Circle Area Formulas – Class 10 Complete Guide

The circle area formulas calculate the space enclosed by a circle or a portion of it — covering the full circle, semicircle, sector, ring, and segment.

Quick Reference:

Full circle: $A = \pi r^2$ Semicircle: $A = \dfrac{\pi r^2}{2}$

Sector (angle in degrees): $A = \dfrac{\theta}{360} \times \pi r^2$

Sector (angle in radians): $A = \dfrac{1}{2}r^2\theta$ Ring /

Annulus: $A = \pi(R^2 - r^2)$ Segment: $A = \dfrac{r^2}{2}(\theta - \sin\theta)$ (angle in radians)

Constant: $\pi \approx 3.14159$

Used in: Geometry, mensuration, engineering, Class 10 board exams

1. Area of a Circle

The area of a full circle with radius $r$ is:

$$A = \pi r^2$$

This formula measures the total flat space enclosed inside the circle's boundary. It follows from the integral of concentric rings of width $dr$ from 0 to $r$, or from the limit of inscribed polygons.

2. Area of a Semicircle

A semicircle is exactly half a circle. Its area is:

$$A = \frac{\pi r^2}{2}$$

The perimeter of a semicircle (the full boundary, including the diameter) is $\pi r + 2r = r(\pi + 2)$.

3. Area of a Sector

A sector is a "pizza slice" — the region between two radii and the arc they subtend.

When angle $\theta$ is in degrees: $$A = \frac{\theta}{360} \times \pi r^2$$

When angle $\theta$ is in radians: $$A = \frac{1}{2}r^2\theta$$

The two forms are equivalent: substituting $\theta_{\text{rad}} = \frac{\theta_{\text{deg}} \times \pi}{180}$ converts between them.

4. Area of a Ring (annulus)

A ring (annulus) is the region between two concentric circles with radii $R$ (outer) and $r$ (inner), where $R > r$:

$$A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$$

This formula also factors as $\pi(R + r)(R - r)$ — useful when the sum and difference of the radii are given directly.

5. Area of a Circular Segment

A segment is the region between a chord and the arc it cuts off. For a central angle $\theta$ (in radians) and radius $r$:

$$A = \frac{r^2}{2}(\theta - \sin\theta)$$

The segment area equals the sector area minus the triangle area formed by the two radii and the chord.

Variable Key

Worked Examples of Circle

Example 1: Area of a circle

Find the area of a circle with radius 7 cm. (Use $\pi = \frac{22}{7}$.)

$$A = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154 \text{ cm}^2$$

Final answer: 154 cm²

Example 2: Area of a sector

A sector has radius 6 cm and central angle 60°.

$$A = \frac{60}{360} \times \pi \times 6^2 = \frac{1}{6} \times \pi \times 36 = 6\pi \approx 18.85 \text{ cm}^2$$

Final answer: $6\pi \approx 18.85$ cm²

Example 3: Area of a ring

A ring has outer radius 10 cm and inner radius 6 cm.

$$A = \pi(R^2 - r^2) = \pi(100 - 36) = 64\pi \approx 201.06 \text{ cm}^2$$

Final answer: $64\pi \approx 201.06$ cm²

Origin

The area formula $A = \pi r^2$ was rigorously proved by Archimedes of Syracuse (c. 287–212 BCE) in Measurement of a Circle, showing that the area of a circle equals that of a right triangle with legs equal to the circumference and the radius. His method of exhaustion — approximating the circle with inscribed and circumscribed polygons — was a precursor to integral calculus by nearly 2,000 years.

Common Confusions With Circle Area Formulas

The area formula uses radius squared ($r^2$), not diameter squared. Using diameter in place of radius gives four times the correct area. Always halve the diameter before substituting.

Area is in square units (cm², m²); circumference is in plain units (cm, m). A common exam error is writing the area with a non-squared unit.

The sector formula requires the angle to be in the correct form. The degree formula uses $\frac{\theta}{360}$; the radian formula uses $\frac{1}{2}r^2\theta$. Mixing the two forms without converting gives a wrong answer.