What is a segment of a circle? ⌓
If you want to understand what the segment of a circle is, try to imagine cutting part of a circle off with a single, straight cut. And that's it! You've just created two parts of the circle, and the smaller one is called the circular segment. A more formal mathematical definition says that:
A circular segment is a region bounded by a chord and the arc of a circle (of less than 180°)
If it's equal to 180°, then it's simply a half circle – semicircle. According to some definitions, the central angle doesn't need to be smaller than 180° – in that case, you can say that cutting a circle with a line gives you two segments: a major segment and a minor segment.
What is a chord of a circle?
A chord is a line that connects two points on a circle. The infinite line extension of a chord is called a secant. The special case of a chord is one that passes through the center of a circle – and that's the circle diameter, of course!
Formulas for a segment of a circle area
To find the circle segment area, you need to know at least two variables. This segment area calculator has two modes:
Mode 1: Radius + Central Angle
Given the radius r and central angle θ (in degrees), all values are computed as follows:
| Property | Formula |
|---|---|
| Segment Area | A = (r² / 2) × (θrad − sin θrad) |
| Chord length | c = 2r × sin(θ / 2) |
| Arc length | L = r × θrad |
| Sagitta (height) | h = r × (1 − cos(θ / 2)) |
where θrad = θ × π / 180
Mode 2: Radius + Chord Length
Given the radius r and chord length c, the central angle is derived first:
θrad = 2 × arcsin(c / (2r))
Then the same formulas apply to compute segment area, arc length, and sagitta.
Sagitta — the height of a segment
The sagitta (from Latin for "arrow") is the height of the circular segment measured perpendicularly from the midpoint of the chord to the arc. It is sometimes also called the versine.
Units supported
This calculator supports both metric and US Imperial unit systems:
- Metric: mm, cm, m, km — Area is shown in the corresponding square unit (mm², cm², m², km²)
- US Imperial: in, ft, yd, mi — Area is shown in in², ft², yd², or mi²
FAQs
Is a semicircle a segment?
Yes! A semicircle is a special case of a circular segment where the central angle equals exactly 180° and the chord is the diameter of the circle.
Can the chord be larger than the diameter?
No. The maximum chord length equals the diameter (2r). If you enter a chord length greater than or equal to 2r, the calculator will return an error.
What is the difference between a segment and a sector?
A sector is a "pizza slice" — bounded by two radii and an arc. A segment is the region between a chord and the arc it cuts off (without the triangular part connecting the two radii).
How do I find segment area from chord and radius?
Switch to Radius + Chord mode, enter your radius and chord length, and click Calculate. The calculator will derive the central angle and compute the segment area automatically.