Volume And Surface Area Of A Prism

What is a Prism?

A prism is a 3D shape which has a constant cross section. It is a three-dimensional solid object in which the two ends are identical. It is the combination of the flat faces, identical bases and equal cross-sections. The faces of the prism are parallelograms or rectangles without the bases. And the bases of the prism could be triangle, square, rectangle or any n-sided polygon.

Prism Shape

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i - Square Prism

A square prism is a three-dimensional cuboid where the base and top are equal squares and the remaining 4 faces are rectangles. The opposite sides and angles of a square prism are congruent and parallel to each other. A square prism has at least two of its length equal to each other. A real-life example is a tissue box. A square prism has two ends as faces shaped like a square with 4 rectangles or squares in between them as other faces. A square prism can be classified into two types: a) Right Square Prism, and b) Oblique Square Prism.

The various properties of a square prism are as follows:

• A square prism has 6 faces, 8 vertices, and 12 edges.
• The opposite sides of a square prism are parallel to each other.
• The opposite sides of a square prism are congruent to each other.
• In a square prism, the diagonals bisect each other.
• In a square prism, opposite angles are congruent to each other.

ii - Cube

When all three lengths are equal it is called a cube (or hexahedron)

and each face is a square. A cube is still a prism. A cube is one of the Platonic Solids.

• A cube is just a special case of a square prism, and
• A square prism is just a special case of a rectangular prism, and
• They are all cuboids!

A cuboid is a box-shaped object. It has six flat faces and all angles are right angles. All of its faces are rectangles. It is also a prism because it has the same cross-section along a length. In fact, it is a rectangular prism. Cuboids are very common in our world, from boxes to buildings we see them everywhere. We can even fit them inside other cuboids.

iii - Triangular Prism

A triangular prism is a polyhedron made up of two triangular bases and three rectangular sides. It is a three-dimensional shape that has three side faces and two base faces, connected to each other through the edges. If the sides are rectangular, then it is called the right triangular prism else it is said to be an oblique triangular prism. Triangular Prism is a pentahedron and has nine distinct nets. The edges and vertices of the bases are joined with each other via three rectangular sides.

Properties

• It has a total of 9 edges, 5 faces, and 6 vertices (which are joined by the rectangular faces).
• It has two triangular bases and three rectangular sides
• If the triangular bases are equilateral and the other faces are squares, instead of a rectangle, then the triangular prism is said to be semiregular

iv - Pentagonal Prism

A pentagonal prism is a three-dimensional solid that has two pentagonal bases - bottom and top. All the other sides of a pentagonal prism have the shape of a rectangle. It is easy to understand the shape of a pentagonal prism by drawing a pentagon on a piece of paper using straight lines. It has a total of 7 faces, 15 edges, and 10 vertices out of which 2 faces are pentagonal in shape.

What are the Types of Prism?

i - Regular Prism

If the bases of the prism are in the shape of a regular polygon, it is called regular prism.

ii - Irregular Prism

If the bases are in the shape of an irregular polygon, then the prism is called an irregular prism.

What are the differences between Right Prism and Oblique Prism?

Formulas -Surface Area & Volume

(Note: there should be a proper explanation of 3 examples)

i - Surface Area of a Prism

The surface area of the prism is the total area covered by the faces of the prism. For any kind of prism, the surface area can be found using the formula;

Surface Area of a Prism = 2 (Base Area) + (Base perimeter × height)

Example 1:

What is the surface area of a prism where the base area is 25 m², the base perimeter is 24 m, and the length is 12 m

 Surface Area  = 2 × Base Area + Base Perimeter × Length

            = 2 × 25 m² + 24 m × 12 m

            = 50 m² + 288 m²

            = 338 m²

ii - Volume of a Prism

Volume of a Prism

The volume of the prism is defined as the product of the base area and the prism height.

Therefore,

Volume of Prism = Base Area × Height

For example,

if you want to find the volume of a square prism, you must know the area of a square, then its volume can be calculated as follows:

The volume of a square Prism = Area of square × height

V = s² × h cubic units

Where “s” is the side of a square.

Example: What is the volume of a prism where the base area is 25 m² and which is 12 m long:

Volume= Area × Length

 = 25 m² × 12 m

 = 300 m³