How to Calculate the Area of a Sector

Calculating the area of a circular sector is like determining the exact size of a slice of pie. Because a sector is simply a fraction of a full circle, finding its area requires knowing the total area of the circle and the specific angle that the sector occupies (the central angle).

The mathematical approach changes slightly depending on whether your central angle is measured in degrees or radians. Below are the standard formulas and step-by-step examples for both methods.

Sector Area Formula (Degrees)

When working with degrees, a full circle is $360^\circ$. To find the area of a sector, you divide your central angle ($\theta$) by $360$ to find out what fraction of the circle you have, and then multiply that fraction by the standard circle area formula ($\pi r^2$).

The Formula:
$$A = \frac{\theta}{360} \times \pi r^2$$

• A = Area of the sector
• $\theta$ = Central angle in degrees
• r = Radius of the circle
• $\pi$ = Pi (approximately 3.14159)

Example Calculation (Degrees)

Imagine you have a circle with a radius of $5$ cm and a sector with a central angle of $90^\circ$.

1. Identify the variables: $r = 5$, $\theta = 90$.
2. Substitute the values into the formula: $$A = \frac{90}{360} \times \pi (5)^2$$
3. Simplify the fraction and square the radius: $$A = 0.25 \times \pi (25)$$
4. Calculate the final result: $$A \approx 19.63 \text{ cm}^2$$

Sector Area Formula (Radians)

In advanced mathematics and engineering, angles are frequently measured in radians. A full circle is $2\pi$ radians. When using radians, the sector area formula simplifies beautifully because the $\pi$ from the circle formula cancels out with the $\pi$ in the radian conversion.

The Formula:
$$A = \frac{1}{2} r^2 \theta$$

• A = Area of the sector
• r = Radius of the circle
• $\theta$ = Central angle strictly in radians

Example Calculation (Radians)

Suppose you are designing a curved garden bed (a sector) with a radius of $8$ meters and a central angle of $1.5$ radians.

1. Identify the variables: $r = 8$, $\theta = 1.5$.
2. Substitute the values into the formula: $$A = \frac{1}{2} (8)^2 (1.5)$$
3. Square the radius: $$A = 0.5 \times 64 \times 1.5$$
4. Calculate the final result: $$A = 48 \text{ m}^2$$

Checking Your Work

It is incredibly easy to mix up degrees and radians or forget to square the radius. To avoid manual errors, or if you need to calculate complex decimal measurements quickly, you can use our area of the sector of a circle calculator. It instantly applies the correct formula based on your chosen unit and provides a step-by-step breakdown of the math.