The area formula
A = π · r² · (angle / 360)
Where the formula comes from
A sector is the slice of a circle between two radii, like a slice of cake. Its area is the area of the whole circle multiplied by the fraction of a full turn it spans: a ninety degree sector is a quarter of the circle and a sixty degree one a sixth. The arc follows the same proportion of the circumference. What gets confused constantly is the perimeter with the arc: the perimeter of a sector is not just the curve, it is the curve plus the two radii that close it, because the boundary of the figure runs along them. With the angle in radians the formulas simplify — area ½·r²·θ and arc r·θ — and that is exactly what the radian is for.
A worked example, step by step
Sector of radius 10 cm and 60°
- Fraction of the circle: 60 ÷ 360 = 1/6
- Area: π × 10² × 1/6 = 52.36 cm²
- Arc: 2π × 10 × 1/6 = 10.47 cm
- Perimeter: 10.47 + 2 × 10 = 30.47 cm
Watch the units: area is always in square units.
Where it turns up in practice
Slices of cake and pizza, sectors of a pie chart, sprinkler arcs, camera fields of view and sheet metal cut in a fan. The chord is the straight segment joining the ends of the arc, and it belongs not to the sector but to the circular segment, the region between the chord and the arc: in a sixty degree sector the chord is exactly as long as the radius, because the two radii and the chord form an equilateral triangle.