The torus: the doughnut surface

Q: What is the equation of a torus?

(√(x² + y²) − R)² + z² = r², where R is the distance from the centre of the tube to the centre of the torus and r is the radius of the tube.

A torus is the surface of a doughnut: a circle of radius r swept around a larger circle of radius R. Its equations are a neat exercise in 3D geometry, and its volume and surface area follow from a theorem older than calculus.

A torus drawn in 3D from its implicit equation — The torus (√(x² + y²) − 3)² + z² = 1, ray-traced from its implicit equation.

Equations to type

Implicit

(sqrt(x^2 + y^2) - 3)^2 + z^2 = 1

Open

Parametric

((3 + cos(v)) cos(u), (3 + cos(v)) sin(u), sin(v))

Open

At a glance

Implicit equation: (√(x² + y²) − R)² + z² = r²
Volume: 2π²Rr²
Surface area: 4π²Rr

Where the equations come from

√(x² + y²) is a point’s distance from the z-axis. The tube’s centre line is the circle at distance R from that axis, so a point is on the surface when its distance from the centre line is r: (√(x² + y²) − R)² + z² = r². The parametric form sweeps the small circle (R + r cos v, r sin v) around the z-axis by angle u.

Volume and area by Pappus’ theorem

Pappus of Alexandria, around 320 AD, found that a solid of revolution has volume equal to the area of the shape being rotated times the distance its centroid travels. The small disc has area πr² and its centre travels 2πR, so the volume is 2π²Rr². The same idea for the circumference 2πr gives the surface area 4π²Rr.

Ring, horn and spindle tori

When R > r the hole is open: a ring torus, the doughnut. When R = r the hole closes to a point: a horn torus. When R < r the surface intersects itself: a spindle torus. Add a slider for R in the implicit equation to watch the hole close.