Volume of the Regular Tetrahedron and Regular Octahedron

move

Let

d

be the length of an edge of a regular tetrahedron

T

inscribed in a cube

C

of edge length

a=d

2

. Let

vol(F)

stand for the volume of a solid

F

. We get

T

by cutting away four triangular pyramids from

C

. The volume of such a pyramid

P

is

vol(P)=1/6

3

a

=

3

d

12

2

. Then

vol(T)=vol(C)-4vol(P)=

3

d

2

-

3

d

3

2

=

3

d

6

2

. But these four pyramids form one half of the regular octahedron

O

, which therefore has volume

3

d

2

3

and the ratio

vol(O)/vol(T)=4

.

External Links

Regular Tetrahedron (Wolfram MathWorld)

Octahedron (Wolfram MathWorld)

The Volume of the Regular Octahedron Is Four Times the Volume of the Regular Tetrahedron

Permanent Citation

Izidor Hafner

"Volume of the Regular Tetrahedron and Regular Octahedron"
http://demonstrations.wolfram.com/VolumeOfTheRegularTetrahedronAndRegularOctahedron/
Wolfram Demonstrations Project
Published: November 13, 2014