Circular Segment from a Chord
Half the chord over the radius is the sine of the half-angle, which gives the central angle. The segment is the sector minus the triangle, so the area is r²/2 × (angle − sin angle), with the angle in radians.
Your numbers
Area
65.4cm²
Angle (°)
73.74°
Sagitta
4cm
Arc length
25.74cm
Formula
Angle (°) = 2 × asin(Chord length ÷ (2 × Radius))
A chord cannot exceed the diameter. Enter one that does and the angle saturates at 180°.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Radius (cm)Centre to edge — half the diameter. | 20 | 0.01 and up |
| Chord length (cm)The straight line between two points on a circle. | 24 | 0 and up |
Step by step
Quick reference table
Results when only Radius (cm) changes and everything else stays put.
| Radius (cm) | Area (cm²) | Angle (°) (°) | Sagitta (cm) |
|---|---|---|---|
| 10 | 157.08 | 180 | 10 |
| 15 | 100.641 | 106.26 | 6 |
| 20 | 65.4 | 73.74 | 4 |
| 30 | 40.42 | 47.16 | 2.505 |
| 40 | 29.617 | 34.92 | 1.842 |
What each result means
| Result | At default values |
|---|---|
| Area (cm²)The space a shape covers — length squared. | 65.4 |
| Angle (°) (°)An angle in degrees; a full turn is 360°. | 73.74 |
| Sagitta (cm)The height from the middle of the chord to the arc — how much it bulges. | 4 |
| Arc length (cm)The length of the curved piece cut from a circle’s edge. | 25.74 |
Common mistakes
A chord cannot exceed the diameter. Enter one that does and the angle saturates at 180°.
Glossary
- Radius
- Centre to edge — half the diameter.
- Chord length
- The straight line between two points on a circle.
- Area
- The space a shape covers — length squared.
- Angle (°)
- An angle in degrees; a full turn is 360°.
- Sagitta
- The height from the middle of the chord to the arc — how much it bulges.
- Arc length
- The length of the curved piece cut from a circle’s edge.
Frequently asked questions
Q. How is Circular Segment from a Chord calculated?
Angle (°) = 2 × asin(Chord length ÷ (2 × Radius)) — Half the chord over the radius is the sine of the half-angle, which gives the central angle. The segment is the sector minus the triangle, so the area is r²/2 × (angle − sin angle), with the angle in radians.
Q. Can you walk through an example?
With Radius 20cm, Chord length 24cm, the answer is Area 65.4cm².
Q. What do I need to enter?
Enter Radius, Chord length. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Radius (cm) moves the answer to Radius (cm) 10 → Area (cm²) 157.08 and Radius (cm) 40 → Area (cm²) 29.617. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
A chord cannot exceed the diameter. Enter one that does and the angle saturates at 180°.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.