functions

Sector Calculator | Compute Area From Radius and Radian Angle

How the Sector Calculator works

Under the hood, this calculator applies Area = ½ r² θ (θ in radians); arc = rθ; chord = 2r sin(θ/2), so the sector area calculator result you see is genuine math, not a lookup table.

The reference table below covers the most common sector area calculator cases at a glance, the methodology section breaks the calculation into verifiable steps, and the FAQ tackles the edge cases.

Quick reference

Radius	Angle	Area	Arc length
5	60°	13.09	5.24
5	90°	19.63	7.85
10	45°	39.27	7.85
8	120°	67.02	16.76

info Sector Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

Output	Formula
Sector area	½r²θ = (θ/360)×πr²
Arc length	rθ (θ in radians)
Chord	2r·sin(θ/2)
Perimeter	arc + 2r

lightbulb Worked example — one slice of a 16-inch pizza

A 16-inch pizza is cut into 8 equal slices. Each slice is a sector with radius 8 inches and a central angle of 45°. Convert the angle first: 45° = π/4 ≈ 0.7854 radians.

Result: Area = ½ × r² × θ = 0.5 × 64 × 0.7854 = 25.1 in². Arc length = rθ = 8 × 0.7854 = 6.28 in. Chord = 2 × 8 × sin(22.5°) = 16 × 0.3827 = 6.12 in. Perimeter = arc + 2r = 6.28 + 16 = 22.28 in.

What this means: One slice covers about 25 square inches — eight slices total 201 in², matching π × 8² = 201.06 in² for the whole pizza, which is a good sanity check. The chord (6.12 in) is the straight cut edge from crust to tip, slightly shorter than the curved crust (6.28 in).

Formula & methodology

Formula: Area = ½ r² θ (θ in radians); arc = rθ; chord = 2r sin(θ/2)

How the sector measurements are calculated

The tool applies Area = ½r²θ with θ in radians, arc length = rθ, chord length = 2r sin(θ/2), and perimeter = arc + 2r. Angles entered in degrees are converted to radians automatically, and every formula is displayed with the working.

Example: a 60° sector of a circle with radius 6 has area ½ × 36 × (π/3) ≈ 18.85 square units and arc length 6 × (π/3) ≈ 6.28 units.

Reading the result: the sector is the circular slice — pizza portions, gear segments, irrigation coverage, and radar sweeps all use these formulas. The area gives material estimates, the arc gives the curved boundary length for fencing or cutting, and the chord is the straight-line distance across the slice.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do you find the area of a sector?

Enter the required values into the input fields and press the calculate button. The tool applies the formula Area = ½ r² θ (θ in radians); arc = rθ; chord = 2r sin(θ/2) and returns the result with a short explanation of each step, so you can check the working yourself.

What is arc length and how is it different from the chord?

The Sector Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates sector area using Area = ½ r² θ (θ in radians), in real time in your browser. There is no limit on usage, and it works on any device.

What is the sector perimeter?

The perimeter of a sector is the sum of its arc length and the two radii: P = rθ + 2r (with θ in radians). For a 60° sector of radius 6, the arc is about 6.28 units and the perimeter about 18.28 units. The calculator returns the arc length, chord length, and perimeter alongside the area, covering every linear measure of the sector.

Does the calculator accept radians?

Yes — you can enter the angle in degrees or radians, and the tool converts internally since the area formula Area = ½r²θ requires radians. A 60° angle becomes π/3 radians automatically. The results are consistent either way, and the working shows the conversion so you can verify the arithmetic.

What real-world problems use sectors?

Sectors describe circular slices: pie and pizza portions, gear segments, irrigation coverage, radar and sonar sweeps, and circular statistics. The area formula ½r²θ sizes anything from a slice of cake to the coverage of a rotating sprinkler. The chord and arc measures matter for cutting materials, fencing curved boundaries, and layout work.

Why must the area formula use radians instead of degrees?

The formula A = ½r²θ only works when θ is in radians, because a radian is defined so that arc length equals rθ exactly. Degrees need a conversion factor: A = (θ/360) × πr². The conversion is automatic — 60° becomes π/3 before the formula is applied.

When would I use the chord instead of the arc?

The chord is the straight-line distance across the slice, while the arc is the curved boundary. Use the chord when you need the straight cut between two points on the circle — like the length of a straight cable spanning a curved section — and the arc when you need the length along the curve itself.

How does the sector relate to the whole circle?

A sector with angle θ is exactly θ/360 (in degrees) of the full circle. A 90° sector is one quarter: its area is πr²/4 and its arc is πr/2. Checking that your sector results scale up to the whole-circle formulas is the fastest way to verify them.

What happens at angles above 180°?

The formulas still work — a 270° sector is a major sector covering three quarters of the circle. The arc and area keep growing with θ, while the chord shrinks back down as the two endpoints approach each other on the far side. At 360° the arc is the full circumference and the chord returns to zero.

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