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, and dividing by 8 gives the same 19.24. Two routes, one answer.

Worth contrasting with the straight-sided version. Treat the slice as a triangle with a 5.36 in straight base instead of a curved crust and you get 17.32 sq in, about 10% low. The missing 1.92 sq in per slice is the crescent of crust the straight line cuts off, and across eight slices that is 15.3 sq in of pizza unaccounted for.

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.
• Cone and funnel patterns. Unroll a cone and it flattens into a sector, which is how sheet-metal workers, lampshade makers and party-hat manufacturers mark out the blank before rolling it.
• Pizza and pie portions, where the sector area is the honest answer and the straight-sided triangle underestimates it by about 10%.

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.