Why is the area πr² and not 2πr?

2πr is the length of the boundary, while πr² is the surface it encloses. Cutting a circle into thin sectors and lining them up into a shape of base πr and height r produces the πr² formula.

What is the difference between radius and diameter?

The radius runs from the centre to the edge, the diameter crosses the whole circle through the centre, so D = 2r. Mixing them up is the most common reason an area comes out four times too large or too small.

Is π really infinite?

π is irrational: it cannot be written as a fraction of whole numbers and its decimals never repeat. Practice needs only a few digits - NASA uses about 15 for interplanetary navigation.

⭕ Circle Area Calculator

Calculate area and circumference of a circle from its radius.

Radius (r)

What is a Circle?

A circle is the set of all points in a plane equidistant from a fixed point called the center.

How to Calculate Area

Area Formula

S = π × r²

S: Area | r: Radius | π ≈ 3.14159...

radius × radius × pi

How to Calculate Circumference

L = 2πr = πD

L: Circumference | D: Diameter (D = 2r)

radius × 2 × pi = diameter × pi

Pi (π)

The ratio of a circle’s circumference to its diameter. Approximately 3.14159265..., an irrational number with infinite non-repeating decimals.

Archimedes and Circle Area

Ancient mathematician Archimedes used inscribed and circumscribed polygons to compare perimeters and calculate π with remarkable accuracy.

Newton's Infinite Series for π

Isaac Newton devised a method to compute π more precisely using infinite series. Advances in mathematics have since extended π to billions of digits.

March 14 is Pi Day (π Day)

Every year, March 14 is celebrated as “Pi Day”, originating from the value 3.14. Pi plays a central role in mathematics, physics, engineering and beyond.

Worked Examples

Radius 5 gives an area of 78.5398, a circumference of 31.4159 and a diameter of 10.

A 30 cm pizza has r = 15 and therefore 706.86 cm² of surface - four times a 15 cm pizza, because area grows with the square of the radius.

Limits of This Calculation

The calculator uses double-precision π, so rounding only appears at display time. No unit conversion is applied: the result keeps the unit you type, and the area comes out in that unit squared. Sectors, chords and segments need their own formulas, and an ellipse is not a circle - its area is πab.

Frequently Asked Questions

Why is the area πr² and not 2πr?: 2πr is the length of the boundary, while πr² is the surface it encloses. Cutting a circle into thin sectors and lining them up into a shape of base πr and height r produces the πr² formula.
What is the difference between radius and diameter?: The radius runs from the centre to the edge, the diameter crosses the whole circle through the centre, so D = 2r. Mixing them up is the most common reason an area comes out four times too large or too small.
Is π really infinite?: π is irrational: it cannot be written as a fraction of whole numbers and its decimals never repeat. Practice needs only a few digits - NASA uses about 15 for interplanetary navigation.