Torus Volume
Calculate the volume of any torus (doughnut shape) from its inner and outer radius using V = 1/4pi^2(b - a)^2(b + a). Free online 3D geometry calculator for students and engineers with instant results.
About This Calculator
Our Torus Volume Calculator helps you compute the volume of any torus (the doughnut-shaped 3D surface) quickly and accurately. Whether you are a student learning 3D geometry, an engineer designing ring components, a manufacturer calculating material volume for toroidal tanks or O-rings, or a designer working with toroidal shapes, this tool provides instant results based on two simple measurements: the inner radius and the outer radius of the torus.
A torus is a 3D shape created by revolving a circle around an axis that lies in the same plane as the circle. Think of a doughnut, a tire, a lifebuoy, or a bagel. The tube itself has a circular cross-section with radius r (the minor radius), and this tube is bent into a circle of radius R (the major radius, measured from the center of the torus to the center of the tube). The inner radius a and outer radius b relate to these as a = R - r and b = R + r.
The volume is calculated using the formula V = 1/4pi^2(b - a)^2(b + a) where a is the inner radius and b is the outer radius. This is equivalent to the more common form V = 2pi^2Rr^2 where R = (a + b)/2 is the major radius and r = (b - a)/2 is the minor radius. The derivation follows from Pappus's centroid theorem: the volume equals the cross-sectional area (pir^2) multiplied by the distance the centroid travels (2piR), giving V = pir^2 x 2piR = 2pi^2Rr^2.
Regional notes: This geometry calculator works with any unit system -- millimeters, centimeters, meters, inches, or feet -- and the result is always in cubic units matching your input. The same formula applies worldwide and is not specific to any country or region. Students and professionals in the US, UK, India, and everywhere else use the same torus volume formula in their work.
Frequently Asked Questions
What is the formula for the volume of a torus?
The volume of a torus is calculated using V = 1/4pi^2(b - a)^2(b + a), where a is the inner radius and b is the outer radius. Alternatively, V = 2pi^2Rr^2 where R = (a + b)/2 is the major radius (distance from the center of the torus to the center of the tube) and r = (b - a)/2 is the minor radius (radius of the tube cross-section).
What is a torus?
A torus is a 3D shape obtained by revolving a circle around an axis that is coplanar with the circle. It looks like a doughnut or a ring. Common examples include tires, lifebuoys, and bagels. A torus has two radii: the minor radius r (radius of the tube cross-section) and the major radius R (distance from the center of the torus to the center of the tube).
What are the different types of tori?
Based on the relationship between the major radius R and minor radius r, there are three types of tori: Ring type (R > r) where the torus looks like a regular doughnut, Horn type (R = r) where the inner radius becomes zero and there is no hole, and Spindle type (R < r) where the surface self-intersects and the torus looks like a spindle. This calculator supports ring-type and horn-type tori.
How do you calculate the volume of a torus from inner and outer radii?
First enter the inner radius a (distance from the center to the inner edge) and the outer radius b (distance from the center to the outer edge). The calculator then determines the minor radius r = (b - a)/2 and major radius R = (a + b)/2, and computes the volume using V = 2pi^2Rr^2 or V = 1/4pi^2(b - a)^2(b + a). The result is displayed in cubic units matching the input units.
Can I calculate the surface area of a torus as well?
Yes, we have a separate Torus Surface Area calculator available on Calculy. The surface area of a torus is calculated using A = pi^2(b - a)(b + a) where a is the inner radius and b is the outer radius. You can find it under our Math category.
Is the Torus Volume calculator free to use?
Yes, all calculators on Calculy are completely free to use. There are no hidden charges, subscription fees, or usage limits. Simply enter your torus dimensions and get instant volume results with our free online 3D geometry calculator.
What units does the Torus Volume calculator support?
The calculator works with any consistent unit system. If you enter inner and outer radii in centimeters, the volume will be in cubic centimeters. Similarly for meters, inches, feet, or any other unit. The output does not specify a unit -- you interpret the result based on the units you input. This makes the calculator flexible for students, engineers, and designers worldwide.
How is the volume of a torus derived?
The volume of a torus is derived using Pappus's centroid theorem: the volume equals the area of the cross-section (pir^2 where r is the minor radius) multiplied by the distance traveled by its centroid (2piR where R is the major radius). Since the cross-section revolves around the axis, the centroid follows a circular path of circumference 2piR, giving V = pir^2 x 2piR = 2pi^2Rr^2.