Paraboloid — Definition, Formula & Examples

A paraboloid is a bowl-shaped 3D surface where cross-sections parallel to the base are circles (or ellipses), and cross-sections through the central axis are parabolas. The most common type, an elliptic paraboloid, opens upward or downward like an infinite satellite dish.

An elliptic paraboloid is a quadric surface defined by an equation of the form $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ , where $a$ and $b$ are positive constants controlling the curvature along the $x$ - and $y$ -axes respectively. When $a = b$ , the surface is a circular paraboloid with perfectly circular cross-sections. A hyperbolic paraboloid, $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ , is a saddle-shaped surface and constitutes the other major type.

Key Formula

z = \frac{x^2}{a^2} + \frac{y^2}{b^2}

Where:

$a$ = Controls the width of the paraboloid along the x-axis
$b$ = Controls the width of the paraboloid along the y-axis
$z$ = Height above the vertex of the paraboloid

How It Works

To analyze a paraboloid, set

z = k

for various constants

k

to see horizontal cross-sections (called level curves or traces). For the elliptic paraboloid

z = x^2 + y^2

, setting

z = 4

gives the circle

x^2 + y^2 = 4

at height 4. Setting

y = 0

gives the parabola

z = x^2

in the

xz

-plane. These traces help you sketch the surface and set up integrals for volume or surface area calculations.

Worked Example

Problem: Find the radius of the circular cross-section of the paraboloid

z = x^2 + y^2

at height

z = 9

Set z equal to 9: Substitute the given height into the equation of the paraboloid.

9 = x^2 + y^2

Identify the shape: This is the equation of a circle centered at the origin in the xy-plane with radius

r

where

r^2 = 9

r = \sqrt{9} = 3

Answer: The cross-section at height

z = 9

is a circle of radius 3.

Why It Matters

Paraboloids appear throughout multivariable calculus when you set up double and triple integrals to compute volumes, surface areas, and flux. They also model real-world shapes — satellite dishes, reflectors, and telescope mirrors use the reflective property of parabolic surfaces to focus signals to a single point.

Common Mistakes

Mistake: Confusing an elliptic paraboloid (bowl) with a hyperbolic paraboloid (saddle).

Correction: Check the signs: if both squared terms have the same sign (

+

), the surface is an elliptic paraboloid. If the signs differ (

+

and

-

), it is a hyperbolic paraboloid.