Area of a Sector of a Circle (Radians)
(KS3, Year 7)
How to Find the Area of a Sector of a Circle (Radians)
Finding the area of a sector of a circle, when the angle is in radians, is easy.Question
What is the area of the sector with an angle of 2 radians and a radius of 5 cm, as shown below?Step-by-Step:
1
Start with the formula:
Area of sector = 1⁄2 r2θ
Don't forget: r2 = r × r (r squared).
2
Substitute the angle and the radius into the formula. In our example, θ = 2 and r = 5.
Area of sector = 1⁄2 × 52 × 2
Area of sector = 1⁄2 × 5 × 5 × 2
Area of sector = 25 cm2
Answer:
The area of a sector of a circle with a radius of 5 cm, with an angle of 2 radians, is 25 cm2.What Is a Sector?
A sector is a region of a circle bounded by two radii and the arc lying between the radii.What Are Radians?
Radians are a way of measuring angles. 1 radian is the angle found when the radius is wrapped around the circle.Why Does the Formula Work?
The area of a sector is just a fraction of the area of the circle of the same radius. The area is given by πr2, where r is the radius. For example, a sector that is half of a circle is half of the area of a circle. A sector that is quarter of a circle has a quarter of the area of a circle. In each case, the fraction is the angle of the sector divided by the full angle of the circle. When measured in radians, the full angle is 2π. Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 2π multiplied by the area of the circle:
Area of sector = θ⁄2π × πr2
The πs cancel, leaving the simpler formula:
Area of sector = θ⁄2 × r2 = 1⁄2 r2θ
Beware
Is the Angle Given in Degrees or Radians
The formula to find the length of a sector of a circle depends on whether the angle at the center of the sector is given in degrees or radians. Make sure you check what units the angle is given in.Worksheet
This test is printable and sendable