What is the formula for the circumference of a circle?

The circumference of a circle is calculated using C = 2πr, where r is the radius and π (pi) is approximately 3.14159. An equivalent form using diameter is C = πd, where d = 2r. For example, a circle with a radius of 5 cm has a circumference of 2 × π × 5 = 31.416 cm.

What is the formula for the area of a circle?

The area of a circle is calculated using A = πr², where r is the radius and π is approximately 3.14159. For example, a circle with a radius of 5 cm has an area of π × 5² = π × 25 = 78.540 cm². If you know the diameter instead of the radius, use A = π(d/2)² or equivalently A = πd²/4.

How do I find the radius from the circumference?

Rearrange C = 2πr to give r = C / (2π). For example, if the circumference is 31.416 cm, the radius is 31.416 / (2 × 3.14159) = 5 cm. Once you have the radius, you can calculate the area as A = πr². This calculator accepts circumference as an input and computes all other properties automatically.

Circumference and Area of a Circle Calculator

Calculate the circumference and area of a circle from any known measurement. Enter the radius, diameter, or circumference and all other circle properties are computed instantly using the standard formulas C = 2πr and A = πr².

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Standard formula Uses exact value of π (Math.PI in JavaScript, approximately 3.14159265358979).

1. Input

Solve from

Radius (r)

Diameter (d = 2r)

Circumference (C)

Please enter a positive number.

2. Units

Unit of measurement

Decimal places

Circle Measurements

Radius (r)

5.000

Diameter (d)

10.000

Circumference (C)
31.416
cm

Area (A)
78.540
cm²

Full Calculation

Input usedRadius = 5 cm

Radius (r)5.000 cm

Diameter (d = 2r)10.000 cm

Circumference (C = 2πr)31.416 cm

Area (A = πr²)78.540 cm²

Formulas Applied

CircumferenceC = 2πr

AreaA = πr²

Diameter from radiusd = 2r

Radius from diameterr = d / 2

Radius from circumferencer = C / (2π)

Value of π used3.14159265358979…

Worked Example (default: radius = 5 cm)

Step 1: Diameter = 2 × r = 2 × 5 = 10.000 cm

Step 2: Circumference = 2 × π × r = 2 × 3.14159 × 5 = 31.416 cm

Step 3: Area = π × r² = 3.14159 × 25 = 78.540 cm²

Circumference and Area of a Circle: Explained

A circle is defined by a single measurement: its radius (the distance from the centre to the edge). From the radius, every other property of a circle follows exactly, with no approximation beyond the use of π.

The Two Key Formulas

The circumference is the perimeter of a circle: the total distance around the outside. The formula is:

C = 2πr (or equivalently, C = πd, where d is the diameter)

The area is the amount of flat space enclosed by the circle. The formula is:

A = πr²

Both formulas use π (pi), the ratio of any circle's circumference to its diameter. Pi is an irrational number: it cannot be expressed as a simple fraction and its decimal expansion never repeats. To many decimal places, π = 3.14159265358979…

Solving from Different Inputs

Known value	Radius	Diameter	Circumference	Area
Radius (r)	r	2r	2πr	πr²
Diameter (d)	d / 2	d	πd	π(d/2)²
Circumference (C)	C / (2π)	C / π	C	C² / (4π)

Practical Applications

These formulas appear in a wide range of everyday situations:

Fencing a round garden: The circumference tells you how many metres of fencing you need.
Turf or paving: The area tells you how many square metres of material to order.
Engineering and manufacturing: Wheel circumferences determine how far a vehicle travels per revolution. Pipe cross-sections use circle areas to calculate flow rates.
Cooking: A round cake tin's area determines how much batter you need, or how to scale a recipe when switching between tin sizes.
School maths: Circle problems frequently appear in NCEA Level 1 and Level 2 geometry.

Relationship Between Circumference and Area

The circumference and area are related through the radius. If you double the radius of a circle, the circumference doubles (linear relationship) but the area quadruples (squared relationship). This is why small differences in radius can have a large effect on area but a proportionally smaller effect on circumference.

Related Calculators

Maths and Stats Calculators: all geometry and statistics tools.
Circumference Calculator: circumference from radius or diameter.
Area of a Circle Calculator: area from radius, diameter, or circumference.
Arc Length and Sector Area Calculator: partial circle calculations.
Annulus Calculator: area and perimeter of a ring (annulus) between two circles.

Method: Standard Euclidean geometry. Formulas: C = 2πr, A = πr². Pi used: JavaScript Math.PI (IEEE 754 double precision, 15–17 significant digits). Reference: Euclid's Elements, Book III; any standard mathematics textbook.

This calculator uses standard geometric formulas. Results are exact within the precision of floating-point arithmetic. For very large or very small values, verify results with the formulas given.

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