Today's post is more related to mathematics, as you can tell from the title. I'll be telling you about the amazing solid shape, the octahedron.
WHAT YOU WILL LEARN TODAY:
We will have found-
-->The derivation of the formula for the volume of the octahedron.
-->The ratio of the volume of an octahedron when placed inside a cube with its vertices touching the centers of the faces of the cube and the volume of the cube.
So! Let's begin! You will have to understand platonic solids first.
PLATONIC SOLIDS:
Platonic solids are five geometric solids that have congruent and regular sides that meet at three-dimensional angles.
Plato used them to explain the structure of the universe. Hence, the name 'Platonic'.
There are five Platonic solids:
the tetrahedron with four faces,
the cube with six faces,
the octahedron with eight faces,
the dodecahedron with twelve faces,
and the icosahedron with twenty faces.
They are used in many fields, including biomedical research, architecture, art, and
even nanotechnology.
They are found in many places in nature. For example:
-->Crystals can be in the form of
octahedrons, cubes or tetrahedrons.
-->Viruses and algae are icosahedrons.
Sugar and salt come in cubes.
The main focus today is the octahedron.
THE OCTAHEDRON:
The octahedron has a square base, this can be seen in 3 axes.
If we take one of those axes and take the square base with side 'a', it's diagonal will be √2 a (Pythagoras theorem).
The same diagonal when seen from another the adjacent axis, becomes the height of the octahedron.
The octahedron is now divided into 2 square pyramids, therefore the height is halved(√2 a/2).
As you may know, the volumes of a cone and cylinder with same height and radius are in the ratio of 1:3.
Similarly, the volumes of the square pyramid and a cuboid of the same side length and height are in ratio 1:3. Therefore, the volume of the pyramid is base area x height/3.
Base area= a^2 (Square base)
Height= √2 a/2
Volume of oct = 2(volume of py)
Therefore, the volume of octahedron=
2 ((a^2 x √2 a/2) /3)
= √2 a^3/3
After this, we are going to find the relation between the volume of the cube and the volume of the octahedron.
Here, the octahedron is such that it can fit inside the cube with the vertices touching the centres of the faces of the cube.
Using Pythagoras theorem we can find the area of the base of the octahedron.
The octahedron can be divided into two pyramids. So using the base area and the height (which is half the side of the cube) we can get the volume of one pyramid and multiply it by two to get the volume of the octahedron.
Now we can relate it to the volume of the cube and get the ratio (which is 6:1).
And there you go!
NOW YOU KNOW:
-->The derivation of the formula for the volume of the octahedron.
-->The ratio of the volume of an octahedron when placed inside a cube with its vertices touching the centers of the faces of the cube and the volume of the cube.
HOPE YOU LIKED THIS POST AND LEARNED SOMETHING GOOD TODAY
THANKS FOR READING
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