A sector area calculator that solves in four directions: give it the radius and angle, the area and angle, the area and radius, or the arc length and radius — and it returns every remaining property of the sector instantly. It works entirely in your browser, supports both degrees and radians, and shows you the full step-by-step working alongside a scaled SVG diagram of the sector.
What is a circular sector?
A circular sector (also called a pie slice) is the region bounded by two radii of a circle and the arc that connects them. Three quantities fully define a sector: the radius r, the central angle theta, and you can derive everything else from those two. The most important derived properties are:
- Sector area — how much flat space the slice covers
- Arc length — the curved edge of the slice
- Chord length — the straight line connecting the two arc endpoints
- Sector perimeter — the total boundary: arc + two radii
Core formulas
All sector formulas hinge on the central angle expressed in radians. If your angle is in degrees, multiply by pi / 180 first (the tool does this automatically).
| Property | Formula |
|---|---|
| Arc length s | r * theta |
| Sector area A | (1/2) * r^2 * theta |
| Chord length | 2 * r * sin(theta / 2) |
| Sector perimeter | s + 2r |
Because the four quantities are algebraically linked, you can rearrange any formula to solve for an unknown. For example, if you know the area and the angle, then r = sqrt(2A / theta). If you know the arc length and the radius, then theta = s / r. This calculator exposes all four solve paths through the “Solve for” drop-down so you never have to rearrange by hand.
Worked example
Suppose a pizza slice has a radius of 14 cm and a central angle of 60 degrees.
- Convert to radians: theta = 60 * pi / 180 = pi / 3 = 1.0472 rad
- Arc length s = 14 * 1.0472 = 14.661 cm
- Sector area A = 0.5 * 14^2 * 1.0472 = 0.5 * 196 * 1.0472 = 102.62 cm^2
- Chord = 2 * 14 * sin(pi / 6) = 2 * 14 * 0.5 = 14 cm (exactly, since sin(30 degrees) = 0.5)
- Perimeter = 14.661 + 2 * 14 = 42.661 cm
Notice the chord (14 cm) equals the radius here — this is a well-known property of the equilateral triangle inscribed in a 60-degree sector.
Solve-for-variable examples
| Known | Find | Formula used |
|---|---|---|
| r = 10, theta = 90 deg | Area | A = 0.5 * 100 * (pi/2) = 78.540 |
| A = 78.54, theta = 90 deg | Radius | r = sqrt(2 * 78.54 / (pi/2)) = 10 |
| A = 78.54, r = 10 | Angle | theta = 2 * 78.54 / 100 = 1.5708 rad = 90 deg |
| arc = 15.708, r = 10 | Angle | theta = 15.708 / 10 = 1.5708 rad = 90 deg |
Relationship to the full circle
A sector is a fraction of the full circle. The fraction equals theta / (2 * pi), so a 90-degree sector covers exactly one quarter of the circle. The sector area formula A = (1/2) * r^2 * theta becomes the familiar A = pi * r^2 when theta = 2 * pi. The calculator reports the percentage of the full circle as part of its working output, which is handy for checking whether your answer is plausible.
Practical uses
- Pizza and pie charts — working out equal-slice dimensions
- Engineering and machining — cam profiles, sector gears, curved brackets
- Architecture — curved walls, arched windows, rotunda floor plans
- Irrigation and sprinkler layout — coverage area of a rotating sprinkler head
- Trigonometry and exam prep — verify hand calculations against a reliable reference