Area of a Circular Sector Formula

The area of a circular sector is found with the formula A = (θ/360) × πr² when the central angle θ is in degrees, or A = (1/2)r²θ when θ is in radians. Enter the radius and central angle below to get the sector area instantly.

Formula

The sector area formula is a proportional slice of the full circle’s area. For a circle of radius r, the total area is Ï€r². A sector with central angle θ (in degrees) covers θ/360 of the full circle, so:

A = (θ / 360) × π × r²

If the angle is measured in radians, the formula simplifies to A = ½ × r² × θ. This works because a full circle is 2π radians, so the sector fraction is θ/(2π), and (θ/(2π)) × πr² = ½r²θ.

Variables:

• r – radius, any unit (ft, m, cm, etc.). Area comes out in that unit squared.
• θ – central angle, degrees or radians (must match the formula version).
• A – sector area, in square units of the radius.

Worked Example

Using the calculator defaults: radius = 10 ft, central angle = 60°.

1. Full circle area: π × 10² = 3.14159 × 100 = 314.159 ft²
2. Sector fraction: 60 / 360 = 1/6
3. Sector area: 314.159 × (1/6) = 52.3599 ft²

The calculator shows 52.36 ft² (rounded to two decimals). The sector fraction is 0.167 (1/6), and the full circle area is 314.16 ft².

When to Use the Sector Area

You need this formula any time you’re dealing with a slice of a circle, not the whole thing. Common real-world uses:

• Cutting a circular cake or pizza into equal pieces – each slice is a sector.
• Designing a fan blade, a pie chart wedge, or a circular garden bed with an angled path.
• Calculating the area of a portion of a circular lawn or patio that will be covered by sod or pavers.
• Working out the material needed for a conical shape’s lateral surface (the sector is the net of a cone’s side).
• Estimating the area swept by a rotating sprinkler that covers only part of a circle.

If the angle is 360°, the formula gives the full circle area – a good sanity check.

How to Choose the Right Formula Version

The only real choice is degrees vs. radians. Use the table below to avoid errors:

• Angle in degrees → use A = (θ/360) × Ï€r². Most everyday measurements (protractors, blueprints) use degrees.
• Angle in radians → use A = ½r²θ. This is common in calculus, physics, and engineering where angles are naturally in radians.
• If you mix them up – for example, plugging 60 radians into the degree formula – you’ll get a wildly wrong answer (about 57 times too large).

To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.

Common Mistakes and How to Avoid Them

1. Using the diameter instead of the radius – the formula requires r, not d. If you have the diameter, halve it first.
2. Forgetting to square the radius – r² means r × r, not 2r.
3. Angle unit mix-up – always check whether your angle is in degrees or radians before choosing the formula.
4. Rounding too early – if you round Ï€ to 3.14, your result will be slightly off. Use a calculator’s Ï€ button for precision.
5. Confusing sector area with arc length – arc length is (θ/360) × 2πr, not the area. They have different units (length vs. area).

Real-World Example: Sod for a Circular Garden

Imagine you’re laying sod in a circular flower bed that has a radius of 8 feet, but you only want to cover a 90° wedge (a quarter circle). Using the formula:

A = (90/360) × π × 8² = 0.25 × 3.14159 × 64 = 50.27 ft²

Since sod is sold in rolls or pallets, you’d need to round up to the nearest whole unit. If each roll covers 10 ft², you’d need 6 rolls (ceil(50.27/10) = 6). Always check the actual coverage on the product label, as roll sizes vary.

Limitations and Related Formulas

The sector area formula assumes a perfectly circular sector with straight radial sides. If the shape is not a true sector (e.g., an irregular curved boundary), this formula won’t apply. For a segment (the area between a chord and the arc), you need to subtract the triangle area from the sector area. For an annulus sector (a slice of a ring), subtract the inner sector from the outer sector.

The formula is also limited to Euclidean geometry – it doesn’t apply to spherical surfaces or non-flat planes. In those cases, you’d need spherical trigonometry or calculus.

Frequently Asked Questions

What is the sector area formula in radians?

When the central angle θ is in radians, the area is A = ½r²θ. This is the more compact form and is derived from the fact that a full circle has 2π radians, so the fraction is θ/(2π).

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

If you know the arc length s and the radius r, you can find the angle in radians as θ = s/r. Then use A = ½r²θ = ½r²(s/r) = ½rs. So the area is simply half the product of the radius and arc length.

What is the difference between a sector and a segment?

A sector is bounded by two radii and the arc, like a pizza slice. A segment is bounded by a chord and the arc, like the area between a straight cut and the curved edge. To find a segment’s area, subtract the isosceles triangle formed by the two radii from the sector area.

Can the formula work for angles greater than 360°?

Mathematically, the formula still gives a number, but a sector with an angle greater than 360° would overlap itself. In practice, angles are usually between 0° and 360°. The calculator restricts the input to 0–360° to avoid nonsensical results.

What if my angle is in gradians?

Gradians (gons) divide a full circle into 400 units. To use the degree formula, convert gradians to degrees by multiplying by 0.9 (since 400 gradians = 360 degrees). Or convert to radians and use the radian formula.

Why is the sector area proportional to the angle?

A sector is a scaled copy of the full circle, scaled by the fraction of the angle. Doubling the angle doubles the area, because the shape expands linearly in the angular direction while the radius stays constant. This proportionality is the core of the formula.

Related

• Acres and Unit Per Acre Calculator
• Acre Feet to Gallons Conversion Calculator
• Acre Foot to Gallons Calculator

in r | Radius | ft | 10 | 0 | 100000
in th | Central angle | degrees | 60 | 0 | 360
out a | Sector area | ft² | (th / 360) * pi * r^2 | 2
out ar | Full circle area | ft² | pi * r^2 | 2
out frac | Sector fraction | - | th / 360 | 3