Circular Segment Calculator
Enter any two of radius, central angle, chord and height, and the segment — the region between a chord and its arc — is solved and drawn to scale: area, arc length and all the rest. Click any amber value on the drawing to change it. Rotate a segment about its height and you get a spherical cap.
Known values
from radius + angleEnter any two values — the rest update automatically.
Results
Segment formulas
A segment is fixed by its radius R and central angle θ (in radians). The chord, height and area all follow, and any two of the four measurements recover the rest.
Area · A = ½R²(θ − sin θ)
Chord · c = 2R sin(θ⁄2)
Height · h = R(1 − cos(θ⁄2))
Arc · s = Rθ
From chord and height
The pair you can measure directly on a real object gives the radius immediately: R = (c² + 4h²)⁄8h — the same relation a spherical cap uses. The angle then follows from θ = 2·acos(1 − h⁄R).
Tips
Angles can be entered in degrees or radians — switch in the Display row. When you give a radius and a chord, the calculator takes the minor segment (θ ≤ 180°); enter the angle directly for a major segment.
How to find the area of a circular segment
A circular segment is the region a chord cuts off a circle — the shape of water in a horizontal pipe, the flat cut on a shaft, a bridge arch, the risen top of a loaf. Two measurements fix it. In the maths classroom that pair is usually the radius R and central angle θ; in the workshop it's usually the chord c across the flat and the height h (the sagitta) from chord to arc. This calculator accepts any two of the four.
A = ½R²(θ − sin θ)
with θ in radians. Everything else follows: the chord is 2R sin(θ⁄2), the height is R(1 − cos(θ⁄2)), and the arc is Rθ. The formula has a neat reading: ½R²θ is the whole pie-slice sector, and ½R²sin θ is the triangle between the two radii and the chord — the segment is the sector with the triangle removed.
Worked example
Take a segment with radius 10 and a central angle of 120°.
Convert the angle first; the area formula needs radians.
θ = 120° × π⁄180 = 2π⁄3 ≈ 2.0944
Half R squared times (θ − sin θ).
A = ½ × 100 × (2.0944 − 0.8660) = 50 × 1.2284 ≈ 61.42
The remaining measurements come from the half-angle.
c = 20 sin 60° ≈ 17.32 · h = 10(1 − cos 60°) = 5 · s = 10 × 2.0944 ≈ 20.94
Measured the flat instead?
If you have the chord and the rise — say a 20-wide flat with a 4-high crown — the radius is R = (c² + 4h²)⁄8h = (400 + 64)⁄32 = 14.5, and the angle follows. This is exactly the relation the spherical cap calculator uses in 3D: spin a segment about its height line and the cap is what you get.
Common mistakes
The classic one is using degrees in the area formula — θ must be in radians, or the (θ − sin θ) subtraction is meaningless. The second is the chord's ambiguity: one chord cuts a circle into a minor and a major segment, so a radius-plus-chord input is taken as the minor one here; give the angle itself (over 180°) when you want the major piece. And h is the height of the segment, not the distance from the centre to the chord — those two add up to R.
Where it comes up
Segment area is the cross-section of liquid in a horizontal cylindrical tank or partly full pipe — multiply by the length and you have the fill volume, which is exactly what the tank volume calculator does. It also covers flats machined on round bar, arch and vault profiles, chord trusses, dished timber and the strength check of a pipe running part-full. For the full pie slice including the centre, use the sector calculator; for the ring between two circles, the annulus.
Frequently asked questions
How do you find the area of a circular segment?
With the angle in radians: A = ½R²(θ − sin θ). It's the pie-slice sector, ½R²θ, minus the triangle between the radii and the chord, ½R²sin θ. For R = 10 and θ = 120°, that's 50 × (2.0944 − 0.8660) ≈ 61.42.
I only know the chord and the height — can I still solve it?
Yes, and it's the most practical pair: R = (c² + 4h²)⁄8h, then θ = 2·acos(1 − h⁄R). Enter c and h and the calculator does exactly this.
What's the difference between a segment and a sector?
A sector is the full pie slice from the centre out — two radii and an arc. A segment is what's left when you cut straight across with a chord instead: arc and chord only, no centre point.
What are the minor and major segments?
One chord makes two segments: the smaller (minor, θ under 180°) and the larger (major, θ over 180°). From a radius and chord the calculator returns the minor one; enter the reflex angle directly for the major, or read the major area from the results list.
Is the height the same as the distance from the centre?
No — they're complements. The height h runs from the chord to the arc; the distance from the centre to the chord is d = R cos(θ⁄2), and h + d = R for a minor segment. The results list shows both.