Sphere and segment calculator
From any two of radius, base diameter and height the calculator finds the third and gives you the points and the arc for the program. Covers a convex spherical end, a concave dish and a ball on a neck.
Quick answer
Sphere radius from base diameter D and height h: R = (a² + h²) / 2h, a = D/2. Example: D = 2″, h = 0.25″ → R = (1 + 0.0625) / 0.5 = 2.125″; D = 40 mm, h = 5 mm → R = 42.5 mm.
Result
- Sphere radius R
- —
- Base diameter D
- —
- Height h
- —
- —
- Sphere center
- —
- Segment angle (from the axis)
- —
Contour (G7, zero at the face)
Points follow the part contour. Check the toolpath in a simulator before cutting.
Who needs it and why
Drawings often give a sphere not by its radius but by what is easy to measure: "a spherical end 3 mm high on a 20 mm diameter". The program needs the arc radius and its end points. Or the other way round: the radius is given and you need to know where the sphere meets the cylinder.
- Spherical ends — shafts, pushrods, pins, pads: the radius from the dome height and diameter.
- Dishes and concave spheres — for a ball seat, a center, a tip: the depth from the radius and diameter.
- Balls on a neck — knobs, ball studs, handles: where the ball ends and the neck begins.
- Checking a part — find the actual radius from a measured height and diameter.
The formulas
a = D / 2 is half the base diameter:
- radius
R = (a² + h²) / (2h) - height
h = R − √(R² − a²) - base diameter
D = 2 · √(h · (2R − h)) - ball Ø 2R on a neck Ø d: length from tip to neck
R + √(R² − (d/2)²)
The arc from the axis to the diameter for a convex shape, and from the diameter to the axis for a dish, is G3 (counterclockwise in the XZ plane). The Turn G-code Sphere cycle builds the toolpath along the arc — convex or concave, down to a pulley groove — and shows it in the 2D simulator.