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# Center of Mass of a 4D Pyramid

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Find the center of mass of a 4-D pyramid (cube base) using algebraic means (no calculus is allowed). I need a step-by-step solution with all work shown.

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The center of mass of a 4-dimensional pyramid is found without the use of calculus. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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Suppose the cube base has its side of size A (3D volume A^3) and the height of the pyramid in the 4th dimension is H
(let us use the word "height" for the 4th dimension only hereafter).

We assume the material to be homogeneous and identify the center of mass with the "center of volume".

The center of mass of a cube is in cube's center.
Let us slice the pyramid into N parts of equal height (H/N) each.
If we count the slices from the top, the larger base of each slice is a cube of side equal

A_k = (k/N)*A (1)

at height

H_k = (k/N)*H (2)

(counting from the top to the base).
Therefore its 3D volume is

V_k(3D) = A^3k^3/N^3 (3)

If we take N very large, these slices will be very thin.
As N ...

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