Toroid — Definition, Formula & Examples

A toroid is a doughnut-shaped 3D surface formed by revolving a circle around an axis that lies in the same plane as the circle but does not intersect it. The most common toroid is called a torus.

A torus is the surface of revolution generated by rotating a circle of radius $r$ about a coplanar axis at distance $R$ from the center of the circle, where $R > r$ . The resulting solid encloses a volume and has an outer radius of $R + r$ and an inner radius of $R - r$ .

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
$A$ = Surface area of the torus
$R$ = Major radius — distance from the center of the torus to the center of the tube
$r$ = Minor radius — radius of the circular cross-section (the tube)

How It Works

Imagine taking a circular tube and bending it into a ring until the two ends meet — that ring shape is a torus. The distance

R

from the center of the ring to the center of the tube is called the major radius, while the radius

r

of the tube itself is the minor radius. These two measurements are all you need to compute the volume and surface area. Unlike a sphere, a torus has a hole through its center, which changes its topology and means Euler's formula for polyhedra does not apply to it in the standard way.

Worked Example

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

Volume: Substitute into the volume formula.

V = 2\pi^2 R r^2 = 2\pi^2(6)(2^2) = 2\pi^2(6)(4) = 48\pi^2 \approx 473.7 \text{ cm}^3

Surface Area: Substitute into the surface area formula.

A = 4\pi^2 R r = 4\pi^2(6)(2) = 48\pi^2 \approx 473.7 \text{ cm}^2

Answer: The torus has a volume of

48\pi^2 \approx 473.7

cm³ and a surface area of

48\pi^2 \approx 473.7

cm².

Why It Matters

Toroids appear in engineering (transformer cores, O-rings, and magnetic confinement in fusion reactors) and in topology courses where the torus serves as a fundamental example of a surface with a different genus than a sphere. Understanding its geometry builds intuition for surfaces of revolution studied in calculus.

Common Mistakes

Mistake: Confusing R and r, or using the outer radius directly in the formulas.

Correction: R is the distance from the center of the hole to the center of the tube, not the outer edge. If given an outer radius of 8 cm and inner radius of 4 cm, compute R = (8+4)/2 = 6 and r = (8−4)/2 = 2.