Skip to content
Lizely

Sector Area Calculator

Calculate circular sector area, arc length, and circle share from a radius and central angle in degrees or radians.

Privacy: your files never leave your device. All processing happens locally in your browser.

How to use

  1. 1.Enter the positive circle radius using one consistent length unit.
  2. 2.Enter the central angle and choose whether it is expressed in degrees or radians.
  3. 3.Select Calculate sector, then read the area, arc length, and percentage of the full circle.

About Sector Area Calculator

Sector Area Calculator finds the area of a circular sector from its radius and central angle. A sector is the region bounded by two radii and the arc between them, often compared to a slice of a pie. Enter the radius, enter the angle, choose degrees or radians, and the page reports the sector area, the corresponding arc length, and the percentage of the full circle.

The core formula is A = ½θr² when θ is measured in radians. If the input is in degrees, the calculator first converts it with θ radians = θ degrees × π/180. Arc length uses s = θr, and the circle share is θ/(2π). These relationships come from the fraction of a full 2π-radian rotation covered by the central angle.

For example, OpenStax considers a sprinkler with radius 20 feet and a 30-degree sweep. Thirty degrees is π/6 radians, so the area is ½ × π/6 × 20² = 100π/3, approximately 104.72 square feet. Entering radius 20 and angle 30 in degree mode reproduces that result and also reports an arc length of 10π/3.

The calculator accepts any consistent radius unit. If the radius is in centimeters, the area is in square centimeters and arc length is in centimeters. If the radius is in feet, the area is in square feet. The tool does not convert measurement units, so do not mix a radius in one unit with an expected result in another. The displayed numbers use up to 12 significant digits to balance precision and readability.

Input validation rejects zero, negative, non-finite, or extremely large radii. The central angle must be greater than zero and cannot exceed a full circle: 360 degrees or 2π radians. This page calculates the ordinary sector swept by that positive central angle. It does not calculate a circular segment bounded by a chord, an annulus sector with inner and outer radii, an elliptical sector, a reflex sweep beyond one rotation, or an area from chord and sagitta measurements.

Eight external golden cases cover full, half, quarter, and smaller sectors in both radians and degrees. They include the published radius-20, 30-degree example. Independent university material cross-checks both A = ½r²θ and s = rθ. Boundary tests additionally prove that an angle above one full circle and a non-positive radius are rejected instead of producing an unexplained value.

All arithmetic runs in the browser with JavaScript double-precision numbers. Floating-point rounding means a displayed decimal is an approximation whenever π or a non-terminating value is involved. For construction, surveying, machining, legal boundaries, or safety-critical engineering, retain appropriate significant figures and verify the result with the governing professional method and tolerances.

This tool has no uploads, account requirement, or hidden server calculation. Changing an input clears the old result so that a stale area is not mistaken for the new inputs. The formula statement remains visible beside the result, making the calculation boundary easy to audit. Use the related angle, circle area, and circumference tools when the known measurements describe a different geometry problem.

Methodology & sources

Validate a positive finite radius and a positive central angle no greater than 360 degrees or 2π radians. Convert degrees to radians, then compute sector area as 0.5 × θ × r², arc length as θ × r, and full-circle fraction as θ/(2π). Format finite results to 12 significant digits. Verify eight degree/radian fixtures against OpenStax and an independent university derivation.

Frequently asked questions

What units does the sector area use?
The area uses the square of the radius unit, while arc length uses the original radius unit.
Can the angle be greater than 360 degrees?
No. This calculator is bounded to one positive full circle, or 2π radians.
Is a sector the same as a circular segment?
No. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
Why is a result containing pi shown as a decimal?
The browser evaluates π numerically and displays up to 12 significant digits, so irrational results are approximations.

Related tools

Calculators guides

View all