Area of a Sector
Area of a circular sector (θ in radians).
Formula
Variables
Example
Radius 6, angle 2 rad: A = ½·6²·2 = 36 square units.
How it works
A sector is the slice of a circle cut out by two radii and the arc between them. This formula works because the sector’s area is the same fraction of the whole circle as its angle is of a full turn: since a full circle has angle 2 0 radians and area 0r 2, the sector area is ( 0/ 0r 2) times the circle’s area, which simplifies to 12r 2 0. The angle must be in radians for this version of the formula to be correct.
When to use it
Use this when finding the area of a wedge-shaped part of a circle, such as in pie charts, circular segments in design, or geometry problems involving parts of a wheel or pie. It is especially useful whenever the angle is already given in radians.
Worked example, step by step
Radius = 8, angle = 0.75 rad A = 1/2 × r² × 0 A = 1/2 × 8² × 0.75 A = 1/2 × 64 × 0.75 A = 32 × 0.75 A = 24 square units
Common mistakes to avoid
- Using degrees instead of radians without converting first
- Forgetting to square the radius
- Multiplying by the full circle formula instead of taking the correct fraction of it
Did You Know?
A pizza slice is a sector — its area is just a fraction of the whole pie's πr².
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