Circle Area Calculator
The circle area formula, explained and computed: enter the radius to get the area, exact (in terms of π) and decimal, with steps.
How to use this calculator
Enter the radius and press Calculate. The calculator substitutes into A = πr², shows the exact multiple-of-π form and the decimal area, and walks through the substitution.
How it works
A = πr². Intuition: a circle of radius r fits inside a square of side 2r (area 4r²) and contains about 78.5% of it, that fraction is π/4. The formula is one of the oldest in mathematics, and the calculator keeps π symbolic until the final decimal so nothing is lost to early rounding.
Worked example
Example: r = 2
Step 1, Substitute: A = π × 2².
Step 2, Simplify: A = 4π.
Step 3, Decimal: A ≈ 12.566 square units.
Common mistakes
- Squaring the diameter: with diameter d, the area is π(d/2)² = πd²/4, halve first, then square.
- Confusing area with circumference: area is πr² (square units); circumference is 2πr (linear units). The r² vs r is the tell.
- Rounding π too early: using 3.14 in intermediate steps compounds error, keep π symbolic until the end, as the calculator does.
When to use it
Circle area sizes pizzas, pipes (cross-section), sprinkler coverage, wheels, and round tables. It is also the base area inside the cylinder and cone volume formulas, master this and those come free.
Frequently asked questions
A = πr²: pi times the radius squared.
Halve the diameter to get r, then A = πr². Equivalently, A = πd²/4.
Area (πr²) measures the surface inside the circle in square units; circumference (2πr) measures the distance around it in linear units.
It can be shown by rearranging the circle into a near-rectangle (the classic wedge proof) or with calculus. The calculator’s methodology page sketches the reasoning.