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Volume of a Tetrahedron

```Date: 02/27/2003 at 15:48:16
From: Haze
Subject: Tetrahedron

How do I find the volume of a tetrahedron?
```

```
Date: 03/02/2003 at 18:43:35
From: Doctor Ian
Subject: Re: Tetrahedron

Hi Haze,

The most straightforward way to find it is to apply the Pythagorean
theorem several times.

Here is an equilateral triangle, with vertices A, B, and C; center D;
and midpoint of base E.  It's just like the one at the base of your
tetrahedron.

A
. .
. . .
.  .  .
.   .   .
.    .    .
.     D     .
.   .     .   .
.  .          . .
C . . . .E. . . . B

For brevity, we'll use

a = AB = BC = CA

h = AE

d = DE

r = CD

The Pythagorean theorem tells us that

h^2 + (a/2)^2 = a^2

h^2 = a^2 - (a/2)^2

h^2 = 3a^2/4

h = a sqrt(3)/2

So the area of the whole triangle is

area = (1/2) * base * height

= (1/2) * a * a sqrt(3)/2

= a^2 sqrt(3)/4

The area of triangle DBC is 1/3 the area of the whole triangle, so

(1/3) a^2 sqrt(3)/4 = (1/2) * base * height

= (1/2) a d

(2/a)(1/3) a^2 sqrt(3)/4 = d

a sqrt(3)/6 = d

The Pythagorean theorem tells us that

r^2 = (a/2)^2 + d^2

= (a/2)^2 + [a sqrt(3)/6]^2

= a^2/4 + 3a^2/36

= a^2/4 + a^2/12

= (3a^2 + a^2)/12

= 4a^2/12

= a^2/3

r = a/sqrt(3)

So, if we make a tetrahedron from four of these triangles, and drop an
altitude from the apex of the tetrahedron to the center of the base,
we get a right triangle hypotenuse a, base r, and height H.  The
Pythagorean theorem tells us that

r^2 + H^2 = a^2

H^2 = a^2 - r^2

= a^2 - a^2/3

= 2a^2/3

H = a sqrt(2/3)

Since the volume of any pyramid is 1/3 the area of the base times
the height,

Volume of a Pyramid
http://mathforum.org/library/drmath/view/55041.html

the volume is

volume = (1/3) * area of base * height

= (1/3) * [a^2 sqrt(3)/4] * [a sqrt(2/3)]

= a^3 (1/3) [sqrt(3)/4] [sqrt(2/3)]

= a^3 sqrt(2)/12

Happily, this agrees with the formula in our Dr. Math Geometric
Formulas FAQ:

Pyramid & Frustum Formulas
http://mathforum.org/dr.math/faq/formulas/faq.pyramid.html

I hope this helps!

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polyhedra

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