Torus Volume & Surface Area Calculator
Compute the volume and surface area of a torus (a doughnut shape) from its centre-to-tube radius and tube radius.
How it works
Volume = 2·π²·R·r² · Surface area = 4·π²·R·r (R = major radius, r = tube radius)
A torus is the doughnut shape swept out when a circle of radius r is revolved around an axis at distance R from its centre. By Pappus’s theorem the volume equals the tube’s cross-sectional area πr² times the distance its centroid travels, 2πR — giving 2π²Rr². The surface area is the tube’s circumference 2πr times that same travel distance, 4π²Rr.
Worked example
For major radius R = 10 and tube radius r = 3: volume = 2π²·10·3² = 2·9.8696·10·9 = 1776.5 cubic units; surface area = 4π²·10·3 = 1184.4 square units.
Sources: Torus — Wikipedia
Frequently asked questions
What do the major and minor radii mean?
The major radius R is the distance from the centre of the hole to the centre of the tube. The minor radius r is the radius of the tube itself — the thickness of the ring.
Why do the formulas contain π²?
The torus is generated by two independent circular motions — around the tube and around the central axis — so each contributes a factor of π. Pappus’s centroid theorem combines them into π².
What if the tube radius equals the major radius?
When r = R the hole closes to a single point (a horn torus). This calculator allows r up to R; beyond that the shape self-intersects and the simple formulas no longer describe a plain ring.
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This calculator is for educational purposes. Double-check important results before acting on them.