Quick answer
A sector with radius r and central angle θ has area ½r²θ (θ in radians) and arc length rθ. With r = 6 and a 90° angle, the sector area is 28.2743 and the arc length 9.4248.
Enter the radius and central angle in degrees. The calculator returns the sector area and arc length.
Related tools: Arc Length Calculator: Length of a circular arc. Radius Calculator: Radius of a circle from diameter, circumference or area. Circle Diameter Calculator: Diameter of a circle from radius, circumference or area.
How to use the Sector Area Calculator
- Enter radius (above 0).
- Enter central angle (°) (between 0 and 360).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Radius | A number | > 0 |
| Central angle | A number, in ° | > 0, ≤ 360 |
Formula and method
Formula
A = ½r²θ (θ in radians); arc = rθ
IEEE-754 double precision; shown to at most 6 decimal places.
Fraction θ/360° of the full circle.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Radius = 6; Central angle = 90.
| Result | Value |
|---|---|
| Sector area | 28.274334 |
| Arc length | 9.424778 |
Example 2
Inputs: Radius = 1; Central angle = 360.
| Result | Value |
|---|---|
| Sector area | 3.141593 |
| Arc length | 6.283185 |
The first line is the headline result (sector area). Below it the sector Area Calculator lists arc length, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
As a fraction of the circle
90° is a quarter of 360°, so the sector is a quarter of the circle: π × 36 ÷ 4 = 28.274. The arc is a quarter of the circumference, 2π × 6 ÷ 4 = 9.425.
Uses
Pie slices, fan-shaped gardens and radar sweeps. Angle up to 360°. For the area between a chord and the arc, use the Circle Segment Area calculator.
How the result changes with radius
Holding every other input at the first worked example, this table changes only radius. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Radius | Sector area | Arc length |
|---|---|---|
| 3 | 7.068583 | 4.712389 |
| 6 | 28.274334 | 9.424778 |
| 12 | 113.097336 | 18.849556 |
| 24 | 452.389342 | 37.699112 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Central angle: entering "361" shows "Central angle must be at most 360." beside that field and no result.
Choosing between the Sector Area Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Arc Length Calculator](/math/arc-length-calculator/) | radius and central angle | Arc length |
| [Radius Calculator](/math/radius-calculator/) | i know the and value | Radius |
| [Circle Diameter Calculator](/math/circle-diameter/) | i know the and value | Diameter |
| [Circumference Calculator](/math/circumference/) | i know the and value | Circumference |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Angle up to 360°
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for central angle to confirm this.
Frequently asked questions
What is the area of a semicircle sector?
Half the circle, ½πr².
How do I convert degrees to radians?
Multiply degrees by π/180: 90° × π/180 = π/2 ≈ 1.5708 radians.
What is the difference from a segment?
A segment excludes the triangle.
What does the Sector Area Calculator calculate?
Area and arc length of a circular sector. It returns Sector area and Arc length.
What formula does the Sector Area Calculator use?
A = ½r²θ (θ in radians); arc = rθ. Fraction θ/360° of the full circle.
What do I need to enter in the sector Area Calculator?
You enter Radius and Central angle (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angle up to 360°.
What is an example of a sector Area result?
With radius 6 and central angle 90, the sector area is 28.274334.
Related calculators
- Arc Length Calculator — Length of a circular arc.
- Radius Calculator — Radius of a circle from diameter, circumference or area.
- Circle Diameter Calculator — Diameter of a circle from radius, circumference or area.
- Circumference Calculator — Circumference of a circle.
- Area Calculator — Area of common two-dimensional shapes.
- Area of a Circle Calculator — Area of a circle from its radius, diameter or circumference.
- Area of a Trapezoid Calculator — Area of a trapezoid from its parallel sides and height.
- Surface Area Calculator — Surface area of a sphere, cube, cylinder, cone or box.
Sources
- Standard Euclidean geometry formulas (e.g. NIST Digital Library of Mathematical Functions §4, CRC Standard Mathematical Tables)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.