio of central angles.

Solution:

Let A = central angle

Area of a sector = pi r^2 (A/360)

Since both sectors have the same area,

pi r1^2 (A1/360) = pi r2^2 (A2/360)

simplifying

A1 r1^2 = A2 r2^2

Therefore

A1 : A2 = r2^2 : r1^2 = 4^2 : 6^2 = 4 : 9

Answer 2

Thus, the ratio of the central angles is 4 : 9.

Explanation:

Area, A = 12 π square units

radius, R = 6 units

radius, r = 4 units

Area of sector is given by

`A=(\theta )/(360)* \pi r^(2)`

For first sector

`12\pi=(\theta )/(360)* \pi * 6^(2)\\\\\theta = 120^(o)`

For second sector

`12\pi=(\theta' )/(360)* \pi * 4^(2)\\\\\theta' = 270^(o)`

So, the ratio is

`(\theta)/(\theta')=(120)/(270) =4 : 9`