Question

Area of a sector A sector with a radius of \maroonD{8\,\text{cm}}8cmstart color #ca337c, 8, start text, c, m, end text, end color #ca337c has an area of \goldE{56\pi\,\text{cm}^2}56πcm

Answer 1

To find the angle of the sector, follow the steps below.

Step 01: Find the total area of the circle.

The area (A) of a circle with radius r is:

`A=\pi r^2`

Knowing that r = 8 cm, then the area is:

`\begin{gathered} A=8^2\pi \\ A=64\pi\text{ cm}^2 \end{gathered}`

Step 02: Find the central angle.

To find the angle, use proportions.

Knowing that:

When angle = 2π, A = 64π,

Then when angle is x, A = 56π

`\begin{gathered} (x)/(2\pi)=(56\pi)/(64\pi) \\ \\ \text{ Multiplying both sides by 2}\pi: \\ (x)/(2\pi)*2\pi=(56\pi)/(64\pi)*2\pi \\ x=(56*2)/(64)\pi \\ x=(112)/(64)\pi \\ \\ \text{ Dividing both the numerator and the denominator by 16:} \\ x=((112)/(16))/((64)/(16))\pi \\ x=(7\pi)/(4) \end{gathered}`

Answer: The central angle measure is:

`(7\pi)/(4)`