Last modified on January 14th, 2022

chapter outline

 

Prism

Prism is a member of the polyhedron family. It is a three-dimensional solid consisting of two identical and parallel n-sided polygon-shaped bases on parallel planes bounded by three or more non-parallel flat faces.

In general, a prism refers to a transparent solid used to refract or scatter a beam of white light. It is a commonly used instrument in physics. Mostly it is triangular in shape.

Figure 1 (heading: Prisms, filename: Prism)

A prism has the bases, lateral faces, edges, and vertices.

Base – The parallel faces which makes the 2 ends of any prism. They are congruent. The base determines the cross-section of any prism and it remains uniform throughout the shape.

Lateral faces – The non-parallel faces which connects the 2 bases.

Vertices – They are the corners.

Edges – Where any 2 faces meet.

Figure 2 (heading: Parts of a Prism, filename: Prism Parts)

Prism Shape

Depending on its base, a prism can be of any shape, such as triangular, rectangular, pentagonal, hexagonal, or heptagonal. E.g., a triangular prism has a triangular base. A square prism has a square base.

Below given are the prisms of different shapes.

Figure 3 (heading: Prism Shapes, filename: Prism Shape)

[ sample link: https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/7-types-of-prisms-01-1-1622445053.png

Include heptagonal prism ]

  • Triangular Prism – Bases are triangle in shape. It has
    • 5 faces (2 triangular faces and 3 rectangular faces)
    • 9 edges
    • 6 vertices.
  • Rectangular prism – Bases are rectangle in shape. Rectangular prisms are also known as cuboids or cubes as they are cuboidal in shape. Its base edges are not congruent. It has
    • 6 faces (all faces rectangles in shape)
    • 12 edges
    • 8 vertices
  • Square Prism – Bases are square in shape. Its base edges are congruent but height is variable. It has
    • 6 faces (2 square faces and 4 rectangular faces)
    • 12 edges
    • 8 vertices
  • Pentagonal Prism – Bases are pentagon in shape. It has  
    • 7 faces (2 pentagonal faces and 5 rectangular faces)
    • 15 edges
    • 10 vertices
  • Hexagonal Prisms – Bases are hexagon in shape. It has
    • 8 faces (2 hexagonal faces and 6 rectangular faces)
    • 18 edges
    • 12 vertices
  • Heptagonal Prisms – Bases are heptagon in shape. It has
    • 9 faces (2 Heptagonal faces and 7 rectangular faces)
    • 19 edges
    • 14 vertices
  • Octagonal Prism – Bases are octagon in shape. It has
    • 10 faces( 2 octagonal faces and 8 rectangular faces)
    • 24 edges
    • 16 vertices
  • Trapezoidal Prism – It is a 3-D version of a trapezoid. Its bases are trapezoids in shape. It has –
    • 6 faces (2 trapezoidal faces and 4 rectangular faces)
    • 12 edges
    • 8 vertices

All the above examples and images are regular prisms. They all have their base edges of equal measure. however, sometimes a prism might have unequal base edges. Those are irregular prisms. Given below is an image showing a regular and irregular triangular prism.

Figure 4 (heading: Regular vs Irregular Prisms, filename: Regular vs Irregular Prisms)

Based on the type of polygon, prisms can be regular or irregular.

Regular Prism -The base is a regular polygon with equal side lengths. Regular prisms have identical bases and identical lateral faces.

Irregular Prism – The base is an irregular polygon with unequal side lengths. Irregular prisms have identical bases. However, the lateral faces are not identical.

Based on the alignment of the bases, prisms can be right or oblique.

Like all other polyhedrons, a prism also has a surface area and a volume.

Let’s see the basic formulas.

Formulas

Surface Area

The surface area of any prism is the entire amount of space occupied by all its outer surfaces (or faces). It is twice its base area added to the product of its perimeter and height. The area is always in square units such as m2, cm2, mm2, and in2.

There are 2 types of surface areas- (1) total surface area (TSA), and (2) lateral surface area (LSA)

Total Surface Area (TSA) = (2 × Base Area) + (Perimeter × Height)

Lateral Surface Area (LSA) = Perimeter × Height

Volume

The volume of any prism is the space it occupies in the three-dimensional plane. It is the product of its base area and height. It is always in cubic units such as m3, cm3, mm3, ft3.

Volume = Base Area × Height

Let us solve some examples involving prisms and the above formulas

Solved Examples

Find the surface area of a triangular prism given in the figure.

Solution:

Here, the 2 bases of the triangular base are equal, so it is an isosceles triangle.
As we know,
Total Surface Area (TSA) = (2 × Base Area) + (Perimeter × Height)
= b × h + (a + b + c) × l, here b = 6 in, a = c = 5 in, h = 4 in, l = 7 in
∴ TSA = 6 × 4 + (5 + 6 + 5) × 7
= 136 in2

Figure 5 (heading: no heading, filename: Prism Example 1)

Find the total and lateral surface area of a rectangular prism given in the figure.

Solution:

As we know,
The figure is a rectangular prism whose adjacent base edges are 4 in, and 6 in, and height is 11 in
Total Surface Area (TSA) = (2 × Base Area) + (Perimeter × Height)
∴ TSA = 2(lw + wh + hl) here, l = 6 in, w = 4 in, h = 11 in
= 2(6 × 4 + 4 × 11 + 11 × 6)
= 268 cm2
Lateral Surface Area (LSA) = 2(wh + hl)
∴ LSA = 2(4 × 11 + 11 × 6)
= 220 cm2

Figure 6 (heading: no heading, filename: Prism Example 2)

Find the volume of a triangular prism whose base area is 64 cm2 and height is 7 cm.

Solution:

As we know,
Volume (V) = Base Area × Height
∴ V= B × h, here B = 64 cm2, h = 7 cm
= 64 × 7
= 448 cm3

Last modified on January 14th, 2022

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