Circles • Topic 3 of 4
Arcs & Sectors
An arc is a portion of the circumference; a sector is the pie-slice region between two radii. Both come from one proportion: the piece is the same fraction of the whole circle as its central angle is of 360°. So arc length = (θ/360) × 2πr, and sector area = (θ/360) × πr². Recognise the friendly fractions — a 90° angle is a quarter of the circle, 120° is a third, 60° is a sixth, 180° is a semicircle — and the arithmetic becomes trivial. The single most common GRE slip is using the arc formula when the question wants the sector area, or vice versa; read whether it asks for a length (arc) or a region (sector) before you compute.
✅ Solved examples
1. A circle has a radius of 6. Find the length of a 90° arc.
Arc = (90/360) × 2πr = ¼ × 12π = 3π.
2. A circle has a radius of 6. Find the area of a 90° sector.
Sector = (90/360) × πr² = ¼ × 36π = 9π.
3. A circle has a radius of 10. Find the length of a 72° arc.
Arc = (72/360) × 2πr = (1/5) × 20π = 4π.
4. A quarter-circle has a radius of 8. Find its area.
A quarter-circle is a 90° sector: ¼ × π × 8² = ¼ × 64π = 16π.
✏️ Practice — try these, take hints as needed
1. A circle has a radius of 4. Find the length of a 90° arc.
90° is ¼ of the circle.
Full circumference = 8π.
¼ × 8π.
2π
2. A circle has a radius of 9. Find the area of a 120° sector.
120° is ⅓ of the circle.
Full area = 81π.
⅓ × 81π.
27π
3. A circle has a radius of 12. Find the length of a 30° arc.
30/360 = 1/12.
Circumference = 24π.
(1/12) × 24π.
2π
4. A semicircle has a radius of 5. Find the length of its arc.
A semicircle is half the circumference.
Full C = 10π.
Half of 10π.
5π
5. A circle has a radius of 6. Find the area of a 60° sector.
60° is 1/6 of the circle.
Full area = 36π.
(1/6) × 36π.
6π
📝 Topic test — 8 questions
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Formula Reference Sheet
This chapter
Circle basics
| Diameter | d = 2r |
|---|---|
| Circumference | C = 2πr = πd |
| Area | A = πr² |
Arcs & sectors (θ = central angle)
| Arc length | (θ / 360) × 2πr |
|---|---|
| Sector area | (θ / 360) × πr² |
Angles in a circle
| Central angle | = measure of the arc it cuts off |
|---|---|
| Inscribed angle | = ½ × its intercepted arc (½ of the central angle on the same arc) |
| Angle in a semicircle | = 90° (diameter subtends a right angle) |
GRE reference
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