Finding the area of a sector of a circle is a key skill in geometry because it helps you understand how much of a circle’s space is being measured. A sector is often described as a “slice” of a circle, formed by two radii and the arc between them. To find its area, you need the radius of the circle and the central angle. With those two measurements, you can use a simple formula and move from a full circle’s area to just a part of it.
What Is a Sector?
A sector of a circle is the region inside the circle that is bounded by two straight lines from the center, called radii, and the curved line between them, called an arc. Think of a pizza slice: the crust is the arc, and the two cut lines are the radii. The angle between those two radii is the central angle Easy to understand, harder to ignore..
Counterintuitive, but true.
The entire circle represents a full sector with a central angle of 360 degrees or 2π radians. Practically speaking, the area of the full circle is πr², where r is the radius. A sector is simply a fraction of that full area.
The Formula for the Area of a Sector
There are two common versions of the sector area formula, depending on whether the central angle is measured in degrees or radians.
Formula Using Degrees
When the central angle is in degrees, the formula is:
Area of Sector = (θ / 360) × πr²
In this formula:
- θ = central angle in degrees
- r = radius of the circle
- π ≈ 3.14159
So if the central angle is 90 degrees, the sector takes up 90/360 = 1/4 of the circle’s area That's the part that actually makes a difference..
Formula Using Radians
When the central angle is in radians, the formula is:
Area of Sector = (θ / 2) × r²
Or more simply:
**Area of Sector = 0
The Formula for the Area of a Sector (Continued)
Formula Using Radians
When the central angle is in radians, the formula simplifies to:
Area of Sector = (1/2) × r² × θ
Here, θ is the central angle in radians. On top of that, this formula is derived from the fact that a full circle corresponds to 2π radians, so a sector with angle θ (in radians) represents a fraction θ / 2π of the entire circle. Multiplying this fraction by the total area πr² gives:
(θ / 2π) × πr² = (1/2) × r² × θ.
Solving Problems with Sector Areas
Example 1: Using Degrees
A sector has a radius of 10 cm and a central angle of 45°. Find its area.
Solution:
Using the degrees formula:
Area = (θ / 360) × πr²
= (45 / 360) × π × 10²
= (1/8) × π × 100
= **12.5π cm²
