Understanding Dimensions and Finding the Center of Mass of a Spheroid

Last Updated on October 17, 2025 by Rajeev Bagra

When we study geometry and physics, we often hear about one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) objects. Understanding these dimensions helps us determine how to locate an object’s center of mass — the point where its entire mass can be thought to be concentrated.

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🌐 What 1D, 2D, and 3D Mean

1. One-Dimensional (1D):
   A 1D object such as a straight line or a uniform rod has only length. Its center of mass lies at the midpoint, assuming uniform density.
2. Two-Dimensional (2D):
   A 2D object such as a rectangle, triangle, or circle has length and breadth. It lies entirely on a plane, and we find the centroid of area instead of the center of mass.
3. Three-Dimensional (3D):
   A 3D object such as a cuboid, sphere, or spheroid has length, breadth, and height (or depth). In this case, we find the center of mass of volume — the true spatial balance point.

A straight line is 1D, a rectangle is 2D, and a rectangular solid or spheroid is 3D. If the density is uniform, the geometric center equals the center of mass.

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⚙️ Center of Mass and the Disk–Shell Analogy

To understand the center of mass of a spheroid, we can use calculus methods similar to finding the volume of solids of revolution.

When a 2D ellipse is rotated around the x-axis, it generates a spheroid (a 3D object).

The disk method and shell method are two equivalent ways to calculate such volumes.

Disk Method: Revolve vertical slices around an axis. Each slice forms a thin disk of radius and height .
Shell Method: Revolve vertical rectangles around an axis. Each forms a cylindrical shell of radius and thickness .

In both cases, symmetry ensures that the center of mass of a uniform spheroid lies exactly at the center — along the x, y, and z axes.

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🧠 Comparing Dimensions and Centers

Object	Dimension	Quantities Involved	Center Type	Method Used
Line	1D	Length	Midpoint	Average of endpoints
Rectangle	2D	Length & Breadth	Centroid of area	Integration over area
Spheroid	3D	Length, Breadth, Height	Center of mass	Disk or shell method

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🖼️ Visual Representation

Below is a simple visual comparison showing how dimensions evolve:

• A straight line → 1D → midpoint as center of mass.
• A flat rectangle → 2D → centroid of area.
• A rotated ellipse (spheroid) → 3D → center of mass of volume.