A sector is the pie-slice part of a circle enclosed by two radii and the arc between them. Its size depends on just two things, the radius and the central angle, so knowing any two related measurements is enough to find the rest. This calculator starts from the radius and angle, the radius and arc length, the radius and sector area, the arc length and angle, or the radius and chord. It also reports the chord, the segment between the chord and the arc, the segment height and the sector’s share of the full circle.
How to use the sector calculator
- Choose from the I know the menu: Radius and central angle, Radius and arc length, Radius and sector area, Arc length and central angle, or Radius and chord length.
- Enter the values that appear: Radius (r), Central angle (θ), Arc length (s), Sector area (A) or Chord length (c). Angles can go up to 360° (2π rad).
- Pick the length Units and the Angle unit, either Degrees (°) or Radians (rad). The angle unit applies to the angle you enter and the angle shown in the results.
- Read the radius, central angle, arc length, sector area, chord, sector perimeter (two radii plus the arc), segment area, segment height (sagitta) and share of the full circle.
Sector and arc formulas
With the central angle θ in radians:
The form A = ½rs is worth remembering. It is the area of a triangle with base s and height r, and that’s no coincidence: slice a sector into many thin wedges and each one is very nearly a triangle whose height is the radius. Running the formulas backwards gives θ = s ÷ r, θ = 2A ÷ r² and θ = 2 arcsin(c ÷ 2r).
Worked example: pizza slices and a garden bed
A 16-inch pizza (r = 8 in) is cut into 8 slices, so each slice is a 45° sector. Choose Radius and central angle and enter r = 8 and θ = 45 in degrees.
- Sector area: ½ × 8² × 0.7854 = 25.13 in², 12.5% of the pie
- Arc length (the crust): 8 × 0.7854 = 6.28 in
- Chord (the straight line between the crust ends): 6.12 in
Compare a 12-inch pizza cut into 6: with r = 6 in and θ = 60°, each slice is only 18.85 in². One eighth of the larger pie is a third bigger than one sixth of the smaller.
Now a garden bed: you have 20 ft of flexible edging for the curved front of a fan-shaped bed with a 10 ft radius. Choose Radius and arc length and enter r = 10 and s = 20. The central angle is 20 ÷ 10 = 2 rad, or 114.59°, and the bed covers ½ × 10 × 20 = 100 ft².
Degrees, radians and common sector values
The most common mistake with sectors is mixing angle units. The formulas s = rθ and A = ½r²θ only hold in radians. In degrees you need the longer forms, s = (θ ÷ 360°) × 2πr and A = (θ ÷ 360°) × πr². A slip here is easy to spot: a 60° arc on a 10 cm circle should be about 10.47 cm, not 600 cm.
For a circle with radius 1, multiply the arc and chord columns by r and the area column by r²:
| Angle | Radians | Share of circle | Arc length | Sector area | Chord |
|---|---|---|---|---|---|
| 30° | 0.5236 | 8.33% | 0.5236 | 0.2618 | 0.5176 |
| 45° | 0.7854 | 12.5% | 0.7854 | 0.3927 | 0.7654 |
| 60° | 1.0472 | 16.67% | 1.0472 | 0.5236 | 1.0000 |
| 90° | 1.5708 | 25% | 1.5708 | 0.7854 | 1.4142 |
| 120° | 2.0944 | 33.33% | 2.0944 | 1.0472 | 1.7321 |
| 180° | 3.1416 | 50% | 3.1416 | 1.5708 | 2.0000 |
| 270° | 4.7124 | 75% | 4.7124 | 2.3562 | 1.4142 |
Notice that for radius 1 the arc length equals the angle in radians. That is the definition of a radian, and one radian is about 57.2958°.
Where sectors and arcs show up
- Food and charts: pizza and pie slices, cake portions, and pie charts, where each category’s angle is its share of 360°.
- Landscaping: fan-shaped beds, curved paths and the area a rotating sprinkler head waters when set to a partial arc.
- Building: the sagitta, or rise of an arc above its chord, sets the height of an arched doorway or a curved beam from its span.
- Roads and rails: curve lengths are arc lengths, while surveyors lay out the straight chord between stakes.
For a full circle use the circle calculator, and to switch between degrees and radians try the angle converter.
Frequently asked questions
How do I find the area of a sector?
Take the fraction of a full turn that the sector covers and multiply it by the circle's area: A = (θ ÷ 360°) × πr². With θ in radians this simplifies to A = ½r²θ. A 45° slice of a 16-inch pizza (r = 8 in) has an area of about 25.13 in².
What is the formula for arc length?
s = rθ with θ in radians, or s = (θ ÷ 360°) × 2πr with θ in degrees. A 60° arc on a circle with a 10 cm radius is 10 × 1.0472 ≈ 10.47 cm long.
What is the difference between a sector and a segment?
A sector is the pie slice bounded by two radii and the arc. A segment is the region between the arc and the straight chord joining its ends. For angles up to 180° the segment is the sector minus the triangle formed by the two radii and the chord; for larger angles that triangle is added instead.
Why do the formulas use radians?
A radian is defined so that an arc one radius long spans an angle of exactly 1, which makes s = rθ and A = ½r²θ work without any conversion factor. If you put degrees into s = rθ by mistake, the answer comes out about 57.3 times too large. The calculator accepts either unit and converts degrees for you.
How do I find the central angle from a chord?
Use θ = 2 arcsin(c ÷ 2r). A straight cut 6.12 inches long across a pizza with an 8-inch radius gives θ = 2 arcsin(6.12 ÷ 16) ≈ 45°. A chord matches two sectors, a small one and a large one, and the calculator returns the smaller, so the angle is never more than 180°.