Direct answer

What are the main circle formulas?

The main circle formulas are area $A = π r^{2}$ , circumference $C = 2 π r$ or $C = π d$ , diameter $d = 2 r$ , and radius $r = d /2$ . Use radius for area; if diameter or circumference is given, convert to radius first.

Short answer

Circle formulas at a glance

Area: $A = π r^{2}$ Circumference: $C = 2 π r = π d$ Diameter: $d = 2 r$

Definition

What is a circle?

A circle is a 2D shape made of all points that are the same distance from a center point.

Diagram

Circle diagram

Circle variables: radius r, diameter d, circumference C, and center.

Quick formula table

Circle Formulas quick table

Formula	Equation	Variables
Area of a circle	$A = π r^{2}$	r = radius
Circumference from radius	$C = 2 π r$	r = radius
Circumference from diameter	$C = π d$	d = diameter
Diameter	$d = 2 r$	r = radius
Radius from diameter	$r = d /2$	d = diameter

Formula explanation

How to use these formulas

The radius is the key circle measurement. If you know diameter or circumference, convert it to radius first, then use the area formula.

Calculator

Circle Calculator

Enter measurements to get the result, unit, formula, and calculation steps.

Examples

Step-by-step examples

Given radius, find area

Given r = 5 cm.
Use $A = π r^{2}$ .
$A = π \times 5^{2} = 25 π \approx 78.54 c m^{2}$ .

Given diameter, find circumference

Given d = 12 m.
Use $C = π d$ .
$C = 12 π \approx 37.70 m$ .

Given circumference, find radius

Given C = 31.4 in.
Use $r = C /2 π$ .
$r = 31.4/6.283... \approx 5 in$ .

Given area, find radius

Given A = 100π cm².
Use $r = A / π$ .
$r = 100 π / π = 10 c m$ .

Real-world circle problem

A round table has diameter 4 ft.
Use $A = π (d /2)^{2}$ .
$A = π \times 2^{2} = 4 π \approx 12.57 f t^{2}$ .

Reverse calculations

Solve backwards from known values

Task	Formula	When to use it
Find radius from area	$r = A / π$	Use when the area is known.
Find radius from circumference	$r = C /2 π$	Use when the circumference is known.
Find diameter from circumference	$d = C / π$	Use when the circumference is known.
Find area from diameter	$A = π (d /2)^{2}$	Use when the diameter is known.

Common mistakes

Using diameter in the area formula without dividing by 2 first.
Writing circumference in square units. Circumference uses linear units.
Rounding pi too early when the problem asks for an exact answer.

Practice

Practice problems with answers

1. A circle has radius 7 cm. Find its area.

$A = π \times 7^{2} = 49 π \approx 153.94 c m^{2}$ .

2. A circle has diameter 18 m. Find its circumference.

$C = π d = 18 π \approx 56.55 m$ .

3. A circle has circumference 20π in. Find its radius.

$r = C /2 π = 20 π /2 π = 10 in$ .

4. A circle has area 64π ft². Find its diameter.

$r = 64 π / π = 8 f t$ , so $d = 16 f t$ .

5. A round rug has radius 3 ft. How much floor area does it cover?

$A = 9 π \approx 28.27 f t^{2}$ .

Related formulas

FAQ

Circle Formulas FAQ

What is the formula for area of a circle?

The area formula is $A = π r^{2}$ , where r is the radius.

What is the circumference formula?

Circumference can be found with $C = 2 π r$ or $C = π d$ .

How do you find radius from diameter?

Divide the diameter by 2: $r = d /2$ .