Circle Sector Calculator
This circle sector calculator works from the radius and one more measurement: angle, arc length or area. For r = 10 and 90° the area is 78.54 and the arc length 15.71 – along with the chord, the perimeter of the sector, the circle segment and the working.
Example: Radius r 10 · Second measurement: Central angle α · Angle α 90 ° → Result: Sector area A = 78.53982 · arc length b = 15.70796. Source: Wikipedia – Kreissektor (Fläche, Bogenlänge). Updated: .
How it is calculated
A circle sector is the part of a disc between two radii – like a slice of pie. Its size is set by the central angle α, which decides what share of the full circle (α ÷ 360°) is left.
The formulas
- Area: A = π · r² · α ÷ 360° = ½ · b · r
- Arc length: b = π · r · α ÷ 180°
- Perimeter of the sector: P = 2 · r + b
- Chord: c = 2 · r · sin(α ÷ 2)
- Circle segment: the sector area minus the triangle formed by the two radii and the chord, A = r² ÷ 2 · (α in radians − sin α).
How to use the tool
Enter the radius and choose which second measurement you know. If the arc length or area is given, the tool first works back to the angle and then to all other measurements. Example: r = 10 and α = 90° give A = π · 100 ÷ 4 = 78.54 and b = π · 10 ÷ 2 = 15.71. A semicircle (180°) has area ½ · π · r² and the full 360° circle π · r². For the circumference and area of a whole circle use the circumference calculator, and for sine and cosine the trigonometry calculator.
Frequently asked questions
How do you find the area of a circle sector?
A = π · r² · α ÷ 360°. For r = 10 and α = 90° that is a quarter of the circle’s area, 78.54.
How do you find the arc length?
b = π · r · α ÷ 180°. For r = 10 and α = 90° that gives 15.71, a quarter of the circumference.
What is the difference between a sector and a segment?
A sector is bounded by two radii and the arc (a pie slice). A segment lies between the chord and the arc; its area is the sector minus the triangle.
How do I find the angle from the arc length?
α = b ÷ r × 180° ÷ π. The tool does this step when you choose arc length as the second measurement.
What is an angle in radians?
The ratio of arc length to radius: α = b ÷ r. 180° equals π and a full circle 2π.
Sources and legal basis
- Wikipedia – Kreissektor (Fläche, Bogenlänge)
- Wikipedia – Circular sector
- Wikipedia – Circular segment (Kreissegment, Sehne)
As of: · Editorial policy
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