Area of a Circle Calculator

Calculate the area of a circle from its radius, diameter, or circumference. Get all measurements with step-by-step solutions.

Select Input & Enter Value

r d

Result

Area of Circle
--
square units
Area--
Area (exact)--
Circumference--
Circumference (exact)--
Radius (r)--
Diameter (d)--

Step-by-Step Solution

A = π × r²

Understanding Circle Area

A circle is a perfectly round shape defined as the set of all points in a plane that are equidistant from a fixed center point. This constant distance is called the radius.

The Number Pi (π)

π (pi) is the ratio of a circle's circumference to its diameter. It is an irrational number:

π ≈ 3.14159265358979323846...

Area Formula Derivation

A = πr²

Imagine cutting a circle into many thin wedges and rearranging them into a rectangle with height r and width πr. Area = r × πr = πr².

Related Formulas

From diameter: A = πd²/4
From circumference: A = C²/(4π)
Circumference: C = 2πr = πd

Real-World Applications

  • Construction: Calculating circular patios, pools, and garden beds for materials estimation.
  • Pizza Math: A 14-inch pizza has ~twice the area of a 10-inch pizza (153.9 vs 78.5 sq inches).
  • Engineering: Cross-sectional areas of pipes, cylinders, and shafts for flow and pressure calculations.
  • Agriculture: Center-pivot irrigation coverage area.
  • Astronomy: Cross-sectional area of celestial bodies and telescope mirrors.

Common Circle Measurements

For a unit circle (r = 1): Area = π ≈ 3.1416, Circumference = 2π ≈ 6.2832, Diameter = 2. The unit circle is fundamental in trigonometry.