Geometry is one of the Ancient branches of mathematics. It is concerned with the properties of space that are related to distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Platonic Solids

A Platonic solid is an ordinary polyhedron in three-layered Euclidean space. Being a regular polyhedron implies that the faces are compatible and are indistinguishable in shape and size, regular polygons are where all angles are identical and all edges are the same, and a similar number of faces meet at every vertex. There are only five polyhedral.

The Platonic solids are unmistakable in the way of thinking of Plato, their namesake. In Timaeus c.360 B.C. Plato wrote about them, in which he related every one of the four traditional components (earth, air, water, and fire) with a regular solid. Cube was related to the earth, Octahedron was related to the air, icosahedron was related to the water, tetrahedron was related to the fire, and dodecahedron was related as heavens were made with this. Five types of platonic solids are,

1. Cube
2. Octahedron
3. Icosahedron
4. Tetrahedron
5. Dodecahedron

Platonic Solids Formula

There are multiple formulae for platonic solids as there are a variety of solids as discussed earlier, there is the cube, octahedron, icosahedron, etc. Then there are different formulae, for instance, total surface area, lateral surface area, etc. Let’s take a look at these formulae in more detail,

Cube 

In the cube formulas, we will get to know how to find the diagonals, volume of the cube, and surface area of the cube. The cube contains twelve edges, eight vertices, and six faces.

 

Surface Area of the cube

The surface area of the cube is divided into two types, 

1. Lateral Surface Area of Cube
2. Total Surface Area of Cube

Lateral Surface Area of a cube is the sum of all side faces, so in the cube, there are 4 side faces. Hence the  Lateral Surface area of the cube is,

Lateral Surface area = 4a²

Where a = length of the side.

The total Surface area of the cube is sum of the base area and vertical surface area. So the cube is of the same dimensions as squares. Hence the Total surface area of the cube is,

Total surface area = 6a²

Where a = length of the side.

Volume of the cube

In the volume formula of a cube, we can specify the Volume in two ways,

Volume of the cube by side length = a³

Where a = length of the side

Or

Volume of the cube by diagonal length = (√3× d³)/9

Where d = Diagonal length of the cube 

Diagonal of a cube

The line that joins the two opposite vertices of a cube is called the diagonal of the cube. The diagonal of the cube helps us to find the main and face diagonals’ lengths.

Length of the main diagonal = √3 × a units.

Length of the face diagonal  = √ 2 × a units.

Where a = Length of the side.

Octahedron

In the Octahedron formulas, we will get to know how to find the volume of the Octahedron and the surface area of the Octahedron. Octahedron contains twelve edges, six vertices, and 4 edges that meet at each vertex, eight faces and having equilateral triangle shape.

 

Surface area of the Octahedron

The octahedron area of one side is the area of the equilateral triangle, so the whole surface area of the octahedron is the area of all sides. Since octahedron contains 8 equilateral sides,

Area of the equilateral side = (√3/4) × a²

Surface area of Octahedron = 8 × (√3/4) × a² = 2 × √3× a²

Where a = length of the side

Volume of an Octahedron

The octahedron is made up of two pyramids, so we can calculate the volume of one pyramid and multiply it by two to get the volume of the octahedron.

1. The volume of the pyramid = (Height × Base)/3
2. Height of pyramid = √(a² – (a/√2))² 
3. Base of pyramid = a²

Volume of the octahedron = 2× volume of the pyramid = 2 × (a³)/3√2 = √2/3 × a³

Where a = length of the side 

Icosahedron

In the Icosahedron formulas, we will get to know how to find the volume of the Icosahedron and the surface area of the Icosahedron. Icosahedron contains twenty faces, twelve vertices, and thirty edges.

 

Volume formula

Volume formula of Icosahedron = (5/12) × (3 + √5) × a³

Where a = length of the side

Surface area formula

Surface area formula of Icosahedron = (5√3 × a²)

Where a = length of the side

Tetrahedron

In the Tetrahedron formulas, we will get to know how to find the volume of the Tetrahedron and the surface area of the Tetrahedron. Tetrahedron contains four faces which are having equilateral triangles as its faces, four vertices that are equidistant with each other, and six edges.

 

Volume formula

Volume formula of Tetrahedron = (Area of the base × height) ×  (1/3) = (√2/12) × a³ cubic units

Where a = length of the side 

Surface area formula

Total Surface area for Tetrahedron = Sum of the 4 Equilateral triangles = 4 × (√3)/4 × a² = √3 × a² square units.

Where a = length of the side 

Dodecahedron

In the Dodecahedron formula, we will get to know how to find the volume of the Dodecahedron and the surface area of the Dodecahedron. Dodecahedron contains twelve Pentagonal sides, twenty vertices where at each vertex 3 edges meet, and thirty edges.

DODECAHEDRON

Volume formula

Volume formula of Dodecahedron = 7.66 × a³ cubic units

Where a = length of the side 

Surface area formula

Surface area formula of Dodecahedron = 20.64 ×  a² square units

Where a = length of the side

Sample Questions

Question 1: What is the amount of Rainwater stored in a cube-shaped container having a side length of 10 inches?

Solution:  

Given,

The Length of the Container is = 10 inches

The stored amount of rainwater in the container is =?

The volume of the container will give the amount of rainwater stored in the container. So as given the shape of the container is a cube, calculate the cube volume.

The volume of the cube = a³ = 10³= 1000 inches³

The stored amount of rainwater in the container is = 1000 inches³

Question 2: Yaswanthi has a pair of jewellery boxes which are having the shape of an octahedron. In a curiosity, she wants to find the surface area of each jewellery box. Calculate the surface area of each jewellery box where the length is 0.8 inches?

Solution:   

Given,

Length of side (a) = 0.8 inches.

The surface area of the Octahedron = 2 × √3 ×  a²

Substitute the a value in the above formula,

The surface area of the Octahedron = 2 × √3 × (0.8)²

The surface area of the jewelry box = 2.2170 inches²

Question 3: A artwork shaped like an Icosahedron is having the length of the side as 9 inches. Find the volume of the Artwork.

Solution:  

Given,

Length of the side (a) = 9 inches

The volume of the Icosahedron = (5/12) × (3+√5) × a³

The Volume of Art work = (5/12) × (3 + √5) ×  (9)³ = 1590.45

Question 4: Calculate the Total surface area of the Tetrahedron where the length is given as 4 units?

Solution:  

Given,

Length of the side = 4 units

The total surface area of  Tetrahedron = √3 × a²

By substituting the value of the a in the above formula,

Total surface area = √3 × 4² = 16√3 = 27.71 square units.

Question 5: Find the Surface area of the Dodecahedron where the sum of the length of all sides is 120 inches?

Solution: 

Given,

Sum of the length of all sides of  Dodecahedron are = 120 inches

Since the Dodecahedron is having the 30 edges, so the length of each edge is (a) = 120/30 = 4 inches

The surface area of the Dodecahedron = 20.64 × a² square units

By substituting the a value in the above formula,

Surface area of Dodecahedron = 20.64 × (4²)  = 330.24 square inches.

Question 6: Find the volume of an octahedron with a side length of  2.1 inches?

Solution: 

Given,

Side length of the Octahedron (a) = 2.1 inches

The volume of the Octahedron = (√2/3) × a³

By substituting the a value in the above formula,

The volume of the Octahedron = (√2/3) × (2.1)³ = 9.261 inches³

Question 7: Find the Volume of a Tetrahedron with a side length that measures 3 inches?

Solution: 

Given,

The side length of the Tetrahedron(a) = 3 inches

The volume of the Tetrahedron = (1/6√2) a³

By substituting the a value in the above formula,

The volume of the Tetrahedron = (1/6√2) (3)³ = 3.181 inches³