How to use the circle sector area calculator
- Enter the radius of the circle.
- Enter the central angle that defines the sector.
- Select whether the angle is in degrees or radians.
- Read the computed sector area, arc length, chord length and segment area below.
Formulas used by this circle sector area calculator
A circle sector is the region bounded by two radii and the arc between them. When the central angle theta is given in radians and the radius is r, the sector area is calculated as:
A = (1/2) * r^2 * theta
If the angle is entered in degrees, it is first converted: theta_rad = theta_deg * pi / 180. For example, with r = 10 and an angle of 90 degrees, theta in radians is 1.5707963. The sector area is then 0.5 * 100 * 1.5707963 = 78.5398.
The arc length uses L = r * theta, and the chord connecting the two endpoints of the arc uses c = 2 * r * sin(theta / 2). The circular segment area is simply the sector area minus the triangle formed by the two radii and the chord: A_seg = A_sector - (1/2) * r^2 * sin(theta).
Common sector values reference table
The table below shows exact and approximate results for a circle with radius 1 at several standard angles. Multiply the area by r squared and the lengths by r to scale to any circle.
| Angle | Radians | Sector area (r=1) | Arc length (r=1) | Chord (r=1) |
|---|---|---|---|---|
| 30 deg | pi/6 | 0.2618 | 0.5236 | 0.5176 |
| 45 deg | pi/4 | 0.3927 | 0.7854 | 0.7654 |
| 60 deg | pi/3 | 0.5236 | 1.0472 | 1.0000 |
| 90 deg | pi/2 | 0.7854 | 1.5708 | 1.4142 |
| 180 deg | pi | 1.5708 | 3.1416 | 2.0000 |
| 360 deg | 2*pi | 3.1416 | 6.2832 | 0.0000 |
Practical applications of sector calculations
Knowing how to find the area of a sector is useful far beyond geometry homework. Engineers use sector area when designing curved structural elements, gears and cam profiles where material volume depends on the swept angle. In agriculture, pivot irrigation systems water a circular sector, so calculating the exact area determines water usage per rotation.
Pie charts are perhaps the most familiar visual application. Each slice is a sector whose area represents a proportion of the whole. If a category accounts for 25 percent of the total, its central angle is 90 degrees, and its sector area is exactly one quarter of the full circle area. Surveyors also rely on arc length and chord measurements when laying out curved roads or property boundaries defined by a radius and sweep angle.
Sector versus segment: understanding the difference
A common source of confusion is mixing up the sector and the segment. The sector includes the triangular region formed by the two radii plus the curved cap. The segment is only the curved cap, the region between the chord and the arc.
To get the segment area, subtract the triangle area from the sector area. The triangle formed by the two radii and the chord has area (1/2) * r^2 * sin(theta). So the segment area formula is A_segment = (1/2) * r^2 * (theta - sin(theta)), where theta must be in radians. For small angles the segment area is much smaller than the sector area, but as the angle approaches 180 degrees the triangle shrinks relative to the sector and the two areas converge.
Edge cases and common mistakes
When using any circle sector area calculator, watch out for these pitfalls:
- Degrees versus radians: Entering 90 when the tool expects radians gives a completely wrong answer because 90 radians wraps around the circle over 14 times. Always verify the selected unit.
- Angles above 360 degrees: A sector is conventionally defined for angles between 0 and 360 degrees (or 0 and 2 pi radians). Angles beyond that describe overlapping sweeps rather than a simple sector.
- Zero angle: A sector with a zero degree angle has zero area, zero arc length and zero chord length. It degenerates into a single radius line.
- Full circle: At exactly 360 degrees the chord length returns to zero because the two radii overlap, and the segment area equals the full circle area since the triangle collapses.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Quarter circle (90 degrees) | Radius: 10; Central angle: 90; Angle unit: deg | Sector area: 78.53981634; Arc length: 15.70796327; Chord length: 14.14213562; Segment area: 28.53981634 |
| Semicircle (pi radians) | Radius: 5; Central angle: 3.14159265358979; Angle unit: rad | Sector area: 39.26990817; Arc length: 15.70796327; Chord length: 10; Segment area: 39.26990817 |
| 60 degree sector | Radius: 12; Central angle: 60; Angle unit: deg | Sector area: 75.39822369; Arc length: 12.56637061; Chord length: 12; Segment area: 13.04439461 |
| Full circle (360 degrees) | Radius: 7; Central angle: 360; Angle unit: deg | Sector area: 153.93804; Arc length: 43.98229715; Chord length: 1.714505519e-15; Segment area: 153.93804 |
FAQ
What is the formula for the area of a circle sector?
The sector area is A = (1/2) * r^2 * theta, where r is the radius and theta is the central angle in radians. If your angle is in degrees, convert it first by multiplying by pi / 180.
How do I calculate arc length from radius and angle?
Arc length equals L = r * theta, with theta in radians. For degrees, use L = r * theta_deg * pi / 180.
What is the difference between a sector and a segment?
A sector is the pie-slice shape bounded by two radii and an arc. A segment is the region between the arc and the chord connecting its endpoints, excluding the triangular part.
How is chord length calculated for a sector?
The chord length is c = 2 * r * sin(theta / 2), where theta is the central angle in radians. This comes from splitting the isosceles triangle formed by the two radii into two right triangles.
Can the central angle be greater than 360 degrees?
In standard geometry a sector is defined for angles between 0 and 360 degrees. Angles larger than 360 degrees imply multiple full rotations and do not form a simple planar sector.
Does this tool work offline?
Yes. This circle sector area calculator runs entirely in your browser. No data is sent to any server, so it works offline once the page has loaded.