Torus Volume Calculator
Our free triangle calculator solves torus volume problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Volume = 2(pi^2)(R)(r^2) | Surface Area = 4(pi^2)(R)(r)
Where R is the major radius (distance from the center of the torus to the center of the tube) and r is the minor radius (radius of the tube). These formulas are derived using Pappus centroid theorem by rotating a circle of radius r around an axis at distance R.
Worked Examples
Example 1: Calculating Volume of a Donut-Shaped Object
Problem:A donut has a major radius (center to tube center) of 5 cm and a tube radius of 2 cm. Find its volume and surface area.
Solution:Volume = 2 x pi^2 x R x r^2\nV = 2 x (9.8696) x 5 x (2^2)\nV = 2 x 9.8696 x 5 x 4\nV = 394.784 cubic cm\n\nSurface Area = 4 x pi^2 x R x r\nSA = 4 x 9.8696 x 5 x 2\nSA = 394.784 square cm\n\nOuter radius = 5 + 2 = 7 cm\nInner radius = 5 - 2 = 3 cm (hole diameter = 6 cm)
Result:Volume = 394.78 cubic cm | Surface Area = 394.78 sq cm | Hole diameter = 6 cm
Example 2: O-Ring Material Calculation
Problem:An O-ring seal has an outer diameter of 30 mm and a tube cross-section diameter of 4 mm. Calculate the volume of rubber needed.
Solution:Minor radius r = 4/2 = 2 mm\nOuter radius = 30/2 = 15 mm\nMajor radius R = outer radius - r = 15 - 2 = 13 mm\n\nVolume = 2 x pi^2 x R x r^2\nV = 2 x 9.8696 x 13 x 4\nV = 1024.46 cubic mm\nV = 1.024 cubic cm\n\nSurface Area = 4 x pi^2 x 13 x 2 = 1024.46 sq mm
Result:Volume = 1024.46 cubic mm (1.024 cc) | Surface Area = 1024.46 sq mm
Frequently Asked Questions
What is a torus and what does it look like?
A torus is a three-dimensional geometric shape that resembles a donut or inner tube. It is formed by rotating a circle around an axis that lies in the same plane as the circle but does not intersect it. The shape has a hole in the middle, which is what distinguishes it from a sphere. A torus is defined by two radii: the major radius R (the distance from the center of the torus to the center of the tube) and the minor radius r (the radius of the tube itself). Tori appear in many real-world objects including donuts, bagels, life preservers, O-rings used in engineering seals, and the shape of some magnetic confinement devices used in nuclear fusion research like tokamaks.
What is the formula for the volume of a torus?
The volume of a torus is calculated using the formula V = 2 times pi squared times R times r squared, often written as V = 2(pi^2)(R)(r^2). Here R is the major radius (distance from the center of the torus hole to the center of the circular cross-section) and r is the minor radius (radius of the circular cross-section or tube). This formula can be derived using Pappus theorem, which states that the volume of a solid of revolution equals the area of the cross-section multiplied by the distance traveled by its centroid. Since the cross-section is a circle with area pi times r squared, and the centroid travels a distance of 2 times pi times R, the volume equals pi times r squared times 2 times pi times R.
What is the surface area formula for a torus?
The surface area of a torus is calculated using the formula SA = 4 times pi squared times R times r, written as SA = 4(pi^2)(R)(r). This formula is also derived from Pappus theorem for surface areas. The circumference of the circular cross-section (the tube) is 2 times pi times r, and this circumference is rotated around the central axis at a distance R, traveling a path of length 2 times pi times R. Multiplying these gives 2(pi)(r) times 2(pi)(R) = 4(pi^2)(R)(r). For a torus with major radius 10 and minor radius 3, the surface area would be 4 times 9.8696 times 10 times 3 = approximately 1184.35 square units. The surface area is always proportional to both radii linearly.
How is a torus different from a sphere or an ellipsoid?
A torus is fundamentally different from a sphere or ellipsoid because it has a hole through its center, giving it a different topological structure. A sphere has genus 0 (no holes), while a torus has genus 1 (one hole). This means you cannot continuously deform a sphere into a torus without cutting or puncturing it. Mathematically, a sphere is defined by a single radius from one center point, while a torus requires two radii and involves rotation around an axis. The Euler characteristic of a sphere is 2, while for a torus it is 0. In practical terms, a sphere encloses a simply connected volume, while the interior of a torus forms a more complex topology that allows paths to loop through the hole without leaving the surface.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy