The Area of A Sector Calculator computes the exact area of a pie-slice (a sector) using your radius and central angle. Enter the angle in degrees or radians, and the calculator returns the sector area in square units.
What Is a Sector and What “Area” Means
A sector is the region between two radii of a circle, plus the arc between them. The central angle is the angle where those two radii meet at the circle’s center. The sector’s area is the “slice” you’d get if you cut a pizza from the center.
To calculate sector area, you need two inputs:
- Radius (r): distance from the center to the arc.
- Central angle (θ): the angle that opens the sector.
Core Formula (Degrees vs. Radians)
The area of a sector is a fraction of the full circle’s area. The full circle area is πr². The fraction depends on the angle.
Angle Given in Degrees
If your angle is in degrees, use:
A = (θ / 360) × πr²
Here, θ is the central angle in degrees, and r is the radius in your chosen length unit.
Angle Given in Radians
If your angle is in radians, use:
A = (1/2) × r² × θ
In this form, θ is the central angle in radians (where a full circle is 2π radians).
Variable Guide (So You Know What to Plug In)
| Symbol | Meaning | Typical Units |
|---|---|---|
| r | Radius | cm, m, in, ft, etc. |
| θ | Central angle | degrees (°) or radians (rad) |
| A | Sector area | square of the radius unit |
How Unit Conversion Works (Square Units)
If you enter radius in one length unit (like meters) and want area in another (like square centimeters), you must convert correctly.
Because area scales with the square of length, conversions use squared factors. For example:
- 1 meter = 100 centimeters
- So 1 square meter = (100)² = 10,000 square centimeters
The calculator handles this by converting the radius to the target length unit first, then applying the sector area formula.
Common Edge Cases (Quick Checks)
- θ = 360° (or 2π rad): the sector is the whole circle, so A = πr².
- θ = 0° (or 0 rad): the sector collapses to a line, so A = 0.
- θ > 360°: the formula still works, but the “sector” wraps more than one full turn. Many problems restrict θ to 0–360°.
- Negative angles or radius: these are not valid for typical geometry problems. The calculator flags invalid inputs.
Practical Examples (Use Cases)
Example 1: Pizza Slice for a Party
You have a pizza with radius r = 12 in. You want the area of a slice with θ = 45°. Use the degrees formula:
A = (45 / 360) × π × 12²
This gives the slice area in square inches, which you can use to estimate how many slices you need for a given number of guests.
Example 2: Garden Path Curved Section
A curved garden path forms a sector with radius r = 2.5 m and central angle θ = 1.2 rad. Use the radians formula:
A = (1/2) × r² × θ
This returns the area in square meters, helping you estimate materials like gravel or paving stones.
How to Use the Area of A Sector Calculator
- Enter the radius value.
- Select the radius unit (cm, m, in, ft, etc.).
- Choose whether your angle is in degrees or radians.
- Enter the central angle.
- Select the area output unit (square of your preferred unit).
- Click Calculate to get the sector area.
Frequently Asked Questions
How do I find the area of a sector if the angle is in degrees?
Use A = (θ/360) × πr². Here θ is the central angle in degrees and r is the radius. This works because the sector is a fraction of the full circle area πr². Make sure r is in the length unit you want squared.
What formula should I use if my angle is in radians?
Use A = (1/2) × r² × θ when θ is in radians. This comes directly from the circle’s arc-length relationship and the fact that a full circle is 2π. If you have degrees, convert them to radians first, or use the degrees formula.
Why does area use squared units?
Area is a two-dimensional measure, so it scales with the square of length. If you convert radius from meters to centimeters, the area must use the square conversion factor. That is why 1 m² equals 10,000 cm², not just 100 cm.
Can the sector area be larger than the circle’s area?
Yes, if the angle θ is greater than 360° (or greater than 2π radians), the formula still returns a value. In real geometry problems, sectors typically use 0° to 360°. For θ above one full turn, interpret the result as a wrapped sector area.
What happens when the angle is zero or the radius is zero?
If θ = 0, the sector collapses to a line and the area is 0. If r = 0, the circle’s center and arc coincide, so the sector has no area. These cases are valid inputs and should produce A = 0.
Bottom Line
The Area of A Sector Calculator gives you sector area instantly and accurately. Provide a valid radius and central angle, and choose your units to get the result in the square units you need.
