Spheroid formula for volume and surface area with examples
In-plane geometry, a spheroid shape, or ellipsoid of revolution, refers to a quadric surface obtained by rotating an ellipse about one of its principal axes. This is to say, a spheroid is an ellipsoid with 2 equal semi-diameters with a circular symmetry.
If the ellipse is revolved about its major axis, the outcome is a prolate (elongated) spheroid, shaped like a rugby ball. An oblate or flattened spheroid is an outcome of a rotating ellipse around its minor axis. If the producing ellipse is a circle, the outcome is a sphere.
Applications of Spheroid
The neutrons and protons of an atomic nucleus are mostly found in the spherical, oblate, or prolate spherical shapes along with having a spinning axis. Deformed nuclear shapes typically form as an outcome of the competition between electromagnetic repulsion between protons, quantum shell effects, and surface tension.
Oblate Spheroids Meaning
A ''squashed'' Spheroid of which the equatorial radius ‘a’ is larger than the polar radius ‘c’, such that a > c. To a first approximation, the shape assumed by a rotating fluid (inclusive of the Earth, that is ``fluid'' over astronomical time scales) is known as an oblate spheroid. The oblate spheroid can be identified parametrically by the general Spheroid equations (for a Spheroid having z-Axis as the symmetry axis).
Oblate Spheroid Shape
The oblate spheroid shape can be defined as an assumed shape for various celestial bodies and planets that are rotating rapidly at their own axis. For example, the Earth, Saturn, and Jupiter. The English mathematician, Enlightenment scientist, and astronomer –Isaac Newton reasoned that Earth and Jupiter are oblate spheroids owing to their centrifugal force. Earth's diverse geodetic and cartographic systems are formed on reference ellipsoids, all of which are oblate.
Example of Oblate Spheroids
An example of an oblate spheroid is the planet Jupiter with a flattening spheroid of 0.06487. Another science-fiction of an extremely oblate spheroid earth is Mesklin from Hal Clement's novel Mission of Gravity.
Prolate Spheroids
The prolate spheroid is an estimated shape of the ball in various sports, such as rugby football. The word is also used to depict the shape of some nebulae like the Crab Nebula.
Several moons of the Solar System estimate prolate spheroids in shape, although they are really triaxial ellipsoids.
On contrary to being distorted into oblate spheroids through quick rotation, celestial objects distort little into prolate spheroids through tidal forces when they orbit a massive body in a close orbit. An utmost example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit because of little eccentricity, inducing intense volcanism. However, the major axis of a prolate spheroid does not run across the satellite's poles, but through the two points on its equator straightaway facing toward and away from the primary.
Example of a Prolate Spheroids
An Australian rules football is an example of a prolate spheroid.
Many submarines have a shape that can be depicted as a prolate spheroid.
Another example includes the atomic nucleus of the element belonging to the lanthanide and actinide groups.
In anatomy, near-spheroid organs such as testis.
Fresnel zones used to evaluate wave propagation and interference in space.
Other Examples of prolates include Saturn's satellites Enceladus, Mimas, and Uranus' and Tethys satellite Miranda.
Geoid and Spheroid
The geoid is essentially described as the surface of the earth's gravity field that approximates mean sea level. The geoid is perpendicular to the direction of the force of gravity. Because the mass of the Earth is not uniform in nature at all points, the magnitude of gravity thus differs, and the shape of the geoid is irregular.
Fun Facts
Saturn is considered to be the most oblate planet in the Solar System. It has a flattening of 0.09796.
FAQs on Spheroid in Geometry Definition and Key Concepts
1. What is a spheroid in mathematics?
A spheroid is a three-dimensional surface obtained by rotating an ellipse about one of its principal axes. It is a special type of ellipsoid where two of the three axes are equal.
- If the ellipse rotates about its major axis, it forms a prolate spheroid.
- If it rotates about its minor axis, it forms an oblate spheroid.
- A sphere is a special case of a spheroid where all three axes are equal.
2. What is the difference between a sphere and a spheroid?
The main difference is that a sphere has all radii equal, while a spheroid has two equal radii and one different radius.
- Sphere: All axes equal (a = b = c).
- Spheroid: Two equal axes (a = b ≠ c).
- A sphere is a special case of a spheroid.
3. What are the types of spheroids?
There are two main types of spheroids: oblate spheroid and prolate spheroid.
- Oblate spheroid: Flattened at the poles (equatorial radius > polar radius), like Earth.
- Prolate spheroid: Stretched along the axis (polar radius > equatorial radius), like a rugby ball.
4. What is the volume formula for a spheroid?
The volume of a spheroid is given by V = (4/3)πa²c, where a is the equatorial radius and c is the polar radius.
- This formula applies to both oblate and prolate spheroids.
- If a = c, the formula becomes (4/3)πr³, which is the volume of a sphere.
5. What is the surface area of a spheroid?
The surface area of a spheroid depends on whether it is oblate or prolate and involves its eccentricity.
- For an oblate spheroid: S = 2πa²(1 + (1 − e²)/e · atanh(e))
- For a prolate spheroid: S = 2πa²(1 + c/(ae) · sin⁻¹(e))
- Here, e is the eccentricity and a, c are radii.
6. How do you calculate the volume of a spheroid step by step?
To calculate the volume of a spheroid, use the formula V = (4/3)πa²c.
- Step 1: Identify the equatorial radius (a).
- Step 2: Identify the polar radius (c).
- Step 3: Substitute into the formula.
- Example: If a = 3 and c = 5,
V = (4/3)π × 3² × 5 = (4/3)π × 9 × 5 = 60π cubic units.
7. What is the equation of a spheroid?
The standard equation of a spheroid is x²/a² + y²/a² + z²/c² = 1.
- a = equatorial radius (x and y directions)
- c = polar radius (z direction)
- If a = c, the equation becomes that of a sphere.
8. Why is the Earth called an oblate spheroid?
The Earth is called an oblate spheroid because it is slightly flattened at the poles and bulges at the equator.
- Equatorial radius ≈ 6378 km
- Polar radius ≈ 6357 km
- Since equatorial radius > polar radius, Earth is not a perfect sphere.
9. What is the eccentricity of a spheroid?
The eccentricity (e) of a spheroid measures how much it deviates from a sphere.
- For oblate spheroid: e = √(1 − c²/a²), where a > c.
- For prolate spheroid: e = √(1 − a²/c²), where c > a.
- If e = 0, the shape is a perfect sphere.
10. What are real-life examples of spheroids?
Common real-life examples of spheroids include planets, sports balls, and rotating objects.
- Earth – oblate spheroid due to rotation.
- Rugby ball – prolate spheroid.
- Some stars and rotating celestial bodies also form spheroidal shapes.
