Area of a Circle
Formal Definition
The area of a circle is the measure of the region enclosed by the circle in a two-dimensional plane. It represents the total number of square units that fit inside the circle.
$$ A = \pi r^2 $$
where:
- $A$ = area of the circle (in square units)
- $\pi$ (pi) ≈ 3.14159265... (an irrational constant — the ratio of circumference to diameter)
- $r$ = radius of the circle (distance from center to any point on the circle)
Equivalent Forms of the Formula
Since the diameter $d = 2r$, we can also express the area as:
Using radius:
$A = \pi r^2$
$A = \pi r^2$
Using diameter:
$A = \frac{\pi d^2}{4}$
$A = \frac{\pi d^2}{4}$
Using circumference:
$A = \frac{C^2}{4\pi}$
$A = \frac{C^2}{4\pi}$
These are all algebraically equivalent. The first form ($A = \pi r^2$) is the most commonly used.