Area of a Circle
Calculate the area of a circle using A = πr².
Learn how to find the area from the radius or diameter with step-by-step examples.
The Formula
A = πr²
This formula calculates the area enclosed by a circle.
If you know the diameter instead, use r = d / 2 first.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the circle |
| π | Pi, approximately 3.14159 |
| r | Radius — the distance from the center to the edge |
Example 1
Find the area of a circle with radius 7 cm
A = πr² = π × 7²
A = π × 49
A ≈ 153.94 cm²
Example 2
A circular garden has a diameter of 12 meters. What is its area?
Diameter = 12 m, so radius = 12 / 2 = 6 m
A = πr² = π × 6²
A = π × 36
A ≈ 113.10 m²
When to Use It
Use the area of a circle formula when:
- Calculating the surface area of circular objects (plates, wheels, tables)
- Determining how much material is needed to cover a circular area
- Comparing the size of circles with different radii
- Working with circular cross-sections in engineering or science
Key Notes
- Area scales as r² — doubling the radius quadruples the area, not doubles it; this surprises many people when comparing circle sizes
- Units must be consistent and will be squared in the result: if r is in meters, A is in m²; if r is in feet, A is in ft²
- When working from diameter: use A = π(d/2)² = πd²/4 — it's easy to forget to halve the diameter before squaring
- Do not confuse area (πr²) with circumference (2πr) — area covers the interior; circumference measures the boundary
Key Notes
- Formulas: A = πr²; C = 2πr = πd: The area and circumference are related by dA/dr = 2πr = C — the circumference is the derivative of the area with respect to radius. Adding a thin ring of width dr adds area dA = C × dr = 2πr dr.
- Proof by integration: Dividing the circle into concentric rings of width dr, each with area 2πr × dr, and integrating from 0 to R gives A = ∫₀ᴿ 2πr dr = πR². This derivation makes the formula feel inevitable rather than arbitrary.
- Area of an annulus (ring): A = π(R² − r²) = π(R + r)(R − r): The factored form is useful when the ring width (R − r) and average radius (R + r)/2 are known. A thin annulus (R ≈ r) has A ≈ 2πr × (R − r) = circumference × width.
- Squaring the circle — proven impossible: Constructing a square with the same area as a circle using only compass and straightedge is impossible — proved in 1882 when Lindemann showed π is transcendental (not a root of any polynomial with integer coefficients).
- Applications: Circle area appears in pipe and cable cross-section calculations (flow capacity ∝ A = πr²), radar and radio coverage areas, circular field irrigation, lens aperture calculations, wheel contact patch estimation, and statistical distributions (the normal distribution bell curve integrates to πA through completing the square).