Area of a Sector Calculator
An area of a sector calculator finds the area of a portion of a circle formed by two radii and the arc between them. It is useful in geometry and in calculations involving circular sections, angles, and curved shapes.
What Is a Sector of a Circle?
A sector is a region of a circle bounded by two radii and an arc. It can be thought of as a slice of a circle, similar in shape to a piece of pie.
The size of the sector depends on the circle's radius and the angle at the center.
Area of a Sector Formula
When the central angle is measured in degrees, the formula is:
A = (θ ÷ 360) × πr2
Here, A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle.
Example of Finding Sector Area
Suppose a circle has a radius of 8 cm and a central angle of 90°.
Substitute the values into the formula:
A = (90 ÷ 360) × π × 82
A = 1/4 × π × 64
A ≈ 50.27 cm2
So, the area of the sector is approximately 50.27 square centimeters.
Using Radians
If the central angle is given in radians instead of degrees, a different form of the formula is used:
A = 1/2 r2θ
In this formula, θ must be measured in radians. For example, a quarter circle has an angle of π/2 radians.
Common Sector Areas
| Central Angle | Fraction of Circle | Sector Area |
|---|---|---|
| 30° | 1/12 | πr2 ÷ 12 |
| 60° | 1/6 | πr2 ÷ 6 |
| 90° | 1/4 | πr2 ÷ 4 |
| 180° | 1/2 | πr2 ÷ 2 |
| 360° | 1 | πr2 |
Sector Area vs. Area of a Circle
The area of a complete circle is πr2. A sector represents only a fraction of that circle, determined by its central angle.
For example, a 90° sector represents one-quarter of a full 360° circle, so its area is one-quarter of the total circle area.
Where Is Sector Area Used?
Sector area calculations can be useful in geometry problems, circular designs, land measurements, engineering diagrams, and situations where only part of a circular region needs to be measured.
Common Mistakes to Avoid
Make sure the angle is in degrees when using A = (θ ÷ 360) × πr2. If the angle is in radians, use the radian formula instead.
Also check that the given measurement is the radius, not the diameter. If the diameter is provided, divide it by 2 before using the formula.
Quick Summary
To find the area of a sector using a degree measure, multiply the area of the full circle by the fraction represented by the central angle: A = (θ ÷ 360) × πr2. The result is expressed in square units.