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Volumes of Polyhedra

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I need help writing a proof of the formulas of the volumes of two regular polyhedra (Platonic solids):

(1) a tetrahedron and
(2) an octahedron.

I then have to use those formulas to find the volumes given a side length of 1.

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Solution Summary

The solution shows you how to derive the volumes of a regular tetrahedron and a regular octahedron.

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Volumes of Polyhedra

(1) Regular tetrahedron:

A tetrahedron is a 3-sided pyramid with equilateral triangles for its sides.

The volume of any pyramid with height h and area of base A is:

V = h · A / 3

So, to find the volume of the tetrahedron, we have to find its height and the area of its base.

(a) Let's find the height first. For now, we're going to denote the lengths of the edges as s.

To find the height, draw a line from a vertex to the midpoint of the opposite side. This line has length h, the height of the tetrahedron. It forms a right triangle along with one of the edges (of length s) and part of the base. By the Pythagorean theorem,

s2 = h2 + (s/√3)2

Therefore, h = √(2/3) * s:

s2 = h2 + (s/√3)2
h2 = -s2/3 + s2
h2 = s2(-1/3 ...

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