Question

The diameter of a circle is 10m. What is the angle measure of an arc bounding a sector area 5pi square meters?

Answer 1

Angle measure of an arc is 72 °.

Explanation:

Given : The diameter of a circle is 10m and sector area 5pi square meters.

To find : What is the angle measure of an arc .

Solution : We have given that Diameter = 10 cm .

Radius =
`(10)/(2)` = 5 cm.

Area of sector =
`(theta)/(360) *pi (r^(2) )`.

Plugging the values of r = 5cm , Area of sector = 5 pi.

5 pi =
`(theta)/(360) *pi (5^(2) )`.

5 pi =
`(theta)/(360) *pi (25 )`.

On dividing by 25 pi

`(5\ pi)/(25\ pi)` =
`(theta)/(360))`.

`(1)/(5)` =
`(theta)/(360))`.

On multiplying both sides by 360 and swtiching sides.

Theta =
`(360)/(5)`.

Theta = 72 °

Therefore, angle measure of an arc is 72 °.

Answer 2

The area of the complete circle is:

`A = pi * r ^ 2`
Where,
r: radius of the circle.
Substituting values we have:

`A = \pi * (10/2) ^ 2 A = \pi * (5) ^ 2 A = 25 \pi`
Then, the measure of the angle of the arc whose area is 5pi is given by:

`theta = A '/ A * (360)`
Where,
A '/ A: ratio of areas
Substituting values:

`theta = (5 \pi / 25 \pi ) * (360) theta = (5/25) * (360) theta = (1/5) * (360)  theta = 72 degrees`
Answer:
the angle measure of an arc bounding to sector area 5pi square meters is:
theta = 72 degrees