Torus calculator

Calculate the volume and surface area of a torus from its major radius R and minor radius r, then see how the geometry changes as the tube gets thicker.

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Volume
—
V = 2π²Rr²
Surface area
—
S = 4π²Rr
Hole radius
—
a = R − r

Volume sensitivity

How volume changes as the tube radius grows
Live
R stays fixedUpdates with every input

Radius profile

Inner, major and outer radii in the same unit
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Inner → major → outerLive geometry
LIVE GEOMETRY INTELLIGENCE

What the current torus is telling you

Updates live
Aspect ratio — major radius ÷ tube radius
Outer radius — R + r
Inner diameter — 2(R − r)
Surface / volume — surface area ÷ volume
Enter valid radii to read the geometry.

Torus dimensions and derived values

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QuantitySymbolFormulaValue

Parameters

cm
Center of the torus to the center of the tube.
cm
Radius of the circular tube cross-section.
For a standard ring torus, use R > r. The boundary case R = r is a horn torus; R < r creates a self-intersecting spindle torus.
THE CONCEPT

What is a torus?

A torus is the three-dimensional surface and solid you get when a circle of radius r is revolved around an external coplanar axis at a distance R from the circle's center. The familiar doughnut shape is the standard ring torus.

The key is to keep the two radii separate. R measures the large circular path followed by the tube's center. r measures the tube itself. Many mistakes come from using the outer radius R + r where the formula needs the major radius R.

R r major radius minor radius A circle swept around a larger circle
Read the picture from the center outward: R locates the tube's centerline, while r controls the tube thickness. The inner and outer radii are then a = R − r and b = R + r.
Major radiusRcenter → tube center
Minor radiusrtube center → tube edge
Aspect ratioR / rhow open or thick the ring is

The two core formulas

For a standard ring torus, the geometry collapses neatly into two formulas. The volume depends on the cross-sectional circle's area and the circular path travelled by its centroid. Surface area follows the same sweep idea, using circumference instead of area.

V = 2π²Rr²
V
enclosed volume
R
major radius
r
minor or tube radius
S = 4π²Rr
S
total surface area
R
major radius
r
minor radius
Pappus gives the intuition: cross-section area πr² multiplied by the centroid path 2πR gives volume. For surface area, cross-section circumference 2πr multiplied by the same path 2πR gives 4π²Rr. This is the geometric engine behind both formulas.
01

Draw the tube

Start with a circle of radius r. Its area is πr².

02

Find the path

The tube center travels one full circle of length 2πR.

03

Sweep the circle

Multiply πr² by 2πR to get volume.

04

Repeat for area

Use 2πr instead of πr² to obtain surface area.

Inner radius, outer radius and diameters

Once R and r are known, the rest of the torus can be read directly from the same cross-section. These quantities are especially useful when the torus must fit inside a housing, shaft, cavity or opening.

R − r r R R + r hole radius tube radius outer radius
Inner radiusa = R − rThe radius of the central opening.
Outer radiusb = R + rThe distance from the center to the outer edge.
Inner diameter2(R − r)Twice the hole radius.
Outer diameter2(R + r)Overall diameter across the torus.

Ring, horn and spindle: the geometry changes at R = r

The relationship between R and r does more than change the look. It changes the type of torus. The standard calculator is intended for the ring case and the boundary horn case; R < r creates a self-intersecting spindle torus.

R > r

Ring torus

A visible hole remains in the center. This is the standard doughnut geometry.

R = r

Horn torus

The central hole closes to a single point. It is the boundary case of the ring torus.

R < r

Spindle torus

The tube crosses itself. It is not the ordinary ring-shaped torus used for this calculator.

Worked example

Suppose a torus has R = 10 cm and r = 3 cm. The tube is comfortably smaller than the major radius, so the shape is a ring torus.

Volume

V = 2π²(10)(3²)

That is 180π² ≈ 1,776.53 cm³.

Surface area

S = 4π²(10)(3)

That is 120π² ≈ 1,184.35 cm².

Where torus geometry appears in the real world

The torus is not only a textbook shape. The same geometry appears when a circular section follows a closed circular path: seals and O-rings, ring-shaped tanks and process vessels, and the toroidal chambers used in fusion research are all useful examples.

O-ring used as a static seal
O-rings and seals An O-ring is a real-world toroidal object. Its cross-section and overall ring dimensions are naturally described using a tube radius and a larger ring radius. Image: Wikimedia Commons, CC0.
Sectional view of the ITER tokamak reactor
Tokamak fusion systems Tokamak devices use toroidal geometry to confine plasma. The physical machine is far more complex than a mathematical torus, but the toroidal form is the starting geometric idea. Image: Wikimedia Commons, CC BY 4.0.

Common mistakes

1. Using the outer radius as R

If the tube reaches from R − r to R + r, then R is the centerline radius, not the outside edge.

2. Losing the π²

Volume contains two π factors: πr² from the cross-section and 2πR from the sweep.

3. Mixing diameter and radius

If a drawing gives a diameter, divide by two before inserting it into the formulas.

4. Allowing R < r

That produces a self-intersecting spindle torus rather than the standard ring torus.

Torus calculator FAQ

What is the formula for the volume of a torus?
For a standard torus, V = 2π²Rr², where R is the major radius and r is the minor radius.
What is the surface area of a torus?
The total surface area is S = 4π²Rr.
What is the difference between major and minor radius?
R runs from the torus center to the center of the tube. r runs from the center of the tube to its edge.
How do I find the inner and outer radius?
Inner radius is R − r. Outer radius is R + r. The corresponding diameters are twice those values.
What happens when R = r?
The hole closes to a point, giving the boundary horn torus. When R < r, the shape becomes a self-intersecting spindle torus.
Why does the volume grow with r²?
Because the tube's cross-section is a circle whose area is πr². Doubling r therefore makes that cross-sectional area four times larger before the sweep around the major circle is applied.