Torus Calculator

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Walk-through

How to Use This Calculator

3 steps

1

Enter the major and minor radii

Type the major radius (R) — the distance from the center of the whole torus to the center of the tube — and the minor radius (r) — the radius of the tube itself. Choose a unit; it's a display label only, since the math is unit-agnostic.

2

Read the volume and surface area

The Volume tab leads with V = 2π²Rr² and shows surface area as a supporting stat; the Surface Area tab flips that around, leading with SA = 4π²Rr. Both update instantly as you change R or r.

3

Solve backward if you need to

Switch to the Radii tab to go the other direction: enter a target volume or surface area plus whichever radius you already know, and the calculator solves for the missing radius, showing the worked steps.

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Reference

Formula & Methodology

4 formulas

Volume of a torus

V = 2π² · R · r²

R is the major radius (center of torus to center of tube), r is the minor radius (radius of the tube). Volume scales with the square of the minor radius, so doubling the tube thickness quadruples the volume. Units are cubic (e.g., cubic feet, cubic centimeters).

Surface area of a torus

SA = 4π² · R · r

The total area of the torus's outer surface. Surface area scales linearly with both radii, so doubling either one doubles the surface area. Units are square (e.g., square feet, square centimeters).

Solving backward

r = √(V ÷ 2π²R) or R = V ÷ (2π²r²) or r = SA ÷ (4π²R) or R = SA ÷ (4π²r)

Each forward formula can be inverted to recover a missing radius from a target volume or surface area, given the other radius. This calculator's Radii tab performs whichever inversion matches your inputs.

Pappus's centroid theorem

V = A · d, SA = P · d

Both torus formulas are a special case of Pappus's theorem: sweeping a shape (here, a circle of radius r) around an external axis produces a volume equal to the shape's area (A = πr²) times the distance its centroid travels in one revolution (d = 2πR). The same logic gives surface area from the circle's perimeter (P = 2πr).

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Glossary

Key Terms Explained

6 terms

Torus A ring-shaped three-dimensional surface formed by revolving a circle around an axis that lies in the same plane as the circle but doesn't intersect it — the mathematical name for a donut shape.

Major radius The distance (R) from the center of the entire torus to the center of the tube — essentially, the radius of the donut's central hole plus half the tube's thickness.

Minor radius The radius (r) of the tube itself — how thick the donut ring is, measured from the center of the tube's cross-section to its outer edge.

Donut The everyday name for a torus — a ring shape with a hole through the middle, whose volume and surface area follow the same formulas whether you're describing a pastry or a mathematical surface.

Revolution The act of rotating a two-dimensional shape (here, a circle) 360° around an axis to sweep out a three-dimensional solid — the torus is generated by revolving a circle of radius r around an axis at distance R from its center.

Tube The circular cross-section that, when revolved around the central axis, forms the torus's ring. The tube's radius is the minor radius (r).

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Scenarios

Real-World Examples

3 worked examples

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Forward calculation

Major radius = 5, minor radius = 2

Major radius 5 ft Minor radius 2 ft

With a major radius of 5 ft and a minor radius of 2 ft, the volume is about 394.78 ft³ and the surface area is about 394.78 ft². This is the calculator's default example — a clean check that both formulas agree with the standard reference values for R=5, r=2.

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Solving backward from volume

Target volume = 394.78, major radius known = 5

Target measure Volume = 394.78 ft³ Known radius Major radius = 5 ft

Given a target volume of 394.78 cubic feet and a known major radius of 5 ft, solving r = √(V ÷ 2π²R) recovers a minor radius of about 2.00 ft — the same torus as the first example, worked in reverse. This pattern applies to problems like sizing a ring-shaped tank or gasket to hit a required capacity.

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Real-world donut framing

A donut with a 4-inch hole-to-hole span and a 1-inch-thick ring

Major radius 2 in Minor radius 1 in

For a donut-sized torus with major radius 2 in and minor radius 1 in, the volume works out to about 39.48 in³ and the surface area to about 78.96 in². Because r is not smaller than R here, this particular shape sits at the self-intersection boundary — a reminder that real donuts always keep the minor radius meaningfully smaller than the major radius to avoid a pinched or overlapping hole.

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Deep Dive

Torus volume and surface area, and how they relate to a donut

3 chapters · 3 min read

A torus has two free parameters — the major radius R and the minor radius r — and its volume and surface area follow directly from both by way of Pappus's centroid theorem. This calculator handles the forward direction (radii in, volume and surface area out) and the backward direction (a target measure plus one known radius, solving for the other).

Why the formulas look like a circle's area and circumference, times 2πR

A torus is generated by revolving a circle of radius r around an axis at distance R from the circle's center. Pappus's centroid theorem says the resulting volume equals the circle's area (πr²) times the distance its centroid travels in one full revolution (2πR) — giving V = πr² × 2πR = 2π²Rr². The surface area works the same way using the circle's perimeter (2πr) instead of its area: SA = 2πr × 2πR = 4π²Rr. Both formulas are really just "shape measure times sweep distance."

The self-intersection boundary: when r ≥ R

A standard donut-shaped torus requires the minor radius to be smaller than the major radius (r < R) — otherwise the tube would have to pass through the central axis, producing a pinched "horn torus" (r = R) or a self-intersecting "spindle torus" (r > R) instead of a normal ring. The volume and surface area formulas still compute a number in either case, but the shape they describe is no longer a simple donut, so this calculator flags the condition when it occurs.

When you need to solve backward

Most textbook problems hand you both radii and ask for volume or surface area. Real-world problems often run the other way: you know a ring-shaped part needs to hold a certain volume, or you know how much material covers its surface, and you need to find the missing dimension. Solving backward means inverting the forward formula for whichever radius is unknown — this calculator's Radii tab does that automatically based on which measure and which known radius you select.

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Questions

Frequently Asked Questions

6 questions

What is the volume formula for a torus?+

V = 2π²Rr², where R is the major radius (center of the tube to the center of the torus) and r is the minor radius (radius of the tube). For R = 5 and r = 2, that's about 394.78 cubic units.

What is the surface area formula for a torus?+

SA = 4π²Rr, using the same major radius R and minor radius r. For R = 5 and r = 2, that's about 394.78 square units — coincidentally equal to the volume in this particular example.

What's the difference between major and minor radius?+

The major radius (R) is the distance from the center of the whole torus to the center of the tube. The minor radius (r) is the radius of the tube itself — how thick the donut ring is. R measures the size of the ring; r measures the thickness of the material forming it.

Is a torus the same as a donut?+

Yes — a torus is the mathematical name for a donut shape: a ring formed by revolving a circle of radius r around an axis at distance R from the circle's center, as long as r stays smaller than R.

What units does the calculator use?+

Both the major and minor radius use whichever linear unit you select (inches, feet, yards, centimeters, or meters). Volume automatically uses that unit cubed, and surface area automatically uses that unit squared.

How does Pappus's theorem relate to these formulas?+

Both formulas are a direct application of Pappus's centroid theorem: revolving a circle of area πr² and perimeter 2πr around an external axis a distance R away sweeps out a volume of (πr²)(2πR) = 2π²Rr² and a surface area of (2πr)(2πR) = 4π²Rr.

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