Sector Area Calculator
A 90 degree sector of a circle with radius 8 has an area of 50.27, exactly a quarter of the full circle's 201.06. This sector area calculator works from the radius and the central angle, and it also returns the arc length and the full perimeter of the slice. A sector is the pizza-slice shape between two radii, and its area is just the circle's area scaled by the angle.
Quick answer
A sector is the region between two radii and the arc connecting them.
Sector area
50.2655
Arc length
12.5664
Perimeter
28.5664
Fraction of circle
0.25
What this tells you
- •A sector is the region between two radii and the arc connecting them.
- •Its area is the fraction of the circle the angle covers: (angle / 360) x pi r squared.
- •The arc length scales the same way: (angle / 360) x 2 pi r.
- •The perimeter of a sector is the arc plus the two straight radii.
How to Use
- 1Enter the radius of the circle.
- 2Enter the central angle in degrees, anything above 0 up to 360.
- 3Read the sector area, with arc length and perimeter below.
- 4The fraction-of-circle figure shows what share of the full circle your slice covers.
How It Works
Formula
A = (angle / 360) x pi r^2The angle over 360 tells you what fraction of the whole circle the sector spans, and multiplying the full circle area by that fraction gives the sector area. For a 90 degree sector of radius 8, the fraction is 0.25 and the circle area is 201.06, so the sector holds 50.27. In radians the same formula reads A = r squared x angle / 2.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Quarter circle (90 degrees), radius 8
A quarter of pi x 64. The arc length is 12.57 and the perimeter 28.57.
Pizza slice (45 degrees), radius 10
One eighth of a 10-inch-radius pizza, with a 7.85 inch crust edge.
Sprinkler coverage (120 degrees), radius 15
A sprinkler sweeping a third of a circle waters a third of pi x 225.
Sector Areas for Radius 10
How the area grows with the central angle at radius 10.
| Angle | Fraction of circle | Area |
|---|---|---|
| 30 | 1/12 | 26.18 |
| 45 | 1/8 | 39.27 |
| 60 | 1/6 | 52.36 |
| 90 | 1/4 | 78.54 |
| 180 | 1/2 | 157.08 |
| 360 | 1 | 314.16 |
Common mistakes
- Mixing degrees and radians. This calculator expects degrees, and a 1.57 entry meant as radians reads as a sliver instead of a quarter circle.
- Using the diameter for r. The formula squares the radius, so entering a diameter quadruples the area.
- Confusing sector with segment. A sector is the pizza slice from the center, a segment is the region cut off by a chord.