Area of a Circle
The area of a circle is $\pi$ times the square of its radius, representing the total space enclosed within the circle.
Formula
A = \pi r^2
Definition
The area of a circle with radius $r$ (the distance from the center to the edge) is $A = \pi r^2$, where pi is approximately $3.14$; the diameter form is $A = \pi (d/2)^2 = \pi d^2/4$. This can be derived by integrating the area of thin concentric rings, $$A = \int_0^r 2\pi t \, dt = \pi r^2,$$ or equivalently via polar integration over the full disk: $$A = \int_0^{2\pi}\int_0^r r \, dr \, d\theta = \left(\frac{r^2}{2}\right)(2\pi) = \pi r^2.$$ In higher dimensions, the volume of an n-ball of radius $r$ scales as $r^n$ times a constant depending only on $n$.
Example
A pizza with radius $6$ inches has $A = 3.14 \times 6^2 = 3.14 \times 36 = 113$ in$^2$, and a circle with diameter $14$ cm (radius $7$ cm) has $A = \pi (7)^2 = 49\pi = 153.94$ cm$^2$; doubling the radius quadruples the area because $r$ is squared. The 3-D analog is the volume of a solid sphere, $(4/3)\pi r^3$, whose surface area $4\pi r^2 = d/dr[(4/3)\pi r^3]$ illustrates the general principle that the derivative of volume with respect to radius gives surface area.
Key Insight
Why r squared? Slice a circle into many thin triangles pointing to the center and rearrange them: they form a near-rectangle with width $r$ and length equal to half the circumference ($\pi r$), giving area $\pi r \cdot r$. Among all shapes with a given perimeter, the circle has the greatest area (the isoperimetric inequality), which is why circular pipes carry the most fluid for a given amount of material and why cells and bubbles tend toward circular cross-sections. The relationship $d/dr(\text{volume}) = \text{surface area}$ holds for both disks and spheres, reflecting the geometric fact that a thin shell of thickness $dr$ contributes surface area times $dr$ to the volume.