Circle Calculator – Find Radius, Diameter, Area & Circumference

Calculate any property of a circle instantly with our free online circle calculator. Enter radius, diameter, area, or circumference and find all other measurements with formulas shown.

I know the...

Radius

Diameter

d = 2r

Area

A = πr²

Circumference

C = 2πr

Radius (r)

Examples:

Understanding Circle Properties

A circle is defined by its center point and radius – the distance from center to any point on the edge. From just the radius, you can derive everything else: diameter (twice the radius), circumference (the distance around), and area (the space inside). All these properties are connected through π (pi).

This circle calculator works backwards and forwards. Enter any one property – radius, diameter, area, or circumference – and it calculates all the others. The formulas are shown step by step so you see exactly how each value is derived.

Circle Formulas

Basic Relationships

d = 2r

Diameter equals twice the radius

r = d/2

Radius equals half the diameter

Circumference Formulas

C = 2πr

Using radius

C = πd

Using diameter

Area Formulas

A = πr²

Using radius

A = πd²/4

Using diameter

Finding Radius from Other Properties

r = d/2

From diameter

r = √(A/π)

From area

r = C/(2π)

From circumference

What Is Pi (π)?

π (pi) is the ratio of a circle's circumference to its diameter. For any circle, no matter the size, C/d always equals the same number: approximately 3.14159. This constant appears in every circle formula.

3.14159...

Decimal approximation

22/7

Common fraction

Irrational

Never repeats, never ends

Why π Appears in Circle Formulas

Since circumference = π × diameter, and diameter = 2 × radius, we get C = 2πr. For area, imagine cutting a circle into many thin wedges and rearranging them into a shape approaching a rectangle. The rectangle has height r and width πr, giving area = πr².

Pi Day

March 14 (3/14) is Pi Day, celebrating the first three digits of π. The date also happens to be Albert Einstein's birthday. Some enthusiasts celebrate at 1:59 PM for 3.14159.

Worked Examples

Example 1: Given Radius

Find diameter, area, and circumference of a circle with radius 5

Given: r = 5

Diameter: d = 2r = 2 × 5 = 10

Area: A = πr² = π × 25 ≈ 78.54

Circumference: C = 2πr = 2π × 5 ≈ 31.42

Example 2: Given Diameter

Find radius, area, and circumference of a circle with diameter 14

Given: d = 14

Radius: r = d/2 = 14/2 = 7

Area: A = πr² = π × 49 ≈ 153.94

Circumference: C = πd = π × 14 ≈ 43.98

Example 3: Given Area

Find radius, diameter, and circumference of a circle with area 50

Given: A = 50

r = √(A/π) = √(50/π) ≈ √15.92 ≈ 3.99

d = 2r ≈ 7.98

C = 2πr ≈ 2π × 3.99 ≈ 25.06

Example 4: Given Circumference

Find radius, diameter, and area of a circle with circumference 100

Given: C = 100

r = C/(2π) = 100/(2π) ≈ 15.92

d = 2r ≈ 31.83

A = πr² ≈ π × 253.3 ≈ 795.77

Real-World Applications

Wheels and Gears

The circumference tells you how far a wheel travels in one revolution. A bicycle wheel with 70 cm diameter has circumference about 220 cm – that's how far you move forward with each complete rotation. Gear ratios work the same way.

Pizza and Food

A 16-inch pizza has area π × 8² ≈ 201 square inches. A 12-inch pizza has area π × 6² ≈ 113 square inches. The larger pizza isn't 33% bigger (16 vs 12) – it's 78% bigger in actual food. Area scales with the square of the radius.

Pipes and Flow Rate

The cross-sectional area of a pipe determines how much fluid can flow through. Doubling the pipe diameter quadruples the area (and potential flow rate). This is why main water lines are much wider than branch lines.

Circular Gardens and Lawns

Planning a circular garden? The area tells you how much soil or mulch you need. A 10-foot diameter garden has area about 78.5 square feet. At 2 inches of mulch depth, you need about 13 cubic feet of material.

Clocks and Angles

The minute hand traces a circle. In 15 minutes, it covers 1/4 of the circumference. In 1 minute, it moves 6 degrees (360°/60). The relationship between time and circular motion is fundamental to how clocks work.

Frequently Asked Questions

What is the formula for the area of a circle?

Area = πr², where r is the radius. Square the radius and multiply by π. For a circle with radius 5, area = π × 25 ≈ 78.54 square units.

How do I find the circumference?

Circumference = 2πr or C = πd. Use whichever is convenient based on whether you know radius or diameter. Both give the same result since d = 2r.

What's the difference between radius and diameter?

Radius is the distance from center to edge. Diameter is the distance across the circle through the center – exactly twice the radius. Diameter is the longest possible chord in a circle.

How do I find radius from area?

Rearrange A = πr² to get r = √(A/π). Take the square root of (area divided by π). For area 100, radius = √(100/π) ≈ √31.83 ≈ 5.64.

Is circumference the same as perimeter?

Conceptually, yes – both measure the boundary length. "Perimeter" is used for polygons (straight sides). "Circumference" is specific to circles. The formulas are completely different.

Why is π used in circle formulas?

π is the fundamental ratio of circumference to diameter for any circle. It's approximately 3.14159 but continues infinitely without repeating. This constant appears naturally whenever circles are involved.

Other Free Tools

Search tools