Question:
A circle has a sector with area 24π/5 and central angle of 48°.
What is the area of the circle?
Answer:
The area of the circle is 36π
Step-by-step explanation:
Given
Area of sector = 24π/5
Central angle, θ = 48°
Required:
Area of the circle
First we need to calculate radius of the circle.
This can be solved from area of a sector
Area of a sector = θ * [tex]\frac{\pi r^2}{360}[/tex]
By substituting 24π/5 for area and 48° for θ, we have
[tex]\frac{24 * \pi}{5} = \frac{\pi * r^2 * 48}{360}[/tex]
Multiply both sides by [tex]\frac{360}{48}[/tex]
[tex]\frac{24 * \pi}{5} * \frac{360}{48} = \frac{\pi * r^2 * 48}{360} \frac{360}{48}[/tex]
[tex]\frac{24 * \pi}{5} * \frac{360}{48} = \pi * r^2[/tex]
[tex]\frac{8640 \pi}{240} = \pi r^2[/tex]
36π = πr²
Divide both sides by π
[tex]\frac{36 \pi}{\pi} = \frac{\pi r^2}{\pi}[/tex]
36 = r²
Take square root of both sides
√36 = √r²
6 = r
Hence, radius = 6
Then, the area of the circle can now be calculated.
Area = πr²
Substitute 6 for r
Area = π * 6²
Area = π * 36
Area = 36π
Hence, the area of the circle is 36π
