Circle Formulas
Reference for circle formulas: area A = πr², circumference C = 2πr, arc length, sector area, and chord length.
Covers unit circle with worked examples.
Basic Circle Formulas
Area = π × r²
Circumference = 2 × π × r = π × d
Diameter = 2 × r
Circumference = 2 × π × r = π × d
Diameter = 2 × r
Arc and Sector Formulas
Arc Length = (θ / 360) × 2πr (degrees)
Arc Length = θ × r (radians)
Sector Area = (θ / 360) × πr² (degrees)
Sector Area = ½ × θ × r² (radians)
Arc Length = θ × r (radians)
Sector Area = (θ / 360) × πr² (degrees)
Sector Area = ½ × θ × r² (radians)
Variables
| Symbol | Meaning |
|---|---|
| r | Radius (distance from center to edge) |
| d | Diameter (distance across the circle through center, d = 2r) |
| π | Pi, approximately 3.14159 |
| θ | Central angle (in degrees or radians) |
| C | Circumference (perimeter of the circle) |
| A | Area enclosed by the circle |
Example 1 — Area and Circumference
Find the area and circumference of a circle with radius 7 cm.
Area = π × 7² = π × 49
Area = 153.94 cm²
Circumference = 2 × π × 7
Circumference = 43.98 cm
Example 2 — Arc Length
Find the arc length for a 60° angle in a circle with radius 10 m.
Arc Length = (60 / 360) × 2π × 10
= (1/6) × 62.83
Arc Length = 10.47 m
Example 3 — Sector Area
Find the area of a sector with a 90° angle and radius 8 inches.
Sector Area = (90 / 360) × π × 8²
= (1/4) × π × 64
Sector Area = 50.27 in²
When to Use These
- Area: Finding how much space a circular region covers (gardens, pools, pizza sizes)
- Circumference: Finding the distance around a circle (fencing, trim, wire)
- Arc length: Measuring a portion of the circumference (curved paths, clock hands)
- Sector area: Measuring a "slice" of the circle (pie slices, radar coverage)
Key Notes
- Core formulas: A = πr²; C = 2πr; sector area = r²θ/2 (radians); arc length = rθ: All circle measurements flow from a single radius. The sector area and arc length formulas are proportional to the central angle θ — doubling the angle doubles both.
- Chord length: chord = 2r sin(θ/2): The straight-line distance between two points on the circle. For θ = π (semicircle), chord = 2r = diameter. Chord length is always ≤ arc length; they are equal only at θ = 0.
- Inscribed angle theorem: An inscribed angle (vertex on the circle) is exactly half the central angle that subtends the same arc. Thales' theorem is a special case: any inscribed angle in a semicircle is 90°.
- Power of a point: For a point P and a circle, if two chords through P have segments a,b and c,d respectively, then a×b = c×d. For a point outside the circle: (tangent length)² = external segment × whole secant. These properties unify many circle problems.
- Applications: Circle formulas are used in engineering (pipe and gear geometry), architecture (arch and dome design), satellite coverage area calculation (a circle on a sphere's surface), GPS accuracy circles, and road/railway curve design (radius of curvature determines safe speed).