Area of a Sector
Simple Explanation
A sector is the pie-slice-shaped region between two radii and the arc connecting them. Its area is A = (1/2)r²θ, where θ is the central angle in radians.
Why Do We Need It?
This lets you find the area of any "slice" of a circle — used for anything from pizza slices to pie charts to the area swept by a rotating machine part.
See It
A shaded pie-slice-shaped sector of a circle with a 60-degree central angle
Formula
Area of a Sector
A = (1/2) r² θ (θ in radians)
The area of a pie-slice-shaped sector equals half the radius squared times the central angle, when that angle is in radians.
- A
- — the area of the sector
- r
- — the radius of the circle
- θ
- — the central angle of the sector, in radians
When to use it: Whenever you need the area of a pie-slice-shaped portion of a circle, given its radius and central angle.
Worked Example
Find the area of a sector
A circle has radius 10 cm. Find the area of a sector with central angle π/3 radians.
Why Does This Work?
A full circle (θ = 2π) has area πr². A sector with angle θ is exactly the fraction θ/(2π) of the full circle, so its area is (θ/2π) × πr² = (1/2)r²θ — the π cancels neatly, leaving this simplified formula.
Real-Life Example
Calculating a pie chart wedge's area
A designer needs the exact area of one wedge of a circular pie chart, given the chart's radius and the wedge's angle.
The sector area formula gives that wedge's exact area directly from its central angle, without needing to measure it by hand.
Practice
A circle has radius 4. Find the area of a sector with central angle 2 radians.
MediumCommon mistake
Forgetting the (1/2) factor, or using the diameter instead of the radius — the formula specifically uses r² (radius squared), not the diameter.
Quick Review
- A = (1/2)r²θ, with θ in radians.
- A sector is the fraction θ/(2π) of the full circle's area, πr².
- Always use the radius, not the diameter.