The Lesson
The area of a sector of a circle is given by the formula: In this formula, r is the radius of the circle and θ is the angle (in radians) of the sector. The image below shows what we mean by the area of a sector: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