Hyperboloid — Definition, Formula & Examples

A hyperboloid is a three-dimensional surface that comes in two types: a hyperboloid of one sheet, which looks like a cooling tower or hourglass with connected sides, and a hyperboloid of two sheets, which consists of two separate bowl-shaped pieces facing away from each other.

A hyperboloid is a quadric surface in $\mathbb{R}^3$ . The hyperboloid of one sheet satisfies $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ , and the hyperboloid of two sheets satisfies $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ . Each can be obtained by rotating a hyperbola about one of its axes of symmetry.

Key Formula

\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \quad \text{(one sheet)}$$ $$-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \quad \text{(two sheets)}

Where:

$a$ = Semi-axis length along the x-direction
$b$ = Semi-axis length along the y-direction
$c$ = Semi-axis length along the z-direction

How It Works

To identify which type of hyperboloid you have, examine the signs in the standard-form equation. If two variables have positive coefficients and one is negative (with the equation equal to 1), you have a hyperboloid of one sheet. If two are negative and one is positive, it is a hyperboloid of two sheets. Cross-sections parallel to the coordinate planes reveal the structure: slicing a one-sheet hyperboloid horizontally produces ellipses at every height, while slicing a two-sheet hyperboloid horizontally produces ellipses only beyond a certain distance from the center, with a gap in between.

Worked Example

Problem: Identify the surface given by

x^2 + y^2 - \frac{z^2}{4} = 1

and describe its cross-section at

z = 2

Identify the type: Two positive terms (

x^2

and

y^2

) and one negative term (

-z^2/4

), with the equation equal to 1. This is a hyperboloid of one sheet.

\frac{x^2}{1} + \frac{y^2}{1} - \frac{z^2}{4} = 1

Find the cross-section at z = 2: Substitute

z = 2

into the equation and simplify.

x^2 + y^2 - \frac{4}{4} = 1 \implies x^2 + y^2 = 2

Interpret: This is a circle of radius

\sqrt{2}

centered at the origin in the plane

z = 2

Answer: The surface is a hyperboloid of one sheet with

a = 1

b = 1

c = 2

. At

z = 2

, the cross-section is a circle of radius

\sqrt{2}

Why It Matters

Hyperboloids appear in structural engineering because the hyperboloid of one sheet is a doubly ruled surface, meaning it can be built from straight beams. Cooling towers, telescopes, and architectural structures like the Canton Tower use this geometry. In multivariable calculus, classifying quadric surfaces is a core skill for understanding functions of several variables.

Common Mistakes

Mistake: Confusing one sheet vs. two sheets based on the number of negative signs.

Correction: One negative sign (with the rest positive) gives one sheet. Two negative signs give two sheets. The surface with fewer negative terms is the connected one.