Spheroid

Q: What is the difference between a spheroid and a sphere?

A sphere has all three semi-axes equal ($a = c$), so every cross-section through the center is the same circle. A spheroid has two equal semi-axes but the third differs, making it either flattened (oblate, $c a$). A sphere is a special case of a spheroid where $a = c$.

Spheroid

Usually this means oblate spheroid. Sometimes, however, spheroid refers to any ellipsoid that is approximately a sphere.

See also

Prolate spheroid

Key Formula

\frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2} = 1

Where:

$a$ = The equatorial radius (semi-axis in the x- and y-directions, which are equal by rotational symmetry)
$c$ = The polar radius (semi-axis in the z-direction, the axis of rotation)

Worked Example

Problem: An oblate spheroid has equatorial radius a = 6 and polar radius c = 4. Find its volume.

Step 1: Recall the volume formula for a spheroid. Since a spheroid is an ellipsoid with two equal semi-axes, the volume simplifies from the general ellipsoid formula.

V = \frac{4}{3}\pi a^2 c

Step 2: Substitute a = 6 and c = 4 into the formula.

V = \frac{4}{3}\pi (6)^2 (4) = \frac{4}{3}\pi (36)(4)

Step 3: Multiply the numerical values.

V = \frac{4}{3}\pi \cdot 144 = \frac{576}{3}\pi = 192\pi

Step 4: Approximate the result.

V \approx 603.19 \text{ cubic units}

Answer: The volume is

192\pi \approx 603.19

cubic units.

Another Example

Problem: A prolate spheroid (like a rugby ball) has equatorial radius a = 3 and polar radius c = 5. Find its volume.

Step 1: Use the same volume formula. It applies to both oblate and prolate spheroids.

V = \frac{4}{3}\pi a^2 c

Step 2: Substitute a = 3 and c = 5.

V = \frac{4}{3}\pi (3)^2 (5) = \frac{4}{3}\pi (9)(5)

Step 3: Compute the result.

V = \frac{4}{3}\pi \cdot 45 = \frac{180}{3}\pi = 60\pi \approx 188.50 \text{ cubic units}

Answer: The volume is

60\pi \approx 188.50

cubic units.

Frequently Asked Questions

What is the difference between a spheroid and a sphere?

A sphere has all three semi-axes equal (

a = c

), so every cross-section through the center is the same circle. A spheroid has two equal semi-axes but the third differs, making it either flattened (oblate,

c < a

) or elongated (prolate,

c > a

). A sphere is a special case of a spheroid where

a = c

Is the Earth a sphere or a spheroid?

The Earth is best modeled as an oblate spheroid. Its equatorial radius is about 6,378 km while its polar radius is about 6,357 km, so it bulges slightly at the equator due to its rotation. The difference is small (about 0.3%), which is why it looks nearly spherical.

Spheroid vs. Ellipsoid

A spheroid is a special case of an ellipsoid. A general ellipsoid has three distinct semi-axes (

a

b

c

), so its equation is

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

. A spheroid constrains two of those axes to be equal (

a = b

), giving it rotational symmetry about the third axis. Every spheroid is an ellipsoid, but not every ellipsoid is a spheroid.

Why It Matters

Spheroids are essential in geodesy and navigation because the Earth's shape is modeled as an oblate spheroid (the WGS 84 reference ellipsoid used by GPS). In physics, many rotating bodies — planets, stars, atomic nuclei — naturally assume spheroidal shapes due to the balance between gravity and centrifugal force. Understanding spheroids also helps in engineering contexts like antenna design and optics, where spheroidal reflectors focus signals along specific axes.

Common Mistakes

Mistake: Confusing oblate and prolate spheroids.

Correction: An oblate spheroid is flattened along its axis of rotation (

c < a

), like the Earth or a doorknob. A prolate spheroid is elongated along its axis (

c > a

), like a rugby ball or an egg. Remember: 'oblate' sounds like 'plate' — flat.

Mistake: Using the sphere volume formula

\frac{4}{3}\pi r^3

for a spheroid.

Correction: A spheroid has two different radii, so you must use

V = \frac{4}{3}\pi a^2 c

. The sphere formula only works when

a = c

Related Terms

Oblate Spheroid — Spheroid flattened at the poles (c < a)
Prolate Spheroid — Spheroid elongated along the polar axis (c > a)
Ellipsoid — General surface with three distinct semi-axes
Sphere — Special spheroid where all semi-axes are equal
Ellipse — 2D curve whose rotation generates a spheroid
Surface of Revolution — Method of creating a spheroid by rotating a curve