How to Calculate the Area of a Sector of a Circle

A sector of a circle is a planar region bounded by two radii and the corresponding arc that connects their endpoints. Understanding the quantitative relation between the area of a sector, the central angle, and the radius is fundamental in elementary and advanced geometry.

Mathematical Definition of the Area of a Sector in Terms of Central Angle

Let a circle have radius $r$ and center $O$. Consider two radii $OA$ and $OB$ forming an angle $\theta$ at the center, and the arc $AB$ between $A$ and $B$. The sector $OAB$ is the region enclosed by $OA$, $OB$, and arc $AB$. The area of the sector depends on the measure of $\theta$ and the radius $r$ of the circle.

Closed-Form Formula for Area of a Sector (Angle in Degrees)

If the central angle $\theta$ is given in degrees, then the area $A$ of the sector is equal to the area of a full circle, multiplied by the ratio of the given angle to the full angle of a circle, i.e., $360^\circ$:

For a circle of radius $r$, the area of the full circle is $A_{\text{circle}} = \pi r^2$.

The fraction of the circle represented by the sector is $\dfrac{\theta}{360^\circ}$.

Formula (Degrees): The area of the sector when $\theta$ is in degrees is \[ A = \frac{\theta}{360^\circ} \pi r^2 \] where $0 < \theta \leq 360^\circ$.

Derivation of the Sector Area Formula with Central Angle in Degrees

The area of the entire circle, corresponding to a central angle of $360^\circ$, is $A_{\text{circle}} = \pi r^2$.

For a central angle of $1^\circ$, the corresponding sector covers $\dfrac{1}{360}$ of the full circle. The area $A_{1^\circ}$ of such a sector is:

\[ A_{1^\circ} = \frac{1}{360} \pi r^2 \]

For a central angle of $\theta$ degrees, the area $A_{\theta^\circ}$ is:

\[ A_{\theta^\circ} = \theta \times \frac{1}{360} \pi r^2 \]

Write this product as a simplified fraction:

\[ A_{\theta^\circ} = \frac{\theta}{360} \pi r^2 \]

Closed-Form Formula for Area of a Sector (Angle in Radians)

When the central angle $\theta$ is given in radians, the relationship adjusts to reflect that the total angle in one circle is $2\pi$ radians. The fraction of the circle represented by $\theta$ radians is $\dfrac{\theta}{2\pi}$. Thus, the area $A$ is:

\[ A = \frac{\theta}{2\pi} \pi r^2 = \frac{1}{2} r^2 \theta \]

Formula (Radians): If $\theta$ is in radians, the area of the sector is \[ A = \frac{1}{2} r^2 \theta \] where $0 < \theta \leq 2\pi$.

Stepwise Derivation of the Sector Area Formula Using Radians

The area of a full circle of radius $r$ is $A_{\text{circle}} = \pi r^2$, corresponding to a central angle of $2\pi$ radians.

The fraction of a full circle occupied by angle $\theta$ (in radians) is $\dfrac{\theta}{2\pi}$.

Multiply the total area by this fraction:

\[ A_{\theta} = \frac{\theta}{2\pi} \pi r^2 \]

Expand numerator and denominator: \[ A_{\theta} = \frac{\theta \cdot \pi r^2}{2\pi} \]

The $\pi$ terms in numerator and denominator cancel: \[ A_{\theta} = \frac{\theta \cdot r^2}{2} \]

Write this as: \[ A_{\theta} = \frac{1}{2} r^2 \theta \]

Area of a Sector Expressed in Terms of Arc Length

Let $l$ denote the length of the arc $AB$ subtending angle $\theta$ at the center of a circle of radius $r$. The area $A$ of the sector can also be related directly to $l$ and $r$.

The arc of a circle subtending angle $\theta$ radians at the center has length $l = r\theta$.

From the previously derived area formula $A = \dfrac{1}{2} r^2 \theta$, and substituting $\theta = \dfrac{l}{r}$, gives:

\[ A = \frac{1}{2} r^2 \cdot \frac{l}{r} = \frac{1}{2} r l \]

Result: The area of a sector specified by arc length $l$ and radius $r$ is $A = \dfrac{1}{2} r l$.

Area of Sector for a Major and Minor Sector

If the central angle $\theta$ satisfies $0 < \theta < 180^\circ$, the corresponding sector is called a minor sector. If $180^\circ < \theta < 360^\circ$, the sector is called a major sector. The area formulas remain unchanged; only the value of $\theta$ varies according to the classification.

Computational Example: Area of Sector with Central Angle in Degrees

Given: A circle with radius $r = 6$ units. Central angle $\theta = 120^\circ$.

Use the degrees formula: \[ A = \frac{\theta}{360^\circ} \pi r^2 \]

Substitute values: $\theta = 120^\circ$, $r = 6$: \[ A = \frac{120}{360} \pi (6^2) \]

Calculate the powers and fraction: \[ A = \frac{1}{3} \pi \cdot 36 \]

\[ A = 12\pi \]

Final result: The area of the sector is $12\pi$ square units.

Computational Example: Area of Sector with Central Angle in Radians

Given: A circle with radius $r = 8$ cm. Central angle $\theta = \dfrac{\pi}{2}$ radians.

Use the radians formula: \[ A = \frac{1}{2} r^2 \theta \]

Substitute: $r = 8$, $\theta = \dfrac{\pi}{2}$: \[ A = \frac{1}{2} \times 8^2 \times \frac{\pi}{2} \]

Calculate the power: \[ A = \frac{1}{2} \times 64 \times \frac{\pi}{2} \]

\[ A = 32 \times \frac{\pi}{2} \]

\[ A = 16\pi \]

Final result: The area of the sector is $16\pi$ square centimeters.

Computational Example: Area of a Sector with Given Arc Length and Radius

Given: A sector in a circle of radius $10$ cm, with arc length $l = 5\pi$ cm.

Use: \[ A = \frac{1}{2} r l \]

Substitute: $r = 10$, $l = 5\pi$: \[ A = \frac{1}{2} \cdot 10 \cdot 5\pi = 5 \cdot 5\pi \]

\[ A = 25\pi \]

Final result: The area of this sector is $25\pi$ square centimeters.

Relationship to Full Circle and Semicircle

When $\theta = 360^\circ$ or $\theta = 2\pi$ radians, the area of the sector coincides with the area of the entire circle, $A = \pi r^2$. When $\theta = 180^\circ$ or $\theta = \pi$ radians, the area is that of a semicircle, $A = \dfrac{1}{2}\pi r^2$. These cases are direct consequences of the general formulas.

Key Formulas for Area of a Sector

Degrees: $A = \dfrac{\theta}{360^\circ} \pi r^2$, with $\theta$ in degrees.

Radians: $A = \dfrac{1}{2} r^2 \theta$, with $\theta$ in radians.

Arc Length: $A = \dfrac{1}{2} r l$, with $l$ as arc length.

For full details on the area of a full circle, refer to the Area Of A Circle Formula page.

For related geometric area results, see the Area Of Equilateral Triangle Formula and Area Of A Rhombus Formula pages.