Area of Sector Calculator
Enter the radius and the central angle or arc length to find the area of a sector.
Updated
Area of Sector Calculator
Our Area of Sector Calculator finds the area of a slice of a circle. Enter the radius and the central angle in degrees or radians, or the radius and the arc length. It also gives the arc length, perimeter, chord and the share of the full circle.
Quick answer
The area of a sector is A = (θ ÷ 360) × πr²: (90 ÷ 360) × π × 10² = 25π ≈ 78.54 for r = 10 and θ = 90°.
- Angle in radians: A = ½r²θ, so r = 10 and θ = 1.5 rad give 75
- From the arc length s: A = ½rs, so r = 10 and s = 15 give 75
- Special angles: 90° is a quarter of the circle, 180° is half and 360° is the whole circle
- Angle from the area: θ = 360A ÷ (πr²)
What Is a Sector of a Circle?
A sector is the part of a circle between two radii and the arc that joins their ends. It looks like a slice of pizza or a piece of a pie chart. Its size depends on two things: the radius r and the central angle θ between the two radii.
A sector with an angle under 180° is a minor sector. The rest of the circle, with an angle of 360° − θ, is the major sector. Together they always make the whole circle.
Area of a Sector Formulas
A = ½r²θ (θ in radians)
A = ½rs (s = arc length)
| You know | r = 10 example |
|---|---|
Angle in degrees(θ ÷ 360) × πr² | θ = 90°: ≈ 78.54 |
Angle in radians½r²θ | θ = 1.5: 75 |
Arc length s½rs | s = 15: 75 |
All three are the same formula. An angle of θ radians makes an arc of s = rθ, and 360° equals 2π radians, so (θ ÷ 360) × πr² turns into ½r²θ and then into ½rs.
Why the Formula Works
A sector is a fraction of the whole circle, and that fraction is its angle out of the full turn: θ ÷ 360. Take that fraction of the circle's area πr² and you have the sector area.
A 90° sector is 90 ÷ 360 = ¼ of the circle, so its area is a quarter of πr². A 120° sector is a third and a 180° sector is a semicircle.
Sector Area From an Angle in Degrees
Divide the angle by 360, then multiply by π and by the radius squared.
A = (90 ÷ 360) × π × 10²
A = ¼ × 100π = 25π
A ≈ 78.539816
| Angle (r = 10) | Area |
|---|---|
| 30° | 25π/3 ≈ 26.18 |
| 45° | 25π/2 ≈ 39.27 |
| 60° | 50π/3 ≈ 52.36 |
| 90° | 25π ≈ 78.54 |
| 120° | 100π/3 ≈ 104.72 |
| 180° | 50π ≈ 157.08 |
| 270° | 75π ≈ 235.62 |
| 360° | 100π ≈ 314.16 |
Sector Area From an Angle in Radians
With the angle in radians the formula is shorter: half the radius squared times the angle. The calculator also accepts angles written with π, such as π/3 or 2pi/3.
A = ½ × 10² × 1.5
A = 50 × 1.5 = 75
With θ = π/2 (the same as 90°) the result is ½ × 100 × π/2 = 25π, the same answer as in degrees.
Sector Area From the Arc Length
When you know the length of the curved edge instead of the angle, multiply it by the radius and halve the result. No π is needed.
A = ½ × 10 × 15
A = 75
The angle follows from θ = s ÷ r = 1.5 rad, about 85.94°. The arc can be at most the whole circumference, 2πr.
Sector vs Segment
A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. The segment is the sector minus the triangle formed by the two radii and the chord.
= ½r²(θ − sin θ)
r = 10, θ = 90°:
25π − 50 ≈ 28.54
Angle or Radius From the Area
Rearrange the formula to go backwards. For the angle in degrees use θ = 360A ÷ (πr²). For the radius use r = √(2A ÷ θ) with θ in radians.
θ = 18000 ÷ (100π)
θ ≈ 57.30° (1 rad)
A = 75, θ = 1.5 rad:
r = √(150 ÷ 1.5) = 10
Real Example: A Pizza Slice
A 14 inch pizza has a radius of 7 inches. Cut into 8 equal slices, each slice is a sector of 360 ÷ 8 = 45°.
A = (45 ÷ 360) × π × 7²
A = 49π/8
A ≈ 19.24 in²
Eight slices give 8 × 49π/8 = 49π ≈ 153.94 in², the area of the whole pizza.
Common Mistakes
- Mixing degrees and radians. ½r²θ needs radians; (θ ÷ 360) × πr² needs degrees.
- Using the diameter as r. The radius is half the width across the circle.
- Forgetting to square r. The area grows with r², not r.
- Confusing sector and segment. The segment leaves out the triangle at the centre.
Key Terms
- Sector
- The part of a circle between two radii and an arc.
- Central angle
- The angle θ at the centre between the two radii.
- Arc (s)
- The curved edge of the sector. s = rθ (θ in radians).
- Radian
- The angle that makes an arc as long as r. 360° = 2π rad.
- Segment
- The area between a chord and its arc.
How to Use the Area of Sector Calculator
- Choose what you know. Pick Radius & angle or Radius & arc length.
- Enter the radius and the angle or arc. Type the radius, such as 10, and pick a unit if you want one. Then type the angle and choose deg or rad, or type the arc length. Decimals, fractions such as 3/4 and radians such as π/3 work.
- Read and copy the result. The sector area appears as you type, with a drawing, the working, the exact value in terms of π, the arc length, perimeter and chord. Press Copy to copy it, or Clear or Esc to start again.
Example: a lawn sprinkler waters a 270° arc with a reach of 3 m. Enter 3, pick m, and type 270. It covers 27π/4 ≈ 21.205750 m².
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the area of a sector?
Divide the central angle by 360 and multiply by the area of the whole circle: A = (θ ÷ 360) × πr². A radius of 10 and an angle of 90° give ¼ × 100π = 25π ≈ 78.54.
What is the formula for the area of a sector in radians?
A = ½r²θ with θ in radians. A radius of 10 and an angle of 1.5 rad give ½ × 100 × 1.5 = 75.
How do you find the area of a sector from the arc length?
Use A = ½rs, half the radius times the arc length. A radius of 10 and an arc of 15 give ½ × 10 × 15 = 75.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, like a pizza slice. A segment is bounded by a chord and an arc. The segment equals the sector minus the triangle at the centre: for r = 10 and 90°, 25π − 50 ≈ 28.54.
How do you find the central angle from the sector area?
Use θ = 360A ÷ (πr²). A sector with an area of 50 and a radius of 10 has an angle of 18000 ÷ (100π) ≈ 57.30°, which is 1 radian.
How do you find the radius from the sector area?
Use r = √(2A ÷ θ) with θ in radians. An area of 75 and an angle of 1.5 rad give √(150 ÷ 1.5) = √100 = 10.
What is the area of a slice of pizza?
A 14 inch pizza cut into 8 slices gives sectors of 45° with a radius of 7 in, so each slice is (45 ÷ 360) × π × 49 = 49π/8 ≈ 19.24 in².
What is the perimeter of a sector?
Two radii plus the arc: P = 2r + s, where s = rθ in radians. For r = 10 and 90°, P = 20 + 5π ≈ 35.71.
Is the Area of Sector Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Area of Sector Calculator to find the area of any slice of a circle from the radius and the angle or arc length.