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,
- Cube
- Octahedron
- Icosahedron
- Tetrahedron
- 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,
CubeÂ
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.
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Surface Area of the cube
The surface area of the cube is divided into two types,Â
- Lateral Surface Area of Cube
- 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 cubeÂ
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.
Â
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.
- The volume of the pyramid = (Height × Base)/3
- Height of pyramid = √(a2 – (a/√2))2Â
- 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 sideÂ
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.
Â
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.
Â
Volume formula
Volume formula of Tetrahedron = (Area of the base × height) ×  (1/3) = (√2/12) × a3 cubic units
Where a = length of the sideÂ
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 sideÂ
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 sideÂ
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: Â
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: Â Â
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: Â
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: Â
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:Â
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:Â
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:Â
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
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