Area of a Sector of a Circle (Radians)
(KS3, Year 7)
The LessonThe 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.
QuestionWhat is the area of the sector with an angle of 2 radians and a radius of 5 cm, as shown below?
Start with the formula:
Area of sector = 1⁄2 r2θDon't forget: r2 = r × r (r squared).
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.
Lesson SlidesThe slider below shows another real example of how to find the area of a sector of a circle when the angle is in radians. Open the slider in a new tab
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 × πr2The πs cancel, leaving the simpler formula:
Area of sector = θ⁄2 × r2 = 1⁄2 r2θ
Is the Angle Given in Degrees or RadiansThe 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.
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