Area of a Circle Sector Calculator & Formula

This calculator works out the area of a sector of a circle – the slice-shaped region between two radii and the arc they enclose. You enter the circle’s radius and the sector’s central angle (in degrees or radians), and it returns the sector area in the same square units as the radius. The result is computed from the standard formula A = (θ/360) × πr² for degrees, or A = (θ/2) × r² for radians.

Formula

The area of a sector is a proportional part of the whole circle’s area. A full circle has area πr², and a sector with central angle θ (in degrees) covers the fraction θ/360 of the full circle. So:

A = (θ / 360) × π × r²

where:

• A = sector area (same square units as r²)
• θ = central angle in degrees (0 to 360)
• r = radius of the circle (any length unit)
• π ≈ 3.14159

If the angle is given in radians, the formula simplifies to A = (θ / 2) × r² because the full circle angle is 2π radians. The calculator above uses degrees by default; to use radians, convert the angle first (radians = degrees × π / 180).

Worked Example

Using the default values: radius r = 5 m, central angle θ = 60°.

1. Find the fraction of the full circle: 60 / 360 = 1/6 ≈ 0.1667.
2. Compute the full circle area: π × 5² = π × 25 ≈ 78.5398 m².
3. Multiply by the fraction: 78.5398 × (1/6) = 13.0899 m².
4. Rounded to 2 decimals: 13.09 m².

The calculator shows 13.09 m² for the first output, matching this step-by-step result. The arc length (the curved edge) is (60/360) × 2π × 5 = (1/6) × 31.4159 ≈ 5.24 m.

When to Use a Sector Area Calculator

Sector area calculations appear in geometry homework, design work, engineering, and everyday tasks like estimating the area of a pizza slice, a pie chart wedge, or a circular garden section. You need only two measurements: the radius (or diameter – divide by 2) and the central angle. The angle can be measured with a protractor, given in a diagram, or derived from the arc length using θ = (arc length / circumference) × 360°.

If you have the arc length s instead of the angle, you can still find the area with A = (s × r) / 2. This is useful when measuring a curved edge directly. If you have the chord length (the straight line between the two arc endpoints), you first need to solve for the angle using trigonometry, which is more involved.

Common Mistakes and How to Avoid Them

• Using diameter instead of radius: The formula requires r, not d. If you have the diameter, divide by 2 first. Forgetting this quadruples the area.
• Mixing degrees and radians: The (θ/360) fraction only works for degrees. If your angle is in radians, use (θ/2) × r² or convert to degrees first.
• Forgetting the angle must be less than 360°: A sector is a portion of a circle; an angle greater than 360° would wrap around and is not a standard sector. If you have a reflex angle (greater than 180°), the calculator still works – it just gives the larger slice.
• Rounding too early: If you round π or intermediate results, the final answer can be off. Use full precision until the last step, then round to the desired decimal places.
• Unit confusion: The area is always in square units of the radius. If r is in cm, area is in cm²; if r is in feet, area is in ft². Convert all lengths to the same unit before calculating.

How to Read the Result

The first output is the sector area – the actual surface of the slice. The second output, the fraction, tells you what portion of the whole circle the sector represents (e.g., 0.1667 means one-sixth). The third output, arc length, is the length of the curved boundary. These three values are linked: the fraction times the full circle area gives the sector area, and the fraction times the circumference gives the arc length.

For a half-circle (θ = 180°), the sector area is exactly half the circle area. For a quarter-circle (θ = 90°), it’s one-quarter. Checking these simple cases is a good way to verify you entered the angle correctly.

Limits of the Formula

The formula assumes a perfect, flat circle and a straight-line radius – it does not account for irregular shapes, curved boundaries that are not circular arcs, or three-dimensional surfaces. If you’re measuring a physical object, the radius must be constant along the arc. For a sector of an ellipse or other curve, the formula does not apply. Also, the angle must be the central angle (vertex at the circle’s center), not an inscribed angle (vertex on the circumference).

For very small angles (near 0°) or very large ones (near 360°), the result approaches 0 or the full circle area respectively, which is correct. The calculator accepts any non-negative radius and any angle from 0 to 360 degrees; angles outside this range will produce results that don’t correspond to a normal sector.

Alternatives and Related Calculations

If you need the area of a full circle, use the simpler formula A = πr² – just set the angle to 360°. If you have the arc length and radius, use A = (s × r) / 2. For a segment (the region between a chord and the arc), subtract the triangle area from the sector area: A_segment = A_sector − (r² × sin θ) / 2 (with θ in radians). If you’re working with a pie chart, the sector area is proportional to the data value – each sector’s angle is (value / total) × 360°.

For a cone, the lateral surface area is a sector of a circle – the radius is the slant height and the arc length equals the base circumference. This calculator can be used to find that area if you know the slant height and the base radius (compute the angle as (base radius / slant height) × 360°).

Frequently Asked Questions

How do I find the sector area if I only have the arc length?

Use the formula A = (s × r) / 2, where s is the arc length and r is the radius. This works because the arc length is proportional to the angle: s = rθ (in radians), so A = (rθ × r) / 2 = (θ/2)r², which matches the radian formula.

What if my angle is in radians?

Convert to degrees by multiplying by 180/π, then use the degree formula. Or use the radian formula directly: A = (θ × r²) / 2. For example, θ = 1 radian with r = 5 m gives A = (1 × 25) / 2 = 12.5 m².

Can I use this for a semicircle or quarter circle?

Yes. A semicircle has θ = 180°, so the area is half the circle’s area. A quarter circle has θ = 90°, giving one-quarter of the circle’s area. The calculator handles these automatically.

What is the difference between a sector and a segment?

A sector is the slice bounded by two radii and the arc. A segment is the region bounded by a chord and the arc – it’s the sector minus the isosceles triangle formed by the two radii and the chord. To find a segment area, subtract the triangle area from the sector area.

Why does the calculator ask for the central angle, not the arc length?

The central angle is the most direct input for the formula, and it’s what most geometry problems provide. If you have the arc length, you can convert: θ = (s / (2πr)) × 360°. The calculator’s third output gives arc length from the angle, so you can verify your input.

Does the sector area depend on the circle’s circumference?

Not directly – it depends only on the radius and the central angle. The circumference is 2πr, and the arc length is a fraction of that, but the area formula uses the radius squared. Two sectors with the same angle and radius always have the same area, regardless of the circle’s total size.

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in r | Radius | m | 5 | 0
in th | Central angle (degrees) | ° | 60 | 0
out area | Sector area | m² | (th / 360) * pi * r^2 | 2
out frac | Fraction of full circle | - | th / 360 | 4
out arc | Arc length | m | (th / 360) * 2 * pi * r | 2