What is the formula for sector area?

When the angle is measured in radians, the sector area formula is A = 1/2 r²θ. If the angle is in degrees, convert it to radians first or use the fraction-of-a-circle method.

How is sector area different from arc length?

Sector area measures the two-dimensional region inside the slice. Arc length measures only the curved edge of that slice. The two values use related formulas but represent different things.

Can I use degrees directly in 1/2 r²θ?

Not unless you convert the degree measure to radians first. The formula A = 1/2 r²θ assumes θ is in radians. This calculator accepts degrees but converts them internally.

What is sector perimeter?

Sector perimeter is the full boundary length of the slice. It equals two radii plus the arc length, so the formula is 2r + rθ when θ is in radians.

Education

Sector Area Calculator

Calculate sector area from radius and central angle, with arc length, sector perimeter, degree/radian conversion, and examples. This page is written for students who need both a fast answer and a clear explanation of the method, so it includes the formula, steps, examples, common mistakes, and practical notes.

What sector area measures

A sector is the “slice” of a circle enclosed by two radii and the arc between them. When the angle is in radians, the clean formula is A = 1/2 r²θ. The OpenStax sector area formula discussion shows this same sector-area relationship in a radius-and-angle setting, which is why the calculator converts degree input to radians before using the formula.

Formula and example

Sector area = 1/2 r²θ Arc length = rθ Sector perimeter = 2r + rθ Example: r = 8, θ = 60° = π/3 Area = 1/2 × 8² × π/3 ≈ 33.51

The area of a sector depends on both the radius and the angle. Doubling the radius has a much larger effect than doubling the angle because radius is squared.

Common questions

When the angle is measured in radians, the sector area formula is A = 1/2 r²θ. If the angle is in degrees, convert it to radians first or use the fraction-of-a-circle method.
Sector area measures the two-dimensional region inside the slice. Arc length measures only the curved edge of that slice. The two values use related formulas but represent different things.
Not unless you convert the degree measure to radians first. The formula A = 1/2 r²θ assumes θ is in radians. This calculator accepts degrees but converts them internally.
Sector perimeter is the full boundary length of the slice. It equals two radii plus the arc length, so the formula is 2r + rθ when θ is in radians.