Circular sector and arc calculator
Enter a radius and a central angle in degrees and this tool returns the arc length, sector area, chord length and the perimeter of the sector — the slice of a circle bounded by two radii and the arc between them. It is handy for geometry homework, drafting and any layout involving curved segments.
How it works
The angle in degrees is first converted to radians with θ = degrees × π ÷ 180,
then each property follows a standard formula:
| Property | Formula |
|---|---|
| Arc length | r × θ |
| Sector area | ½ × r² × θ |
| Chord length | 2 × r × sin(θ ÷ 2) |
| Perimeter | arc length + 2 × r |
At θ = 2π (a full 360°) the sector area reduces to the whole-circle area πr².
Worked example
A radius of 10 and a 90° angle (θ = π/2 ≈ 1.5708):
| Property | Value |
|---|---|
| Arc length | ≈ 15.708 |
| Sector area | ≈ 78.540 |
| Chord length | ≈ 14.142 |
| Perimeter | ≈ 35.708 |
The conversion from degrees happens automatically, all in your browser with nothing uploaded.
Where sector and arc calculations appear in practice
Drafting and CNC: Curved cuts in metal or wood are specified by radius and angle. The arc length tells you how much material the cutter travels along the curve; the chord length is the straight measurement across the opening, which is often the dimension a machinist checks with calipers.
Architecture and landscape design: Circular garden beds, rounded deck sections, and arched doorways are all described as sectors. Knowing the perimeter lets you work out how much edging, trim, or framing material to order; the area tells you how much turf, concrete, or paving to buy.
Pizza and pie serving: A classic circular-sector problem. An 8-slice pizza has each slice at 45°. For a 14-inch pizza (radius 7 inches) each sector has an arc length of about 5.5 inches and an area of about 19.2 square inches — the exact share of crust and topping per person.
Trigonometry and exam problems: Many exam questions give two of the four properties and ask for the others, usually by giving radius and arc length, then requiring the angle. The tool works the other way — angle and radius in, everything out — but the formulas shown let you rearrange algebraically.
Tips on common mistakes
- Degrees versus radians: the formulas use radians. The tool converts for you, but if you apply the formulas manually, make sure you convert first or arc length will be off by a factor of about 57.
- Quarter circle shortcut: a 90° sector has area exactly πr² ÷ 4, which is a useful sanity check for the tool’s output.
- Perimeter is not the arc: perimeter includes the two straight radii. If you need to bend a strip of material around the curved arc only — not the two straight sides — use the arc length figure, not the perimeter.