Area of a Circle
Geometry & MeasurementThe 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 is found using the formula $A = \pi r^2$, where $r$ is the radius (the distance from the center to the edge). Pi is approximately $3.14$.
Example
A pizza with radius $6$ inches: $A = 3.14 \times 6^2 = 3.14 \times 36 = 113$ in$^2$. A circle with radius $10$ cm: $A = 3.14 \times 100 = 314$ cm$^2$.
Key Insight
Why r squared? You can slice a circle into many thin triangles pointing to the center. If you rearrange them, they form a near-rectangle with width $r$ and length equal to half the circumference ($\pi r$), giving area $\pi r \cdot r$.
Definition
The area of a circle with radius $r$ is $A = \pi r^2$. 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: $$A = \int_0^r 2\pi t \, dt = \pi r^2.$$
Example
A circle with diameter $14$ cm has radius $7$ cm: $A = \pi (7)^2 = 49\pi = 153.94$ cm$^2$. Doubling the radius quadruples the area because $r$ is squared.
Key Insight
Among all shapes with a given perimeter, the circle has the greatest area (isoperimetric inequality). This is why circular pipes carry the most fluid for a given amount of material, and why cells and bubbles tend toward circular cross-sections.
Definition
The area of a disk (closed circle) of radius $r$ in $\mathbb{R}^2$ is $\pi r^2$, derived via polar integration: $$A = \int_0^{2\pi}\int_0^r r \, dr \, d\theta = \left(\frac{r^2}{2}\right)(2\pi) = \pi r^2.$$ In higher dimensions, the volume of an n-ball of radius $r$ scales as $r^n$ times a constant depending only on $n$.
Example
The 3-D analog: the volume of a solid sphere is $(4/3)\pi r^3$. The surface area is $4\pi r^2 = d/dr[(4/3)\pi r^3]$, illustrating the general principle that the derivative of volume with respect to radius gives surface area.
Key Insight
The relationship $d/dr(\text{volume}) = \text{surface area}$ holds for spheres and disks ($d/dr[\pi r^2] = 2\pi r = \text{circumference}$). This reflects the geometric fact that a thin shell of thickness $dr$ around a ball contributes surface area times $dr$ to the volume.