Question

Ind the areas of the sectors formed by \angle ACB .

A circle is shown. The measure of central angle A C B is 131 degrees. The radius is 3 centimeters. Point D is on the circle but not on arc A B.

Give the exact answers in terms of \pi . Do not approximate the answers.

Area of small sector =
cm2

Area of large sector =
cm2

Answer 1

The area of the small sector is (131/360) * pi * 3^2 = 3.422 cm^2 (rounded to 3 decimal places).

To find the area of the large sector, we need to subtract the area of triangle ADB from the area of the circle sector ACB. We can use the Law of Cosines to find the length of segment AD:

AD^2 = AB^2 + BD^2 - 2 * AB * BD * cos(ACB)
AD^2 = 3^2 + 2^2 - 2 * 3 * 2 * cos(131)
AD^2 = 13 - 12cos(131)
AD = sqrt(13 - 12cos(131))

The area of triangle ADB is (1/2) * AB * AD * sin(DAB) = (1/2) * 3 * sqrt(13 - 12cos(131)) * sin(49) = 1.786 cm^2 (rounded to 3 decimal places).

The area of the large sector is pi * 3^2 - 1.786 = 23.556 cm^2 (rounded to 3 decimal places).

Therefore, the area of the small sector is 3.422 cm^2 and the area of the large sector is 23.556 cm^2.