Sector Area Calculator

Use A = (1/2) x r^2 x theta when theta is in radians, or A = (theta / 360) x pi x r^2 when theta is in degrees. Both formulas give the same answer; the radian version drops the 360 because a full turn is 2 pi radians. For r = 10 and theta = 90 degrees, A = (90 / 360) x pi x 100 = 25 pi, which is roughly 78.54 square units.

Degrees split a full turn into 360 equal parts; radians measure an angle as the ratio of arc length to radius, so a full turn equals 2 pi radians (about 6.2832). Convert with theta_rad = theta_deg x pi / 180 and theta_deg = theta_rad x 180 / pi. The calculator accepts either, converts internally to radians for the math, and reports both values so you always see the same angle in both units.

A sector is the pie slice bounded by two radii and the arc between them. A segment is the smaller region you get by cutting a sector along the chord and keeping only the part between the chord and the arc. Segment area equals sector area minus the area of the triangle formed by the two radii and the chord: A_segment = (1/2) r^2 (theta - sin(theta)). This calculator reports the sector value, plus the chord and arc length you need to compute the segment if you want it.

Arc length is s = r x theta when theta is in radians, or s = (theta / 180) x pi x r when theta is in degrees. The formula falls straight out of the definition of a radian: one radian is the angle that subtends an arc equal in length to the radius. For a 10-unit radius and a 90-degree angle, s = (90 / 180) x pi x 10 = 5 pi, about 15.71 units. The sector perimeter shown in the result is just 2r + s (the two straight radii plus the arc).

Chord length c = 2 r sin(theta / 2) is the straight-line distance between the two endpoints of the arc, and it matters anywhere you need to build a physical span rather than follow the curve. Examples include the cross-tie on a curved fence section, the rise-and-run of a circular arch, the seating chord on a dome, the chord of an arch bridge, and CNC toolpath planning when a straight cut replaces a short arc. At theta = 180 degrees the chord equals the diameter 2r; at theta = 360 degrees it collapses to zero because the arc loops back to its starting point.