Sector Area Calculator – Find the Area of a Circular Pie Slice
A circular sector is the region of a circle enclosed by two radii and the arc connecting them — essentially a pie slice. Whether you are working on a geometry problem, designing a pie chart, cutting a circular material, or solving an engineering challenge, knowing the sector area is essential. This calculator gives you the area instantly from just two inputs: the radius and the central angle.
The Sector Area Formula
The standard formula for sector area is:
A = ½ × r² × θ
where r is the radius of the circle and θ is the central angle in radians. If you are working in degrees, the equivalent form is:
A = (θ° / 360°) × π × r²
Both expressions are mathematically identical. This tool accepts degrees or radians and applies the conversion automatically so you never have to do it manually.
What Else Does the Calculator Compute?
Beyond the sector area, the tool also returns three related measurements in a single calculation:
- Arc Length — the curved edge of the sector:
L = r × θ(θ in radians). - Perimeter of the Sector — the total boundary including both straight radii and the arc:
P = L + 2r. - Fraction of the Circle — what proportion of the full circle the sector covers, expressed as a decimal and as a percentage:
f = θ / (2π).
Common Sector Sizes
Several sector sizes appear frequently in practice. For a circle with radius r:
- Quarter-circle (90° / π/2 rad): A = π r² / 4 ≈ 0.7854 r²
- Semicircle (180° / π rad): A = π r² / 2 ≈ 1.5708 r²
- Three-quarter circle (270° / 3π/2 rad): A = 3π r² / 4 ≈ 2.3562 r²
- Full circle (360° / 2π rad): A = π r² ≈ 3.1416 r²
Degrees vs Radians – Which Should You Use?
Most everyday users are comfortable with degrees, where a full circle equals 360°. Scientists and engineers often prefer radians because they make many formulas, including the sector area formula, simpler and dimensionally clean. One full circle equals 2π ≈ 6.2832 radians. The calculator accepts both formats.
Sector vs Segment – Know the Difference
A common point of confusion is the difference between a sector and a segment. A sector is the full pie-slice region bounded by two radii and the arc. A segment is the smaller region between a chord (straight line) and the arc — it is the sector with the central triangle removed. This calculator focuses on the sector. The segment area formula is A_seg = ½ r² (θ − sin θ).
Practical Applications
Sector area calculations arise in many fields. In architecture and design, curved floor plans, fan-shaped rooms, and circular staircases require sector measurements. In engineering, rotating machinery, gear teeth, and cam profiles rely on sector geometry. In data visualisation, pie chart slices are literally sectors — the angle of each slice is proportional to the data it represents. In agriculture and surveying, irrigation systems using pivot sprinklers cover a sector-shaped area of land.
Tips for Accurate Results
- Use consistent units — if the radius is in metres, the resulting area will be in square metres and the arc length in metres.
- The central angle must be greater than 0 and no greater than 360° (or 2π radians). Angles larger than a full circle are not geometrically meaningful for a single sector.
- For very small angles, the sector area approaches zero and the arc length approaches the chord length — both converge to the same value as θ → 0.