Key takeaways
- The area of a sector is A = (θ / 360) × π × r².
- θ (theta) is the central angle in degrees, out of 360.
- A 90° sector is always exactly one-quarter of the full circle's area.
- At θ = 360°, the sector formula becomes the regular circle area formula.
In this guide
What is a sector?
Imagine cutting a slice out of a round pizza. That slice, bounded by two straight cuts from the center and the curved crust, is a sector. Its size depends on how wide the angle of the slice is.
Why does the area of a sector matter?
Sectors show up in pizza slices, pie charts, and rotating sprinklers — anywhere a full circle gets divided by angle.
A 90° sector is always exactly one-quarter of the full circle's area, no matter how big the circle is, since 90 ÷ 360 = 0.25 every time.The formula
| Symbol | What it means | Example |
|---|---|---|
| A | The area of the sector you want to find | 52.36 m² |
| θ (theta) | The central angle of the slice, in degrees | 60° |
| r | The radius of the full circle | 10 m |
How to find the area step by step
- 1Measure the radius of the full circle.
- 2Measure or find the central angle of the sector, in degrees.
- 3Divide the angle by 360. This gives the fraction of the full circle the sector represents.
- 4Find the area of the full circle.
π × r². - 5Multiply that fraction by the full circle's area.
Now try it yourself. Change the radius and angle in the calculator below and watch the sector's area update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Confusing sector area with [arc length](/calculator/arc-length-circle). Area is the flat pizza-slice-shaped region; arc length is just the curved crust edge.
- Forgetting to divide by 360. Without this step you'd get the area of the whole circle, not just the slice.
- Squaring the angle instead of the radius. Only r gets squared, never θ.
Tips and tricks
- Recognize common fractions: 90° is a quarter circle, 180° is a half, and 60° is a sixth.
- For the curved edge length of the same slice, rather than its area, use the arc length calculator.
Where you'll use it in real life
- Cooking: working out how much pizza is in one slice.
- Landscaping: calculating the area covered by a rotating lawn sprinkler.
- Data visualization: finding the area of a pie chart segment.
- Engineering: estimating material for a curved, wedge-shaped panel.
Quick summary
To find the area of a sector, divide the central angle by 360 to get a fraction of the full circle, then multiply by the full circle's area (πr²). A 90° sector is always exactly a quarter circle. Use the calculator above to check your own sector instantly.
Worked example
A rotating lawn sprinkler sprays water out to a radius of 10 meters, covering a 60° arc of the lawn at a time. How much lawn area does one arc of spraying cover?
- 01A = (θ / 360) × πr²
- 02A = (60 / 360) × π(10)²
- 03A = 0.1667 × 314.16
- 04A = 52.36 m²
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the area of a sector if the angle is given in radians?
Convert the angle to degrees first using the [degrees to radians calculator](/calculator/degrees-to-radians) in reverse (multiply radians by 180/π), or use a radian version of the formula: A = (θ/2) × r².
What's the difference between sector area and arc length?
Sector area is the flat, pie-slice-shaped region, in square units. Arc length is just the curved outer edge of that slice, in regular units. Use the [arc length calculator](/calculator/arc-length-circle) for the curved edge alone.
What if the sector angle is 360°?
Then the sector is the entire circle, and the formula simplifies to the regular [circle area formula](/calculator/area-circle), πr².
Can the sector angle be more than 180°?
Yes, a sector angle can be anywhere from just above 0° up to 360°. Angles over 180° represent a slice bigger than half the circle.
