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Prolate Spheroid

Prolate Spheroid

A stretched sphere shaped like a watermelon. Formally, a prolate spheroid is a surface of revolution obtained by revolving an ellipse about its major axis.

 

Prolate spheroid: a tall oval (egg-like) 3D shape with a vertical major axis and a dashed horizontal ellipse showing its...

 

 

See also

Oblate spheroid, ellipsoid, spheroid

Key Formula

x2a2+y2a2+z2c2=1where c>a\frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2} = 1 \quad \text{where } c > a
Where:
  • aa = The semi-axis length along the x- and y-directions (the two equal, shorter semi-axes)
  • cc = The semi-axis length along the z-direction (the longer semi-axis, aligned with the axis of revolution)
  • x,y,zx, y, z = Coordinates of any point on the surface of the spheroid

Worked Example

Problem: A prolate spheroid has semi-axes a = 3 cm and c = 5 cm. Find its volume and surface area.
Step 1: Write the volume formula for a prolate spheroid. Since it is an ellipsoid with two equal semi-axes, the volume is:
V=43πa2cV = \frac{4}{3}\pi a^2 c
Step 2: Substitute a = 3 and c = 5:
V=43π(3)2(5)=43π(9)(5)=43π(45)=60π≈188.5 cm3V = \frac{4}{3}\pi (3)^2(5) = \frac{4}{3}\pi (9)(5) = \frac{4}{3}\pi (45) = 60\pi \approx 188.5 \text{ cm}^3
Step 3: For the surface area, first compute the eccentricity. The eccentricity of the generating ellipse is:
e=1−a2c2=1−925=1625=45=0.8e = \sqrt{1 - \frac{a^2}{c^2}} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} = 0.8
Step 4: Apply the prolate spheroid surface area formula:
S=2πa2+2πacesin⁡−1(e)S = 2\pi a^2 + 2\pi \frac{ac}{e}\sin^{-1}(e)
Step 5: Substitute a = 3, c = 5, and e = 0.8. Note that sin⁻¹(0.8) ≈ 0.9273 radians:
S=2π(9)+2π⋅(3)(5)0.8⋅0.9273=18π+2π(18.75)(0.9273)=18π+34.77π≈52.77π≈165.8 cm2S = 2\pi(9) + 2\pi \cdot \frac{(3)(5)}{0.8} \cdot 0.9273 = 18\pi + 2\pi(18.75)(0.9273) = 18\pi + 34.77\pi \approx 52.77\pi \approx 165.8 \text{ cm}^2
Answer: The volume is 60π ≈ 188.5 cm³ and the surface area is approximately 165.8 cm².

Another Example

This edge case shows that a sphere is a degenerate prolate spheroid where the two distinct semi-axis lengths become equal, confirming that the spheroid formula reduces to the sphere formula.

Problem: A prolate spheroid has semi-axes a = 4 m and c = 4 m. Describe the shape and find its volume.
Step 1: Check the relationship between a and c. Here a = c = 4, so the three semi-axes are all equal.
a=c=4a = c = 4
Step 2: When a = c, the prolate spheroid becomes a perfect sphere of radius 4 m. The eccentricity is zero:
e=1−a2c2=1−1=0e = \sqrt{1 - \frac{a^2}{c^2}} = \sqrt{1 - 1} = 0
Step 3: Compute the volume using the spheroid formula (or equivalently, the sphere formula):
V=43πa2c=43π(4)2(4)=43π(64)=2563π≈268.1 m3V = \frac{4}{3}\pi a^2 c = \frac{4}{3}\pi (4)^2(4) = \frac{4}{3}\pi (64) = \frac{256}{3}\pi \approx 268.1 \text{ m}^3
Answer: When a = c = 4, the prolate spheroid is a sphere with volume (256/3)π ≈ 268.1 m³.

Frequently Asked Questions

What is the difference between a prolate spheroid and an oblate spheroid?
A prolate spheroid is elongated along its axis of symmetry (like a football), formed by rotating an ellipse about its major axis, so c > a. An oblate spheroid is flattened along its axis (like Earth), formed by rotating an ellipse about its minor axis, so c < a. Both are special cases of an ellipsoid with two equal semi-axes.
What are real-life examples of a prolate spheroid?
An American football (or rugby ball), a watermelon, and many atomic nuclei are approximately prolate spheroids. Certain planets and moons that spin slowly can also take on a prolate shape due to tidal forces.
How do you find the eccentricity of a prolate spheroid?
The eccentricity is calculated as e = √(1 − a²/c²), where c is the longer semi-axis and a is the shorter one. When e = 0 the shape is a sphere, and as e approaches 1 the shape becomes increasingly elongated and needle-like.

Prolate Spheroid vs. Oblate Spheroid

Prolate SpheroidOblate Spheroid
DefinitionEllipse rotated about its major axis (c > a)Ellipse rotated about its minor axis (c < a)
ShapeElongated, like a footballFlattened, like Earth
Equationx²/a² + y²/a² + z²/c² = 1, c > ax²/a² + y²/a² + z²/c² = 1, c < a
Volume formula(4/3)πa²c (same formula)(4/3)πa²c (same formula)
Surface area formulaUses sin⁻¹(e) (inverse sine)Uses ln[(1+e)/(1−e)] (logarithmic)
Eccentricitye = √(1 − a²/c²)e = √(1 − c²/a²)
Real-world exampleRugby ball, eggEarth, M&M candy

Why It Matters

Prolate spheroids appear in physics (atomic nuclei, antennas), sports (the shape of a football determines its aerodynamics), and astronomy (tidally locked moons). Understanding the volume and surface area formulas lets you solve real engineering and science problems involving elongated rounded objects. In coordinate geometry courses, the prolate spheroid is a key example of a surface of revolution and a special case of the general ellipsoid equation.

Common Mistakes

Mistake: Confusing prolate with oblate by rotating the ellipse about the wrong axis.
Correction: Remember: prolate means stretched (rotate about the major/longer axis, c > a). Oblate means squashed (rotate about the minor/shorter axis, c < a). A mnemonic: 'pro-LATE' — the shape is 'late' getting somewhere because it's stretched out long.
Mistake: Using the oblate surface area formula (which involves a logarithm) for a prolate spheroid.
Correction: The prolate surface area formula uses an inverse sine (arcsin) function, while the oblate formula uses a natural logarithm. Always check whether c > a or c < a before selecting the correct formula.

Related Terms

  • Sphere — Degenerate case when both semi-axes are equal
  • Surface of Revolution — Method used to generate the spheroid shape
  • Ellipse — The 2D curve that is rotated to form the spheroid
  • Major Axis of an Ellipse — The axis about which the ellipse is revolved
  • Oblate Spheroid — Counterpart formed by revolving about the minor axis
  • Ellipsoid — General 3D shape; spheroid is a special case
  • Spheroid — General term covering both prolate and oblate types