Question

The radius of a circle is 4 feet. What is the area of a sector bounded by a 45° arc?

Answer 1

Since the radius of this circle is 4 feet, the area of a sector bounded by a 45° arc is 6.28 square feet.

In Mathematics and Geometry, the area of a sector can be calculated by using the following formula:

Area of sector = π
`r^2` × θ/360

Where:

• r represents the radius of a circle.
• θ represents the central angle.

Note: The measure of an intercepted arc is equal to the central angle of a circle.

By substituting the given parameters into the area of a sector formula, we have the following;

Area of sector = π
`r^2` × θ/360

Area of sector = 3.14 ×
`4^2` × 45/360

Area of sector = 3.14 × 16 × 1/8

Area of sector = 3.14 × 2

Area of sector = 6.28 square feet.

Answer 2

`Area = 6.28\ ft^2` or
`Area = 2 \pi\ ft^2`

Explanation:

Given

`Radius, r = 4ft`

`Angle, \theta = 45`

Required

Calculate the area of the sector

When angle is given in degrees, the area of sector is calculated as thus;

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

`Substitute\ r = 4\ and\ \theta = 45`

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

`Area = (45)/(360) * \pi * 4^2`

`Area = (45)/(360) * \pi * 16`

`Area = (45*16)/(360) * \pi`

`Area = (720)/(360) * \pi`

`Area = 2 * \pi`

`Area = 2 \pi\ ft^2`

The above is the area in terms of π

Solving further.... (Take π as 3.14)

`Area = 2 \pi\ ft^2` becomes

`Area = 2 * 3.14\ ft^2`

`Area = 6.28\ ft^2`

Hence, the area of the sector is

`Area = 6.28\ ft^2` or
`Area = 2 \pi\ ft^2`