Circles — Circumference, Area, Arc & Sector
1. Core Idea
Every circle formula is built from the radius. The first move in any circle problem is to find r — from the diameter, the circumference, or the area.
The second big idea: an arc or sector is just a fraction of the whole circle, and that fraction is central angle / 360.
2. Must-Know Rules
| Concept | Formula |
|---|---|
| Diameter | d = 2r |
| Circumference | C = 2 pi r = pi d |
| Area | A = pi r^2 |
| Arc length (angle t degrees) | (t/360) x 2 pi r |
| Sector area (angle t degrees) | (t/360) x pi r^2 |
| Area from circumference | A = C^2/(4 pi) |
The fraction principle. A 90-degree sector is 90/360 = 1/4 of the circle: one quarter of the area, one quarter of the circumference.
Useful angle fractions:
| Angle | Fraction |
|---|---|
| 30 | 1/12 |
| 45 | 1/8 |
| 60 | 1/6 |
| 90 | 1/4 |
| 120 | 1/3 |
| 180 | 1/2 |
Scaling. If the radius doubles, the circumference doubles but the area quadruples. Area scales with r^2.
Value of pi: about 3.14, or 22/7 for hand computation. GRE answers usually keep pi symbolic — leave it in.
3. Worked Examples
Example 1 — Working backwards to r
A circle has circumference 18 pi. Find its area.
2 pi r = 18 pi -> r = 9.
Area = pi(81) = 81 pi.
Example 2 — Sector and arc
A circle has radius 12. Find the arc length and sector area for a central angle of 60 degrees.
Fraction = 60/360 = 1/6.
Arc = (1/6)(2 pi x 12) = (1/6)(24 pi) = 4 pi.
Sector area = (1/6)(pi x 144) = 24 pi.
Example 3 — Shaded region
A square of side 10 has a circle inscribed in it. What is the area of the region inside the square but outside the circle?
The inscribed circle has diameter 10, so r = 5.
Square area = 100. Circle area = 25 pi.
Shaded = 100 - 25 pi (about 21.5).
4. GRE Traps
- Using the diameter as the radius. If a question gives d = 14, then r = 7. Halving is forgotten constantly.
- Arc length formula with the diameter. Arc uses 2 pi r; get r right first.
- Confusing chord with arc. A chord is a straight segment; an arc is part of the curve. An arc is always longer than its chord.
- Doubling the radius doubles the area. It quadruples it.
- Sector area vs arc length. One is a piece of the area (r^2), the other a piece of the perimeter (r). Different formulas.
- Perimeter of a sector. It is arc + 2 radii, not just the arc.
- Approximating pi too early. Keep it symbolic; the answer choices usually contain pi.
5. Speed Tricks
- Find r first, always. Write it down before touching any other formula.
- Turn the central angle into a fraction of 360 and apply it to the whole-circle quantity. One method covers arcs and sectors alike.
- A = C^2/(4 pi) goes straight from circumference to area with no r step.
- Inscribed circle in a square: diameter = side. Circumscribed circle around a square: diameter = the square's diagonal = s sqrt(2).
- For shaded regions, compute the outer figure minus the inner figure. Never try to compute the odd shape directly.
- Doubling the radius: circumference x2, area x4. Tripling: x3 and x9.
6. Self-Check
Q1. A circle has diameter 20. Find its circumference and area.
Q2. A circle of radius 6 has a sector with central angle 120 degrees. Find the sector area.
Q3. A circle has area 49 pi. Find its circumference.
Answers
A1. C = 20 pi; r = 10 so A = 100 pi.
A2. Fraction 1/3; area = (1/3)(36 pi) = 12 pi.
A3. r = 7, so C = 14 pi.
7. One-Line Summary for the Board
Find r first. Arc and sector are just (angle/360) of the whole. Double the radius, quadruple the area.