Sector Area Calculator
A 90 degree sector of a circle with radius 8 has an area of 50.2655, exactly one quarter of the full circle's area. This sector area calculator uses a radius and central angle to find the area inside a circular slice. It also reports the curved arc length, the sector perimeter, and the decimal fraction of the complete circle. A sector is bounded by two radii and the arc between their endpoints. The angle at the center determines how much of the circle belongs to that sector. A 90 degree angle takes one quarter, a 180 degree angle takes one half, and a 360 degree angle covers the whole circle. The same fraction applies to both area and circumference, which makes the results easy to check.
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.
- •Area uses square units, while arc length and perimeter use the same linear unit as the radius.
- •The calculator accepts positive angles through 360 degrees and positive decimal or whole-number radii.
- •Results are rounded to 4 decimal places, while the calculation uses the full available value of pi before rounding.
How to Use
- 1Enter the radius of the circle using one consistent unit, such as inches, feet, centimeters, or meters.
- 2Enter the central angle in degrees. The value must be greater than 0 and no more than 360.
- 3Calculate the result and read the sector area first. Label it with square units, such as square inches, if your work needs units.
- 4Check the arc length for the curved boundary and the perimeter for the arc plus both straight radius edges.
- 5Use the fraction-of-circle result to confirm the scale. For example, 90 degrees should return 0.25 and 180 degrees should return 0.5.
- 6Keep the displayed precision your task needs. Measurements taken with a ruler rarely justify all 4 decimal places.
How It Works
Formula
A = (angle / 360) x pi r^2The angle divided by 360 gives the fraction of a complete circle. Multiply that fraction by pi times the radius squared to get sector area. For a 90 degree sector with radius 8, 90 / 360 = 0.25 and pi x 8 squared = 201.0619. Multiplying 201.0619 by 0.25 gives 50.2655. Arc length uses the same fraction of the full circumference: 0.25 x 2 x pi x 8 = 12.5664. Adding two radius edges gives a perimeter of 28.5664. In radians, sector area can also be written as A = r squared x angle / 2, but this calculator expects degrees.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Quarter circle (90 degrees), radius 8
One quarter of pi x 64 is 50.2655. The arc length is 12.5664 and the perimeter is 28.5664.
Pizza slice (45 degrees), radius 10
The 45 degree angle is one eighth of a circle. One eighth of pi x 100 is 39.2699, the arc length is 7.854, and the perimeter is 27.854.
Sprinkler coverage (120 degrees), radius 15
The 120 degree sweep is one third of a circle. One third of pi x 225 is 235.6194, with arc length 31.4159 and perimeter 61.4159.
Semicircle (180 degrees), radius 6
A 180 degree sector is half a circle. Half of pi x 36 is 56.5487. Its curved arc is 18.8496, and adding two 6-unit radii gives a perimeter of 30.8496.
Narrow sector (30 degrees), radius 12
Thirty degrees is one twelfth of a circle. One twelfth of pi x 144 is 37.6991. The arc length is 6.2832, and the perimeter including both radius edges is 30.2832.
Three-quarter sector (270 degrees), radius 4
The angle covers three quarters of the circle. Three quarters of pi x 16 is 37.6991. The arc length is 18.8496, and the full sector perimeter is 26.8496.
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 |
The table rounds to 2 decimal places for quick reference. The calculator displays up to 4 decimal places.
Sector area, arc length, and perimeter
Sector area measures the flat region inside the slice, so its unit is squared. If the radius is entered in centimeters, the area is in square centimeters. Arc length measures only the curved edge, while perimeter includes that arc and the two straight radius edges.
Both area and arc length use the angle divided by 360. This shared fraction provides a useful check. A 60 degree sector should contain one sixth of the circle's area and one sixth of its circumference. If either result is larger than the corresponding full-circle value, the angle or radius was probably entered incorrectly.
A circular segment is a different shape. It lies between a chord and an arc and does not include the two lines from the center. Finding segment area requires subtracting the triangle between those radii from the sector area.
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.
- Leaving area in linear units. A radius measured in feet produces sector area in square feet, not feet.
- Rounding pi or intermediate values too early. Keep full precision until the final result when checking the calculator by hand.
- Adding only one radius to the arc when finding perimeter. An ordinary sector has two straight radius edges.
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