What the Sector Area Calculator Does
A circular sector is the "pie slice" shape bounded by two radii and the arc between them. This calculator finds the area of that slice from just two values: the radius of the circle and the central angle measured in degrees. Alongside the sector area, it also reports the arc length, the full circle area, and what percentage of the whole circle the sector represents — so you get the complete picture from a single calculation.
The Inputs You Enter
- Radius – the distance from the centre of the circle to its edge, in any unit you choose (cm, m, inches, etc.).
- Central Angle (in degrees) – the angle at the centre of the circle that opens out to form the slice, from 0° up to 360°.
The Formula Explained
The sector area uses this relationship:
$$A = \frac{\pi \times \text{Radius}^{2} \times \text{Angle}}{360}$$The logic is simple: a full circle has area \(\pi r^{2}\) and spans 360°. A sector is just a fraction of that circle, and the fraction is \(\theta \div 360\). Multiply the full circle area by this fraction and you get the slice. The calculator also computes:
- Arc length = \(2 \times \pi \times r \times \theta \div 360\)
- Circle area = \(\pi \times r^{2}\)
- Sector percentage = \((\theta \div 360) \times 100\)
Worked Example
Suppose you have a radius of 10 and a central angle of 90° (a quarter circle):
- Sector Area = \(\pi \times 10^{2} \times 90 \div 360 = \pi \times 100 \times 0.25 \approx\) 78.54 square units
- Arc Length = \(2 \times \pi \times 10 \times 90 \div 360 \approx\) 15.71 units
- Full Circle Area = \(\pi \times 10^{2} \approx\) 314.16 square units
- Sector Percentage = \((90 \div 360) \times 100 =\) 25%
The sector is exactly one quarter of the circle, which the 25% figure confirms.
Frequently Asked Questions
Does the angle have to be in degrees? Yes. This calculator expects the central angle in degrees and divides by 360. If your angle is in radians, convert it first (degrees = radians × 180 ÷ π).
What units does the answer use? Whatever unit you used for the radius. If the radius is in metres, the sector area is in square metres and the arc length is in metres.
What happens if I enter 360°? The sector becomes the entire circle, so the sector area equals \(\pi r^{2}\) and the percentage shows 100%.
