Torus Surface Area Calculator
Enter two measurements that describe your torus and the calculator returns the exact surface area instantly. You can work with either the tube radius and radius of revolution, or the inner and outer radii; both input pairs are equivalent and the calculator keeps them in sync. Volume is included as a bonus output so you have everything you need for a donut, ring, tire, or any ring-shaped solid.
Formula
Worked example
A donut-shaped torus with tube radius r = 3 cm and radius of revolution R = 8 cm: A = 4 x pi^2 x 3 x 8 = 96 x pi^2 = 946.7 cm^2. The inner radius is a = 8 - 3 = 5 cm and outer radius is b = 8 + 3 = 11 cm. Volume = 2 x pi^2 x 9 x 8 = 144 x pi^2 = 1420.1 cm^3.
What is a torus?
A torus is the three-dimensional surface you get by rotating a circle around an axis that lies in the same plane but does not pass through the circle. Think of a donut, a lifebuoy, a rubber O-ring, or a tire: all of these are torus shapes. The shape is described by two radii. The tube radius r (also called the minor radius) is the radius of the circular cross-section, measured from the center of that circle to the edge. The radius of revolution R (also called the major radius) is the distance from the center of the whole torus to the center of the tube. Equivalently, you can describe the same shape with the inner radius a (the distance from the center to the hole's inner edge) and the outer radius b (the distance from the center to the outermost point). The conversions are: a = R - r and b = R + r.
The surface area formula
The surface area of a torus is derived from Pappus's centroid theorem, which states that the surface area of a solid of revolution equals the arc length of the generating curve multiplied by the distance traveled by its centroid. For a circle of radius r rotated at distance R from its center, the arc length is 2 x pi x r and the centroid travels a circle of circumference 2 x pi x R, giving A = (2 x pi x r) x (2 x pi x R) = 4 x pi^2 x r x R. Written in terms of inner and outer radii: A = pi^2 x (b^2 - a^2), since b^2 - a^2 = (R + r)^2 - (R - r)^2 = 4rR. The two formulas are mathematically identical. A simple worked example: with r = 3 cm and R = 8 cm, the surface area is 4 x 9.8696 x 3 x 8 = 946.7 cm^2.
Ring, horn, and spindle tori
Tori fall into three geometric types depending on how the tube radius r compares to the radius of revolution R. When R is larger than r, you have a ring torus: there is a visible hole in the center and the shape looks like a classic donut or tire. When R equals r, the inner hole collapses to a single point at the center, forming a horn torus. When R is smaller than r, the tube wraps all the way around and the surface self-intersects, producing a spindle torus. Most physical objects are ring tori; the surface area formula A = 4 x pi^2 x r x R applies to all three types but self-intersecting spindle tori are not physically meaningful for solid objects.
Volume and other torus properties
The volume of a torus is V = 2 x pi^2 x r^2 x R. Again by Pappus's theorem, this equals the cross-sectional area (pi x r^2) multiplied by the circumference traveled by the centroid (2 x pi x R). For the example above with r = 3 cm and R = 8 cm, V = 2 x 9.8696 x 9 x 8 = 1420.1 cm^3. The surface-area-to-volume ratio is A / V = 4 x pi^2 x r x R divided by 2 x pi^2 x r^2 x R = 2 / r. This means the ratio depends only on the tube radius, not on how large the overall ring is. Slimmer tubes have a higher surface-area-to-volume ratio, just like thinner wires or smaller spheres.
Torus types by radius ratio
| Type | Condition | Appearance | Central hole |
|---|---|---|---|
| Ring torus | R > r | Donut or tire shape | Yes |
| Horn torus | R = r | Inner edge pinches to a point | No (point) |
| Spindle torus | R < r | Self-intersecting surface | No |
Classification of tori based on the relationship between tube radius (r) and radius of revolution (R).
Frequently asked questions
What is the surface area formula for a torus?
The surface area is A = 4 x pi^2 x r x R, where r is the tube radius and R is the radius of revolution. An equivalent form using the inner radius a and outer radius b is A = pi^2 x (b^2 - a^2). Both give the same result. For example, a torus with r = 3 cm and R = 8 cm has a surface area of 4 x pi^2 x 3 x 8 = approximately 946.7 cm^2.
What is the difference between the tube radius and the radius of revolution?
The tube radius (r, also called the minor radius) is the radius of the circular cross-section of the tube itself. The radius of revolution (R, also called the major radius) is the distance from the center of the whole torus to the center of the tube. Together they fully describe the shape. You can convert to inner and outer radii with a = R - r and b = R + r.
What is a ring torus, and why does it matter?
A ring torus is a torus where the radius of revolution R is greater than the tube radius r. This produces a shape with a visible hole in the center, like a donut or an O-ring. It is the most common physical type. When R equals r you get a horn torus (the hole collapses to a point), and when R is less than r you get a spindle torus (the surface self-intersects). The surface area formula applies to all three, but only ring and horn tori represent physically solid non-intersecting surfaces.
Can I use inner and outer radii instead of tube and major radii?
Yes. If you know the inner radius a and outer radius b, you can compute the tube radius r = (b - a) / 2 and the radius of revolution R = (b + a) / 2. The surface area formula in terms of these is A = pi^2 x (b^2 - a^2). This calculator accepts both input pairs and converts between them automatically.
How does torus surface area differ from sphere or cylinder surface area?
A sphere of radius r has surface area 4 x pi x r^2, which depends on a single radius. A closed cylinder of radius r and height h has surface area 2 x pi x r x (r + h). A torus of tube radius r and revolution radius R has surface area 4 x pi^2 x r x R. The torus formula is the only one of the three that involves two distinct radii multiplied together, and it uses pi^2 rather than pi because the shape involves two independent circular motions.
What units does this calculator use?
You can switch between metric (centimetres) and imperial (inches) using the unit selector. The surface area is returned in the corresponding square unit (cm^2 or in^2) and the volume in cubic units (cm^3 or in^3). All inputs and outputs update automatically when you switch units.