To find the area of a sector in degrees, multiply the area of the full circle by the fraction of the rotation covered by the sector: A = (θ/360) × πr². Here, θ is the central angle in degrees and r is the circle’s radius. The 360-degree denominator represents one complete revolution, so the same proportion works for a small sector or a full circle. For a dependable calculation, the radius must be positive and finite, and the central angle must be greater than zero but no more than 360 degrees. The Sector Area Calculator applies this degree-based relationship after converting the angle to radians, then reports the sector area, corresponding arc length, and percentage of the full circle. Using one length unit consistently matters: a radius entered in feet produces area in square feet and arc length in feet, while a radius in centimeters produces square centimeters for area and centimeters for arc length. This page performs the arithmetic locally in your browser, requires no upload or account, and keeps a formula statement visible beside the result so the calculation can be checked. That makes it useful when a problem supplies degrees but you also need the related boundary length or the sector’s share of the circle.
How degree and radian formulas connect
The formula A = ½θr² requires θ to be measured in radians. If the central angle is already in degrees, the calculator first applies θ radians = θ degrees × π/180. This conversion is consistent with the degree-and-radian discussion in OpenStax Algebra and Trigonometry’s angles chapter.
Because a full circle contains 360 degrees, or 2π radians, the degree form follows directly: A = (θ degrees/360) × πr². The same fraction of a full turn determines the other results. Arc length is s = (θ degrees/360) × 2πr, while the sector’s share of the circle is θ degrees/360. Selecting the correct angle mode ensures that the numerical angle is interpreted properly before these relationships are applied. A separate step-by-step derivation for sector area in radians can also help clarify why the conversion appears in the calculation.
Calculate a sector in degree mode
- Open the Sector Area Calculator and locate the radius, central-angle, angle-unit, and calculation controls.
- Enter a positive, finite circle radius and note the length unit you used. Keep that unit consistent when interpreting the answer. Zero, negative, non-finite, and extremely large radii are rejected.
- Enter the central angle and choose degrees as its unit. A degree-mode angle must be greater than zero and no greater than 360 degrees. The equivalent radian-mode limit is 2π radians.
- Select Calculate sector to apply the degree conversion and calculate the requested measurements.
- Read the sector area, corresponding arc length, and percentage of the full circle from the result. Check that the area uses squared length units while the arc length uses the original radius unit.
Changing an input clears the previous result, preventing an old area from being mistaken for the answer to revised values. Run the calculation again after changing the radius or angle.
Read the three outputs correctly
The calculator reports more than the requested area. These outputs describe different properties of the same sector and should not be treated as interchangeable measurements.
| Output | Degree-based relationship | Unit to expect |
|---|---|---|
| Sector area | A = (θ/360) × πr² | The square of the radius unit |
| Arc length | s = (θ/360) × 2πr | The original radius unit |
| Share of the full circle | θ/360 | Displayed as a percentage |
Internally, the calculator uses the radian angle with A = ½θr², s = θr, and the full-circle fraction θ/(2π). These are equivalent to the degree forms in the table. The angle mode is therefore part of the measurement, not merely a display preference: entering a number under the wrong mode changes its meaning, and an angle beyond the selected mode’s full-circle limit is rejected.
A 20-foot, 30-degree worked example
The OpenStax sprinkler example uses a radius of 20 feet and a central angle of 30 degrees. Convert that angle to radians: 30 × π/180 = π/6 radians.
Substitute the converted angle and radius into the area formula: A = ½ × π/6 × 20² = 100π/3, which is approximately 104.72 square feet.
For the same sector, the arc length is s = π/6 × 20 = 10π/3 feet. Its share of the full circle is 1/12, reported as approximately 8.33333333333%. Entering the original radius and degree value in degree mode reproduces these related results.
Accuracy checks before relying on the result
All arithmetic runs in the browser with JavaScript double-precision numbers. Finite results are formatted to as many as 12 significant digits, but any displayed decimal involving π or another non-terminating value is an approximation rather than an exact symbolic result.
- Check the unit: The calculator does not convert measurement units. If the radius is in centimeters, expect area in square centimeters and arc length in centimeters.
- Check the mode: Degree and radian values are not interchangeable. Make sure the selected unit matches how the central angle was given.
- Check both boundaries: The radius must satisfy the positive, finite input rule, and the central angle must remain within one positive full circle.
- Check the formula: A formula statement remains visible beside the result, allowing the radius, angle basis, and sector boundary to be audited.
- Check the needed precision: For construction, surveying, machining, legal boundaries, or safety-critical engineering, retain appropriate significant figures and verify the result using the governing professional method and tolerances.
When the geometry is not an ordinary sector
A circular sector is bounded by two radii and the arc between them. Similar-looking geometry problems can require different inputs, so confirm the boundary before using the result.
| Problem | Defining information | Sector Area Calculator scope |
|---|---|---|
| Ordinary circular sector | One radius and a positive central angle up to one full circle | Calculates area, arc length, and circle share |
| Circular segment | A chord and the arc between its endpoints | Not calculated |
| Annulus sector | Inner radius, outer radius, and central angle | Not calculated |
| Elliptical sector | An elliptical boundary and central angle | Not calculated |
| Sweep beyond one rotation | A reflex angle greater than a full circle | Not calculated |
| Area from chord and sagitta | Chord length and sagitta rather than radius and central angle | Not calculated |
If the known measurements describe one of these different geometries, use the corresponding method instead of forcing them into the ordinary-sector calculation. For the requested degree-based task, record the radius and its unit, enter the central angle in degree mode, run the calculation, and verify that the returned area carries the squared unit while the arc length carries the original length unit.
If you're weighing options, How to Find Sphere Radius With Surface Area covers this in detail.