A circle is a fundamental geometric figure in the plane, defined as the locus of all points at a fixed distance from a given point, called the centre. One of the most essential quantitative attributes of a circle is its area, representing the region enclosed by its boundary.

Mathematical Structure of the Area of a Circle

Let $C$ be a circle with centre $O$ and radius $r$. The radius is the fixed distance from the centre to any point on the circle. The diameter, denoted $d$, is the segment passing through $O$ with endpoints on the circle, and $d = 2r$.

The area of the circle, denoted $A$, is the measure of the region enclosed by $C$. The area depends solely on the measure of the radius $r$ or, equivalently, the diameter $d$. In all formulae, the constant $\pi$ refers to the mathematical constant (pi), the ratio of the circumference of any circle to its diameter. Standard approximations are $\pi \approx 3.1416$ or $\pi \approx \dfrac{22}{7}$ for computational purposes.

Standard Area Formula in Terms of Radius and Diameter

The area $A$ of a circle of radius $r$ is given by the formula

$A = \pi r^2$

Alternatively, if the diameter $d$ is given, then using $r = \dfrac{d}{2}$, the area can be expressed as

$A = \pi \left(\dfrac{d}{2}\right)^2 = \dfrac{\pi d^2}{4}$

These formulae are always expressed in terms of square units, such as $\mathrm{cm}^2$, $\mathrm{m}^2$, or $\mathrm{in}^2$.

Area Formula in Terms of Circumference

The circumference $C$ of a circle of radius $r$ is $C = 2\pi r$. Given the circumference $C$, the area is obtained as follows:

Start from $r = \dfrac{C}{2\pi}$ and substitute into $A = \pi r^2$:

$A = \pi \left(\dfrac{C}{2\pi}\right)^2$

$A = \pi \cdot \dfrac{C^2}{4\pi^2}$

$A = \dfrac{C^2}{4\pi}$

Result: If the circumference $C$ is known, $A = \dfrac{C^2}{4\pi}$.

Explicit Stepwise Derivation of the Area Formula Using Sectors Rearrangement

To rigorously derive the formula $A = \pi r^2$, consider decomposing the circle into $n$ congruent sectors of central angle $\theta_n = \dfrac{2\pi}{n}$ radians each, where $n$ is a large positive integer.

For large $n$, each sector is a thin "wedge" approximating a small triangle with base equal to the arc length $s_n = r \theta_n$ and height approximately $r$. The area of a single sector is then approximately $\dfrac{1}{2} r \cdot s_n$.

Total area for all $n$ sectors:

$\text{Total area} \approx n \cdot \dfrac{1}{2} r \cdot s_n$

Since $s_n = r\theta_n$ and $\theta_n = \dfrac{2\pi}{n}$, it follows

$s_n = r \left(\dfrac{2\pi}{n}\right)$

Thus,

$\text{Total area} \approx n \cdot \dfrac{1}{2} r \cdot r \left(\dfrac{2\pi}{n}\right)$

$= n \cdot \dfrac{1}{2} r^2 \cdot \dfrac{2\pi}{n}$

$= n \cdot r^2 \cdot \dfrac{\pi}{n}$

$= r^2 \pi$

Taking the limit as $n \to \infty$ makes the approximation exact, and the area of the circle is $A = \pi r^2$.

For an instructive geometric rearrangement, if the $n$ sectors are alternately placed "tip to base" in sequence, as $n \to \infty$ this arrangement approaches a rectangle of length $\pi r$ (half the circumference) and width $r$ (radius). The product gives the same area: $A = (\pi r) \times r = \pi r^2$.

Relationship Between Area and Other Geometric Quantities

Given any one of radius $r$, diameter $d$, or circumference $C$, the area of a circle can be expressed as follows:

In terms of radius: $A = \pi r^2$

In terms of diameter: $A = \frac{\pi d^2}{4}$

In terms of circumference: $A = \frac{C^2}{4\pi}$

For the area of a sector, see Area Of A Sector Of A Circle Formula.

Worked Examples Applying the Area of a Circle Formula

Example 1: A circle has radius $r = 7\, \mathrm{cm}$. Calculate its area.

Given: $r = 7\, \mathrm{cm}$

Substitute in the formula: $A = \pi r^2 = \pi \times 7^2$

$A = \pi \times 49$

If $\pi = \dfrac{22}{7}$, $A = \dfrac{22}{7} \times 49 = 154$

Final result: $A = 154\, \mathrm{cm}^2$

Example 2: The diameter of a circle is $d = 20\,\mathrm{mm}$. Find its area.

Given: $d = 20\,\mathrm{mm}$

$A = \dfrac{\pi d^2}{4} = \dfrac{\pi \times 20^2}{4}$

$= \dfrac{\pi \times 400}{4} = 100\pi$

For $\pi = 3.14$, $A = 314\, \mathrm{mm}^2$

Final result: $A = 314\, \mathrm{mm}^2$

Example 3: The circumference of a circle is $C = 44\,\mathrm{cm}$. Find its area.

Given: $C = 44\,\mathrm{cm}$

$A = \dfrac{C^2}{4\pi} = \dfrac{44^2}{4\pi}$

$= \dfrac{1936}{4\pi} = \dfrac{484}{\pi}$

For $\pi = 3.14$, $A \approx \dfrac{484}{3.14} \approx 154.14\, \mathrm{cm}^2$

Final result: $A \approx 154.14\, \mathrm{cm}^2$

For more formulas involving polygons, refer to Area Of Square Formula.

Distinction Between Area and Circumference

The area of a circle is a measure of the region enclosed within its boundary and is always expressed in square units. The circumference is the measure of the length around the boundary and is expressed in length units. Area grows proportionally to the square of the radius: doubling the radius quadruples the area. In contrast, the circumference grows linearly with the radius.

Units and Notation in Area Calculations

Always express the area of a circle in the appropriate square units: $\mathrm{cm}^2$, $\mathrm{m}^2$, $\mathrm{in}^2$, etc. For computational purposes, use $\pi \approx 3.14$ or $\pi \approx \dfrac{22}{7}$ as required by context.

If direct calculation is required in terms of inches or any other specified unit, be sure to use the correct formula variant and consistent units for $r$, $d$, or $C$. For alternative area formulas with other polygons, see Area Of Triangle Formula and Area Of Hexagon Formula.

Frequently Referenced Questions

Given either the radius, diameter, or circumference of a circle, the area can always be computed using the stated formulae. When more advanced circle problems arise (including sectors, segments, or inscribed figures), refer to relevant pages such as Area Of Isosceles Triangle Formula and Area Of A Rhombus Formula.