Sector Area Calculator

A sector is the slice of a circle between two radii: a pizza slice, a pie chart wedge, a fan blade outline. Two inputs define it: the radius r, and the central angle θ.

In degrees: A = (θ / 360) × π × r² In radians: A = ½ × r² × θ

Both formulas describe the same area. Pick whichever matches your angle measurement: most people work in degrees; engineers and physicists often work in radians.

Worked example, pizza slice: A 14-inch pizza cut into 8 equal slices. Each slice has a 45° central angle. Radius is 7 in. A = (45 / 360) × π × 49 = 0.125 × π × 49 ≈ 19.24 sq in per slice. The whole pizza is π × 49 ≈ 153.94 sq in, divided by 8 = 19.24. Math checks.

Where sectors show up in real measurements:

• Irrigation sprinkler coverage. A 90° sprinkler with a 25 ft throw covers (90/360) × π × 625 ≈ 491 sq ft. A full 360° head with the same throw covers about 1,964 sq ft, four times as much. That’s how you choose between corner heads (90°) and middle-yard heads (full or 180°).
• Garden flower beds. A quarter-circle bed against a fence corner with a 6 ft radius gives (90/360) × π × 36 ≈ 28.3 sq ft of planting area.
• Stadium seating sections. Auditoriums fan outward from a stage. Each seating section is roughly a sector with a known angle of view.
• Fabric or sail panels. Triangular spinnaker sails cut from a sector pattern lay flat after sewing.

Sector vs. segment. Easy to mix up:

A sector is bounded by two radii and the arc between them. Looks like a pizza slice. A segment is bounded by a single chord and the arc cut off by that chord. Looks like a sliver crescent. If a pizza slice is the sector, the crust-end piece you’d cut off with one straight knife cut is the segment.

For reference, the segment area (the sliver between the chord and the arc) is:

A_segment = ½ × r² × (θ − sin θ)

where θ is in radians. The segment is the sector minus the triangle formed by the two radii and the chord. This formula shows up in partially-filled pipe cross-sections and dome design.

Sanity check at the extremes:

• θ = 360°: A = π × r², the full circle. ✓
• θ = 180°: A = ½ × π × r², half the circle. ✓
• θ = 0°: A = 0. ✓

If your sector spans more than 180°, you’re describing the “major sector”. Still works, but you might want to subtract from 360 if you’re really after the small piece.