How do you find the area and circumference of a circle?

The area of a circle is A = πr², where r is the radius and π ≈ 3.14159. The circumference (perimeter) is C = 2πr, or equivalently C = πd where d is the diameter. Both formulas require only the radius — if you have the diameter, divide by 2 first. For a circle with radius 5: • Area: A = π × 5² = π × 25 = 78.5398 square units • Circumference: C = 2 × π × 5 = 31.4159 units • Diameter: d = 2 × 5 = 10 units Common mistakes to avoid: • Using diameter instead of radius in A = πr² (gives 4× the correct area) • Confusing circumference (the distance around) with area (the space inside) • Forgetting that area is in square units and circumference is in linear units • This calculator solves from any known measurement — radius, diameter, circumference, or area

How do you find the radius from the area or circumference?

To find radius from area: rearrange A = πr² → r = √(A/π). To find radius from circumference: rearrange C = 2πr → r = C/(2π). To find radius from diameter: r = d/2. Worked examples: • From area A = 100: r = √(100/π) = √(31.831) = 5.6419 units • From circumference C = 50: r = 50/(2π) = 50/6.2832 = 7.9577 units • From diameter d = 14: r = 14/2 = 7 units Once you have the radius, all other measurements follow: • Area = π × r² • Circumference = 2π × r • Diameter = 2 × r This calculator is bidirectional — select which measurement you know and enter it. The calculator solves for all other properties instantly.

What is the formula for arc length and sector area?

An arc is a portion of the circumference. A sector is the pie-slice area bounded by two radii and an arc. Both depend on the central angle θ (in degrees). Arc length = (θ/360) × 2πr = (θ/360) × C Sector area = (θ/360) × πr² = (θ/360) × A Examples for r = 5 and θ = 90°: • Arc length = (90/360) × 2π × 5 = (1/4) × 31.4159 = 7.8540 units • Sector area = (90/360) × π × 25 = (1/4) × 78.5398 = 19.6350 units² Special cases by angle: • θ = 90°: quarter circle — arc = πr/2, sector = πr²/4 • θ = 180°: semicircle — arc = πr, sector = πr²/2 • θ = 360°: full circle — arc = 2πr (circumference), sector = πr² (area) • Enter any angle in the optional θ field in this calculator to compute arc length and sector area

What is π (pi) and why does it appear in circle formulas?

π (pi) is the ratio of a circle's circumference to its diameter: π = C/d. This ratio is the same for every circle regardless of size — a fundamental constant of geometry. Its value is approximately 3.14159265358979... and it is an irrational number (its decimal never repeats or terminates). Why π appears in circle formulas: • Circumference C = πd: by definition — π is this ratio • Area A = πr²: derived by integration — treating the circle as infinitely many thin rings • Arc length: proportional to the full circumference, so π appears again • Sector area: proportional to the full area, so π appears again Practical approximations: • 3.14159 (5 decimal places) — good for most calculations • 22/7 ≈ 3.14286 — simple fraction approximation • 355/113 ≈ 3.14159292 — very accurate fraction approximation • This calculator uses JavaScript's full precision π (15+ significant figures)

What is the difference between radius, diameter, and circumference?

The radius (r) is the distance from the centre of a circle to any point on its edge. The diameter (d) is the distance across the circle through the centre — exactly twice the radius: d = 2r. The circumference (C) is the total distance around the edge of the circle — the perimeter of the circle: C = 2πr = πd. Key relationships: • d = 2r (diameter is twice the radius) • C = πd = 2πr (circumference uses π) • A = πr² (area uses radius squared) • C/d = π always (defining property of π) Real-world examples: • Pizza with diameter 30cm: r = 15cm, C = 94.25cm, A = 706.86cm² • Wheel with circumference 2m: r = 2/(2π) = 0.318m, d = 0.637m • Swimming pool (circular) with area 200m²: r = √(200/π) = 7.98m, C = 50.13m • This calculator converts between any of these measurements instantly

Circle Calculator — Area, Circumference, Diameter & Arc

The Circle Calculator finds every property of a circle from any single known measurement. Choose what you know — radius, diameter, circumference, or area — enter its value, and the calculator instantly computes all four properties. Enter an optional central angle to also calculate the arc length and...

CIRCLE CALCULATOR

SOLVE FROM

Radius (r)

Must be greater than 0

Central Angle θ (optional) — for arc length & sector area

RADIUS (r)

units

DIAMETER (d)

units

CIRCUMFERENCE (C)

31.415927

units

AREA (A)

78.539816

units²

STEP-BY-STEP

Given: radius r = 5

Diameter: d = 2r = 2 × 5 = 10

Circumference: C = 2πr = 2 × π × 5 = 31.415927

Area: A = πr² = π × 5² = 78.539816

QUICK EXAMPLES

LIVE DIAGRAM

EXACT EXPRESSIONS

Circumference5π × 2 = 31.415927

Areaπ × 5² = 78.539816

C/d ratioπ = 3.14159265...

Created with❤️byeaglecalculator.com

CIRCLE FORMULAS

ALL CIRCLE FORMULAS

NAME	FORMULA	DESCRIPTION
Area	A = πr²	Area enclosed by the circle
Circumference	C = 2πr = πd	Distance around the circle (perimeter)
Diameter	d = 2r	Distance across the circle through the centre
Radius from area	r = √(A/π)	Solve for radius given the area
Radius from circumference	r = C/(2π)	Solve for radius given the circumference
Arc length	L = (θ/360) × 2πr	Length of an arc for central angle θ°
Sector area	S = (θ/360) × πr²	Area of a pie-slice sector for angle θ°
π definition	π = C/d ≈ 3.14159265	Ratio of circumference to diameter — same for every circle

HOW TO USE

1
Select what you know: radius (r), diameter (d), circumference (C), or area (A). The calculator solves from any of these four measurements.
2
Enter your value in the input field. The calculator instantly shows all four circle properties: radius, diameter, circumference, and area.
3
Optionally enter a central angle θ in degrees to calculate the arc length and sector area for that portion of the circle.
4
Check the live diagram — the circle is drawn with labelled radius and circumference. If you entered an angle, the arc is highlighted in red and the sector is shaded.
5
Use the quick example buttons to load common circle sizes and see how the bidirectional solving works across all four input types.

WORKED EXAMPLE

Example 1 (from radius): r=5. d=10, C=2π×5=31.4159, A=π×25=78.5398. Example 2 (from diameter): d=10. r=5, C=π×10=31.4159, A=π×25=78.5398. Example 3 (from area): A=100. r=√(100/π)=5.6419, d=11.2838, C=35.4491. Example 4 (arc/sector): r=5, θ=90°. Arc=7.8540, Sector area=19.6350.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.