Oblate Spheroid — Definition, Formula & Examples

Q: What is the difference between an oblate spheroid and a prolate spheroid?

Both are spheroids formed by revolving an ellipse, but they differ in which axis is used. An oblate spheroid is created by revolving an ellipse about its minor (shorter) axis, producing a shape flattened at the poles — like the Earth. A prolate spheroid is created by revolving an ellipse about its major (longer) axis, producing an elongated shape — like a rugby ball or an American football.

Q: Why is the Earth an oblate spheroid and not a perfect sphere?

The Earth spins on its axis, and this rotation creates a centrifugal effect that pushes material outward near the equator. As a result, the equatorial radius (about 6,378 km) is roughly 21 km larger than the polar radius (about 6,357 km). This makes the Earth slightly flattened at the poles, fitting the definition of an oblate spheroid.

Q: Is an oblate spheroid the same as an ellipsoid?

An oblate spheroid is a special case of an ellipsoid. A general ellipsoid has three distinct semi-axes (a, b, c that can all differ). An oblate spheroid has two equal semi-axes (a = b) with the third axis c shorter, giving it rotational symmetry about the polar axis. Every oblate spheroid is an ellipsoid, but not every ellipsoid is an oblate spheroid.

Oblate Spheroid

A flattened sphere. More formally, an oblate spheroid is a surface of revolution obtained by revolving an ellipse about its minor axis.

Note: The earth is shaped like an oblate spheroid.

Oblate spheroid: a sphere flattened at top and bottom, wider than tall, with a dotted equatorial circle showing its compressed...

See also

Prolate spheroid, ellipsoid

Key Formula

V = \frac{4}{3}\pi a^2 c \qquad\qquad S = 2\pi a^2 + \frac{\pi c^2}{e}\ln\!\left(\frac{1+e}{1-e}\right)

Where:

$V$ = Volume of the oblate spheroid
$S$ = Surface area of the oblate spheroid
$a$ = Semi-major axis (equatorial radius), the longer radius
$c$ = Semi-minor axis (polar radius), the shorter radius, where c < a
$e$ = Eccentricity, defined as e = √(1 − c²/a²)
$\ln$ = Natural logarithm

Worked Example

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

Step 1: Write the volume formula for an oblate spheroid.

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

Step 2: Substitute a = 6 and c = 4.

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: Compute the decimal approximation.

V \approx 192 \times 3.14159 \approx 603.19 \text{ cm}^3

Answer: The volume is

192\pi \approx 603.19

cm³.

Another Example

This example focuses on surface area rather than volume, requiring the student to first compute the eccentricity and work with the natural logarithm term — a more advanced calculation.

Problem: An oblate spheroid has an equatorial radius of a = 5 m and a polar radius of c = 3 m. Find its surface area.

Step 1: First calculate the eccentricity e.

e = \sqrt{1 - \frac{c^2}{a^2}} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} = 0.8

Step 2: Write the surface area formula for an oblate spheroid.

S = 2\pi a^2 + \frac{\pi c^2}{e}\ln\!\left(\frac{1+e}{1-e}\right)

Step 3: Compute the first term: the equatorial contribution.

2\pi a^2 = 2\pi(25) = 50\pi

Step 4: Compute the logarithmic part inside the second term.

\ln\!\left(\frac{1+0.8}{1-0.8}\right) = \ln\!\left(\frac{1.8}{0.2}\right) = \ln(9) \approx 2.1972

Step 5: Compute the second term and add to the first.

\frac{\pi (9)}{0.8} \times 2.1972 = \frac{9\pi}{0.8} \times 2.1972 = 11.25\pi \times 2.1972 \approx 24.72\pi

Step 6: Add both terms for the total surface area.

S \approx 50\pi + 24.72\pi = 74.72\pi \approx 234.76 \text{ m}^2

Answer: The surface area is approximately

74.72\pi \approx 234.76

m².

Frequently Asked Questions

What is the difference between an oblate spheroid and a prolate spheroid?

Both are spheroids formed by revolving an ellipse, but they differ in which axis is used. An oblate spheroid is created by revolving an ellipse about its minor (shorter) axis, producing a shape flattened at the poles — like the Earth. A prolate spheroid is created by revolving an ellipse about its major (longer) axis, producing an elongated shape — like a rugby ball or an American football.

Why is the Earth an oblate spheroid and not a perfect sphere?

The Earth spins on its axis, and this rotation creates a centrifugal effect that pushes material outward near the equator. As a result, the equatorial radius (about 6,378 km) is roughly 21 km larger than the polar radius (about 6,357 km). This makes the Earth slightly flattened at the poles, fitting the definition of an oblate spheroid.

Is an oblate spheroid the same as an ellipsoid?

An oblate spheroid is a special case of an ellipsoid. A general ellipsoid has three distinct semi-axes (a, b, c that can all differ). An oblate spheroid has two equal semi-axes (a = b) with the third axis c shorter, giving it rotational symmetry about the polar axis. Every oblate spheroid is an ellipsoid, but not every ellipsoid is an oblate spheroid.

Oblate Spheroid vs. Prolate Spheroid

	Oblate Spheroid	Prolate Spheroid
Definition	Ellipse revolved about its minor axis	Ellipse revolved about its major axis
Shape	Flattened at the poles, wider at the equator	Elongated along the axis of revolution
Semi-axis relationship	a = b > c (polar axis is shortest)	a = b < c (polar axis is longest)
Volume formula	V = (4/3)πa²c where a > c	V = (4/3)πa²c where c > a
Real-world example	Earth, Jupiter, Saturn	Rugby ball, American football, watermelon

Why It Matters

You encounter oblate spheroids in geography and physics when modeling the shape of the Earth and other rotating planets. GPS systems, satellite orbits, and map projections all rely on an oblate spheroid model of the Earth (specifically the WGS 84 ellipsoid) rather than a perfect sphere for accurate positioning. In geometry courses, oblate spheroids provide a concrete example of surfaces of revolution and connect your understanding of ellipses to three-dimensional solids.

Common Mistakes

Mistake: Confusing which axis to revolve around: students sometimes think 'oblate' means revolving about the major axis.

Correction: Oblate means flattened, so you revolve the ellipse about its minor (shorter) axis. Revolving about the major axis produces a prolate (elongated) spheroid instead.

Mistake: Using the sphere volume formula (4/3)πr³ instead of (4/3)πa²c.

Correction: A sphere has one radius, but an oblate spheroid has two distinct measurements: the equatorial radius a and the polar radius c. You must use both in the formula. Only when a = c does the formula reduce to that of a sphere.

Related Terms

Sphere — Special case where a = c (no flattening)
Spheroid — General term for oblate or prolate types
Prolate Spheroid — Ellipse revolved about its major axis
Ellipsoid — General solid with three distinct semi-axes
Ellipse — The 2D curve that generates the spheroid
Minor Axis of an Ellipse — The axis of revolution for an oblate spheroid
Surface of Revolution — Method used to create the spheroid shape