Torus — Definition, Formula & Examples

Torus

A doughnut shape. Formally, a torus is a surface of revolution obtained by revolving (in three dimensional space) a circle about a line which does not intersect the circle.

Yellow torus (donut shape) with radius r (tube) and R (center to tube center). Volume=2π²Rr², Surface Area=4π²Rr.

See also

Axis of rotation, volume, surface area

Key Formula

V = 2\pi^2 R r^2 \qquad \text{and} \qquad A = 4\pi^2 R r

Where:

$V$ = Volume of the torus (the space enclosed by the surface)
$A$ = Surface area of the torus (the area of the outer skin)
$R$ = Major radius — the distance from the center of the torus to the center of the tube
$r$ = Minor radius — the radius of the circular tube itself
$\pi$ = Pi, approximately 3.14159

Worked Example

Problem: A torus has a major radius of R = 6 cm and a minor radius of r = 2 cm. Find its volume and surface area.

Step 1: Identify the two radii. The major radius R = 6 cm is the distance from the center of the hole to the center of the tube. The minor radius r = 2 cm is the radius of the tube's circular cross-section.

R = 6 \text{ cm}, \quad r = 2 \text{ cm}

Step 2: Apply the volume formula. Substitute the values into the torus volume equation.

V = 2\pi^2 R r^2 = 2\pi^2 (6)(2^2) = 2\pi^2 (6)(4) = 48\pi^2

Step 3: Calculate the numerical volume.

V = 48\pi^2 \approx 48 \times 9.8696 \approx 473.7 \text{ cm}^3

Step 4: Apply the surface area formula.

A = 4\pi^2 R r = 4\pi^2 (6)(2) = 48\pi^2

Step 5: Calculate the numerical surface area.

A = 48\pi^2 \approx 473.7 \text{ cm}^2

Answer: The torus has a volume of approximately 473.7 cm³ and a surface area of approximately 473.7 cm². (It is a coincidence that the numbers match here — this only happens when R·r² = R·r, i.e., when r = 1 or when the particular combination of R and r makes the expressions equal. In this case both expressions simplify to 48π².)

Another Example

This example uses different dimensions to show how quickly the volume grows as the radii increase, and focuses on volume alone to give a streamlined calculation.

Problem: A torus has a major radius of R = 10 cm and a minor radius of r = 3 cm. Find its volume.

Step 1: Write down the known values.

R = 10 \text{ cm}, \quad r = 3 \text{ cm}

Step 2: Substitute into the volume formula.

V = 2\pi^2 R r^2 = 2\pi^2 (10)(3^2) = 2\pi^2 (10)(9) = 180\pi^2

Step 3: Evaluate numerically.

V = 180\pi^2 \approx 180 \times 9.8696 \approx 1{,}776.5 \text{ cm}^3

Answer: The volume of the torus is approximately 1,776.5 cm³.

Frequently Asked Questions

What is the difference between the major radius and minor radius of a torus?

The major radius R is the distance from the center of the entire torus (the center of the hole) to the center of the tube. The minor radius r is the radius of the tube's circular cross-section. Think of R as how wide the doughnut is overall, and r as how thick the tube is. You always need both values to compute volume or surface area.

How do you derive the volume of a torus?

The volume formula comes from Pappus's centroid theorem. The theorem states that the volume of a solid of revolution equals the area of the revolved shape multiplied by the distance its centroid travels. The revolved shape is a circle with area πr², and its centroid travels a circular path of length 2πR. Multiplying gives V = πr² × 2πR = 2π²Rr².

What is a torus in real life?

Common real-life examples of tori include doughnuts, bagels, inner tubes, and life preservers. In engineering, O-rings used to seal pipes and joints are tori. The shape also appears in physics and astronomy — some models of the universe and magnetic confinement devices (tokamaks) use toroidal geometry.

Torus vs. Sphere

	Torus	Sphere
Shape	Doughnut-shaped with a hole through the center	Perfectly round with no hole
Defined by	Two radii: major radius R and minor radius r	One radius: r
Volume formula	V = 2π²Rr²	V = (4/3)πr³
Surface area formula	A = 4π²Rr	A = 4πr²
Generated by	Revolving a circle around an external axis	Revolving a semicircle around its diameter

Why It Matters

The torus appears in multivariable calculus and differential geometry courses as a key example of a surface of revolution and a manifold. It is also central in physics — tokamak fusion reactors and particle accelerators use toroidal geometry. Understanding the torus builds your intuition about how rotating a 2D shape generates a 3D solid, a concept that extends to many engineering applications.

Common Mistakes

Mistake: Confusing the major radius R with the minor radius r, or swapping them in the formula.

Correction: Remember that R (major) is always the larger measurement — the distance from the center of the hole to the center of the tube. The minor radius r measures the tube's thickness. In the volume formula V = 2π²Rr², the minor radius r is squared, not R.

Mistake: Forgetting that the formulas require R > r for a standard (ring) torus with a visible hole.

Correction: If R ≤ r, the torus self-intersects or degenerates (a spindle torus or horn torus). The standard doughnut shape requires R > r. Most textbook problems assume this condition, but check before applying the formulas.

Related Terms

Surface of Revolution — A torus is a specific surface of revolution
Circle — The 2D shape that is revolved to form a torus
Volume — Key measurement computed using the torus formula
Surface Area — The outer area of the torus surface
Axis of Rotation — The external line around which the circle revolves
Three Dimensions — The space in which the torus exists
Line — The axis that the generating circle revolves around