Parts of a circle
An arc is part of the circumference; a sector is the pie-slice region bounded by two radii and an arc. Both depend on the angle θ at the centre, as a fraction θ/360 of the whole circle.
Key idea
Arc length = (θ/360) × 2πr | Sector area = (θ/360) × πr²
Perimeter of a sector
The perimeter of a sector is the arc length plus the two straight radii: perimeter = arc + 2r.
Worked example
A sector has radius 14 cm and angle 90° (π = 22/7). Arc length = (90/360) × 2 × 22/7 × 14 = ¼ × 88 = 22 cm. Sector area = (90/360) × 22/7 × 196 = ¼ × 616 = 154 cm². Perimeter = 22 + 2 × 14 = 50 cm.
Remember
- A semicircle is θ = 180°, a quarter circle is θ = 90°.
- To find the angle from a known arc, rearrange: θ = (arc / circumference) × 360.
From angle to area
The whole circle corresponds to θ = 360°, so a sector is simply the fraction θ/360 of it. This lets you reverse any calculation: given an arc length or a sector area, first work out the fraction of the circle, then multiply by 360 to recover the angle. Always keep the radius in the same length unit as the arc, and give area answers in square units.