The area of a circular sector is found with the formula A = ½θr², where r is the radius of the circle and θ is the central angle measured in radians. When the angle is given in degrees, the same formula works after converting to radians using θ radians = θ degrees × π/180, giving the equivalent expression A = (θ°/360) × πr². A sector is the pie-slice region bounded by two radii and the arc between them, so its area scales with the fraction of a full 2π rotation that the central angle covers. Using that fraction directly gives A = (θ / 2π) × πr², which simplifies to ½θr² when θ is already in radians. The same proportional reasoning gives arc length s = θr and the circle share θ / (2π), which is why these three quantities always appear together. Because all three derive from the same fraction, you can read off the area, the curved edge length, and the percentage of the full circle from a single calculation.
What a Sector Is and Why the Formula Works That Way
A sector is one of the pieces you get when you cut a circle along two straight lines from its center to its edge. The two lines are radii, and the curved outer boundary is the arc that connects their endpoints. Because the whole circle has area πr² and a full sweep covers 2π radians, any sector with central angle θ in radians covers exactly the fraction θ/(2π) of that circle. Multiply the circle area by that fraction and you arrive at A = ½θr². The same logic produces the arc length s = θr (a fraction θ/(2π) of the full circumference 2πr) and the circle share θ/(2π). This is why the Sector Area Calculator always reports all three side by side: they describe the same piece of the same circle, and any one of them can be derived from the other two.
Sector Area Formula in Degrees and Radians
Whether you write the formula with θ in radians or θ in degrees changes only the constant in front. In radians the formula is the cleanest version, A = ½θr², and the arc length is s = θr. In degrees you have to convert first, or use the equivalent forms A = (θ°/360) × πr² and s = (θ°/360) × 2πr. The conversion itself is exact: θ radians = θ degrees × π/180, as confirmed by OpenStax Algebra and Trigonometry on angles. The table below summarizes the formulas in both units.
| Quantity | Formula with θ in radians | Formula with θ in degrees |
|---|---|---|
| Sector area | A = ½θr² | A = (θ°/360) × πr² |
| Arc length | s = θr | s = (θ°/360) × 2πr |
| Fraction of full circle | θ / (2π) | θ° / 360 |
The three rows always agree at the full-circle limit: θ = 2π radians or 360° returns A = πr², s = 2πr, and share = 1, which is a quick sanity check that the formulas are consistent.
How to Find Sector Area Step by Step
To find a sector area with the calculator, you only need a positive radius and a positive central angle no greater than one full rotation.
- Enter the radius of the circle using one consistent length unit. Pick any unit you like — feet, meters, centimeters, inches — but stick to it for the rest of the problem, because the area will be reported in the square of that unit and the arc length in the unit itself.
- Enter the central angle and tell the page whether it is in degrees or radians. If the angle is in degrees, choose Degrees; if it is already in radians (for example, π/6 or 1.5708), choose Radians.
- Select Calculate sector. The page returns the sector area, the matching arc length, and the percentage of the full circle that the sector covers.
Open the Sector Area Calculator and run a 20-foot radius with a 30° angle to reproduce the OpenStax sprinkler example later in this article.
Units, Rounding, and Boundary Rules
The calculator uses whatever radius unit you type, and it does not convert. A radius in centimeters produces an area in square centimeters and an arc length in centimeters; a radius in feet produces an area in square feet and an arc length in feet. Do not mix a radius in one unit with an expected result in another, because the tool will not catch that mistake for you.
All arithmetic runs locally in your browser using JavaScript double-precision numbers. Results are displayed with up to 12 significant digits so you can read enough precision for a calculation log without being overwhelmed by floating-point noise. Whenever the exact answer involves π or another irrational value, the displayed decimal is an approximation of the exact form, and you may want to keep the symbolic form alongside the number when you write the answer down.
Input validation rejects a few values outright so that the result never reflects a meaningless input. A radius of zero or a negative radius is not accepted, and neither is a non-finite or extremely large radius. The central angle must be greater than zero and cannot exceed one full circle, which means 360° in Degrees mode and 2π radians in Radians mode. Boundary tests confirm that an angle just above that limit and a non-positive radius are both rejected rather than producing an unexplained number. For safety-critical engineering, surveying, or legal work, retain the appropriate significant figures and verify the result against the governing professional method.
Sector Versus Segment, Annulus Sector, and Reflex Sweep
The Sector Area Calculator is built specifically for the ordinary sector swept by a positive central angle of at most one full rotation. It does not handle a few shapes that look similar but require different inputs and formulas. Use the table below to decide quickly whether this is the right tool for your geometry.
| Geometry | Calculated here? | What you would need instead |
|---|---|---|
| Ordinary sector (two radii + arc) | Yes | Radius and central angle |
| Circular segment (chord + arc) | No | Radius and chord, or chord and sagitta |
| Annulus sector (inner and outer radii) | No | Inner radius, outer radius, central angle |
| Elliptical sector | No | Semi-axes and central angle |
| Reflex sweep beyond one rotation | No | A different geometric construction |
For an overview of the formulas and worked examples in radians, see Calculate Sector Area and Arc Length: Formula and Steps.
Worked Example: 30-Degree Sprinkler Sweep
A common textbook problem, drawn from OpenStax, asks for the area that a lawn sprinkler with radius 20 feet and a 30° sweep will water. Working it out by hand:
- Convert 30° to radians: 30 × π/180 = π/6.
- Plug into the radian formula: A = ½ × θ × r² = ½ × (π/6) × 20² = ½ × (π/6) × 400 = 100π/3.
- Numerically, 100π/3 ≈ 104.7197552 square feet.
The same inputs give an arc length s = θr = (π/6) × 20 = 20π/6 = 10π/3, which is approximately 10.47197551 feet, and a circle share of (π/6) / (2π) = 1/12, or about 8.3333%. Entering radius 20 and angle 30 with Degrees selected in the Sector Area Calculator reproduces all three numbers in one click.
If your known measurements describe a different geometry — for instance, a chord and a sagitta, or two radii for a ring-shaped sector — switch to a tool that is built for that shape rather than trying to bend the sector formula around it.
Related reading: How to Calculate Sphere Surface Area from Diameter.