Geometry · Circle Properties
Circle
Calculator
Enter any one circle value — radius, diameter, circumference, or area — and instantly get all other properties. Includes sector, arc, chord, inscribed shapes, and a live SVG diagram.
—
Radius
—
Area
—
Circumference
Solve Any Circle
💡 Enter any ONE value below — radius, diameter, circumference, or area. Leave the others blank. All will be calculated automatically.
Quick Load (radius)
Sector / Arc (optional angle)
°
Circle Diagram
Radius r
—
Diameter d
—
Circumference C
—
Area A
—
📐 All Circle Properties
📋 Step-by-Step Derivations
🥧 Sector & Arc Properties (θ = 90°)
π (Pi) — The Heart of Every Circle
Every circle calculation uses π ≈ 3.14159265358979… — a transcendental number with infinite non-repeating decimals. It's the ratio of any circle's circumference to its diameter: C/d = π, always.
π = 3.1415926535 89793 23846 26433 83279 50288 41971
69399 37510 58209 74944 59230 78164 06286 20899
86280 34825 34211 70679 82148 08651 32823 06647
09384 46095 50582 23172 53594 08128 48111 74502
69399 37510 58209 74944 59230 78164 06286 20899
86280 34825 34211 70679 82148 08651 32823 06647
09384 46095 50582 23172 53594 08128 48111 74502
C/d = π
Circumference ÷ Diameter
A/r² = π
Area ÷ Radius²
π = 4·(1−1/3+1/5−1/7+…)
Leibniz series
355/113 ≈ π
Accurate to 6 decimal places
Circle Formulas & Properties
A circle is the set of all points equidistant from a center point. That distance is the radius r. All four fundamental circle measurements — radius, diameter, circumference, and area — can be derived from any single one.
All Circle Formulas
From radius r:
Diameter: d = 2r
Circumference: C = 2πr = πd
Area: A = πr²
From diameter d: r = d/2
From circumference C: r = C/(2π), A = C²/(4π)
From area A: r = √(A/π), C = 2√(πA)
Sector (angle θ in degrees):
Arc length: L = (θ/360) × 2πr
Sector area: A_s = (θ/360) × πr²
Chord length: c = 2r·sin(θ/2)
Inscribed square side: s = r√2
Inscribed equilateral Δ side: s = r√3
How do you find the radius from the area?
From A = πr², solve for r: r = √(A/π). For example, Area = 78.54: r = √(78.54/3.14159) = √25 = 5. You can also go the other direction: if you know the circumference C, radius r = C/(2π). All four circle values are interrelated through π, so knowing any one gives you all others.
What is a sector and how is its area calculated?
A sector is a "pie slice" of a circle — the region between two radii and the arc connecting them. For angle θ (in degrees): Sector Area = (θ/360) × πr². This makes intuitive sense: a 90° sector is 1/4 of the circle, so its area is (90/360) × πr² = πr²/4. The arc length of the sector is L = (θ/360) × 2πr. For θ in radians: Sector Area = r²θ/2, Arc = rθ.
Why is π irrational and transcendental?
π is irrational — it cannot be expressed as a ratio of two integers (proved by Lambert in 1761). It's also transcendental — it's not a root of any polynomial with rational coefficients (proved by Lindemann in 1882). This means you can't "square the circle" (construct a square with equal area to a given circle using compass and straightedge alone) — this was one of antiquity's great unsolved problems, resolved by transcendence of π.
What is the largest square that fits in a circle?
The largest square inscribed in a circle of radius r has side length s = r√2 (the diagonal of the square equals the diameter 2r). Area of inscribed square = 2r². This is 2/π ≈ 63.7% of the circle's area. Conversely, the smallest circle circumscribed around a square of side s has r = s√2/2 = s/√2, circumference = πs√2.