Sector Area Calculator

A sector is the pie-slice-shaped region of a circle bounded by two radii and the arc between them. The angle formed at the center between the two radii is called the central angle. If the central angle is less than 180°, it's a minor sector; if greater than 180°, it's a major sector.

When the central angle is in degrees, the formula is: Area = (θ / 360) × π × r². When the angle is in radians, it simplifies to: Area = ½ × r² × θ. Both formulas calculate what fraction of the full circle the sector represents and multiply by the total circle area.

Arc length is the curved outer edge of the sector. In degrees: Arc Length = (θ / 360) × 2πr. In radians: Arc Length = r × θ. For example, a sector with radius 7 and a 90° angle has an arc length of (90/360) × 2π × 7 ≈ 10.996 units.

No — an arc is just the curved line segment along the edge of a circle. A sector is the full enclosed region that includes two straight radii and the arc between them. Think of an arc as the crust of a pizza slice, and the sector as the entire slice including the dough.

A 90° sector is exactly one quarter of a circle, so its area is (1/4) × π × 1² = π/4 ≈ 0.7854 square units. This is also called a quadrant of the circle.

Rearrange the area formula to solve for θ. In degrees: θ = (Area × 360) / (π × r²). In radians: θ = (2 × Area) / r². Simply plug in the known sector area and radius to find the central angle.

If you know the arc length (s) and the radius (r), you can find the area using: Area = (r × s) / 2. This is derived by substituting the arc length formula into the sector area formula, and it works regardless of whether the angle is in degrees or radians.

Sector area calculations come up in pie chart design (sizing chart slices), engineering (calculating material in circular components), agriculture (irrigation coverage of rotating sprinklers), and architecture (designing arched features or curved floor plans). Any scenario involving a partial circle likely involves sector area.