Circle Segment Calculator
Circle segment area, arc, and chord.
What it does
A segment is the cap cut off a circle by one chord. Lenses, arches, and gauge faces all reduce to radius plus central angle. The same angle also sets arc length along the rim and chord straight across, so one input triple answers three shop questions.
The tool above runs the trio. With radius 10 cm at 90 deg, segment area is 28.54 cm2 with a 15.71 cm arc and a 14.14 cm chord. A 6 cm radius at 60 deg gives 3.26 cm2 with 6.28 cm of arc. The imperial mirror at 3.9 in gives the same shape in inches.
Radians drive the math while degrees drive the display. The area subtracts the triangular wedge (sine term) from the sector, which is why small angles give thin slivers and 180 deg gives exactly half the circle. Chord uses half angle sine for the straight cut.
This page handles circular arcs only. It does not do elliptical segments, heights without angles, or material kerf. For height based layouts, convert height to angle first.
How to use
- Pick metric (cm) or imperial (in).
- Enter the circle radius.
- Enter the central angle in degrees.
- Read segment area first.
- Read arc length, then chord.
- Check 180 and 360 deg cases when relevant.
Formula
- r
- Circle radius in cm or in.
- deg
- Central angle in degrees, converted to radians inside.
- chord
- Straight cut across the arc endpoints.
Minor segment from a central angle up to 360 deg. At zero angle all three outputs are zero; at 360 the area is the full circle and chord returns to zero.
Worked example
Example: radius 10 cm at 90 deg
Defaults: metric radius 10 cm with a 90 deg central angle. Imperial mirror is 3.9 in.
- Inputs (metric default): radius 10 cm with 90 deg.
- Radians: 90 times pi / 180 is 1.5708.
- Area: 100 / 2 times (1.5708 minus 1) is 28.54 cm2.
- Arc: 10 times 1.5708 is 15.71 cm.
- Chord: 2 times 10 times sin 45 deg is 14.14 cm. Tool shows the same.
Reference table
| Radius and angle | Area | Arc |
|---|---|---|
| 10 cm at 90 deg | 28.54 cm2 | 15.71 cm |
| 6 cm at 60 deg | 3.26 cm2 | 6.28 cm |
| 10 cm at 180 deg | 157.08 cm2 | 31.42 cm |
| 3.9 in at 90 deg | 4.34 in2 | 6.13 in |
| 12 cm at 45 deg | 5.64 cm2 | 9.42 cm |
Limits
- Circular arcs only. Ellipses excluded.
- Angle input required. Height only needs conversion.
- No kerf or waste. Shop cuts need extra.
- Exact to floating point. Huge radii need care.
Common mistakes
- Feeding degrees to sine without converting.
- Reading sector as segment without the sine cut.
- Mixing radius with diameter.
- Using chord as arc length.
- Assuming 360 deg chord is the diameter (it is zero).
Try it yourself
Quick checks. Answers sit tucked under each question.
Radius 6 cm at 60 deg?
Theta is 1.0472. Area is 36 / 2 times 0.1812 is 3.26 cm2 with 6.28 cm arc and 6 cm chord.
Radius 10 cm at 180 deg?
Theta is pi. Area is half circle: 50 times pi is 157.08 cm2 with arc 31.42 cm and chord 20 cm.
Questions
What is the segment formula?
Area = r squared / 2 times (theta minus sin theta), theta in radians. Here 28.54 cm2.
How is arc length figured?
Arc = r times theta. Here 15.71 cm.
How is chord figured?
Chord = 2 r sin(theta/2). Here 14.14 cm.
What happens at 180 or 360 deg?
At 360 the segment is the full circle pi r squared. At 180 it is half.
Do I enter degrees or radians?
Convert first: radians = deg times pi / 180. Trig needs radians.
Does this do ellipse segments?
No. Heights and ellipse arcs need other methods.