Known-angle method: in an axial cross-section, the ball is a circle tangent to the two straight sides of the cone. The perpendicular distance from the ball center to either cone side is exactly the ball radius. That tangent relationship determines the ball-center position from the theoretical cone apex.

Example: a 0.2500 in ball in a 90° cone, measured 0.0350 in proud of the reference face, gives a 0.5336 in opening diameter (theoretical apex 0.2668 in below the face). Those are the same values already filled into the calculator above, so you can run it yourself and check the numbers match.

Two-ball method: each ball sits at a different axial position in the same cone. Subtracting the two readings removes the unknown location of the cone relative to the reference face, leaving enough information to solve the half-angle.

Choosing a ball size: pick a ball diameter close to, or a little larger than, the diameter you expect at the depth you're measuring. Too small and the ball can drop farther into the taper than intended or bottom out; too large and it only contacts near the rim, which is more sensitive to edge condition (burrs, chamfers) than the true conical surface.

Important limitation: these equations assume ideal cone-to-ball tangency. Do not use the result if a ball bottoms out, bridges the opening, contacts a chamfer/radius instead of the true cone, or if the reference face/readings are not repeatable. For inspection decisions, use calibrated balls and appropriate measurement uncertainty.

About external tapers: an external “over balls/rollers” measurement requires a defined axial locating method or fixture; the measurement is not unique from ball diameter and cone angle alone. This tool intentionally covers the unambiguous internal-taper methods rather than pretending those setups are interchangeable.