Prisms

1. Introduction

A prism shape is a polyhedron (a 3D shape made from polygons) with a constant cross section through one dimension.

• A prism shape has two congruent faces (identical ends).

• These same two faces are simple polygons like triangles or squares or pentagons. The other remaining faces are rectangles.

• The name of the prism is represented by the shape of its cross section.

Some of the prisms are Triangular Prism, Rectangular Prism, and Pentagonal Prism.

2. Types of Prisms and its Properties

A prism can be characterised using three features: faces, vertices and edges:

• Face: a closed, flat surface surrounded by edges and vertices.

• Edge: connects two vertices with a straight line

• Vertex: a point where two or more edges meet.

2.1 Triangular Prism

It has triangular faces at both ends

• Faces: 5 (2 triangular faces and 3 rectangular faces)

• Edges: 9

• Vertices: 6

2.2 Rectangular Prism

The rectangular prism is actually a cuboid. It has rectangular faces at both ends.

• Faces: 6 (6 Rectangular Faces)

• Edges: 12

• Vertices: 8

2.3 Pentagonal Prism

It has pentagon faces at both ends.

• Faces: 7 (2 pentagonal faces and 5 rectangular faces)

• Edges: 15

• Vertices: 10

3. Surface Area of Prisms

Surface area of a prism is defined as the total area occupied by all the faces of the prism. The surface area of a 3D prism depends on the shape of the base.

The surface area of a prism is the total area of all of the faces.

\( Surface area of a prism = Area of common faces + Area of rectangular faces \)

1. Surface Area of Triangular Prism = Area of two triangular faces + Area of 3 rectangular faces

2. Surface Area of Rectangular Prism = Area of six rectangular faces

3. Surface Area of Pentagonal Prism = Area of two pentagonal faces + Area of five rectangular faces

4. How to bring image

The volume of a prism is how much space there is inside a prism. In order to work out the volume of any type of prism, we need to multiply the area of cross-section(which is the area of the triangle in a triangular prism) by the length of the prism (which is the length of the rectangle face in the prism).

\( Volume of a prism = Area of cross-section x Length of the prism \)

5. Example Questions on the Surface Area and the Volume of the Prisms

Example:

Find the volume of the prism below.

Volume of prism = Area of cross section × Length of the prism

Area of cross-section (trapezium) = 1/2 x (a+b) h = 1/2 x (9+5) x 4 = 1/2 x 14 x 4 = 28 \( cm^2 \)

Volume of the prism = 28 x 10 = 280 \( cm^3 \)

Example: 

Calculate the surface area of the following triangular prism.

Surface area of triangular prism = Area of two triangular faces + Area of 3 rectangular faces

Area of triangular face = 1/2 x base x height = 1/2 x 5 x 12 = 30 \( cm^2 \)

Area of two triangular faces = 2 x 30 = 60 \( cm^2 \)

Area of first rectangular face = length x width = 10 x 13 = 130 \( cm^2 \)

Area of second rectangular face = length x width = 10 x 12 = 120 \( cm^2 \)

Area of third rectangular face = length x width = 10 x 5 = 50 \( cm^2 \)

Surface Area of the given prism = 60 + (130 + 120 + 50) = 60 + 300 = 360 \( cm^2 \)