Prism Formula

A prism is a solid bounded by a number of plane faces; its two faces, called the ends, are congruent parallel plane polygons and other faces, called the side faces, are parallelograms. We can use the concept of prism in both mathematics and science as well. Let us now study about prism formula in detail.

Prism Formula

What is Prism?

In mathematics, a prism is a polyhedron with two polygonal bases parallel to each other. In physics (optics), a prism is defined as the transparent optical element with flat polished surfaces that refract light. Lateral faces join the two polygonal bases. The lateral faces are mostly rectangular. In some cases, it may be a parallelogram. The Prism Formula is as follows,

• The surface area of a prism = (2×BaseArea) +Lateral Surface Area
• The volume of a prism =Base Area× Height

Properties of Prism

1. The base and top are parallel and congruent.
2. Each face is a parallelogram except base and top. These faces are called a Lateral face.
3. With every lateral face, one edge in common with the base and also with the top.
4. The height of the prism is the common edge of two adjacent side faces.

Types of Prism

Prisms are of different types, which are named according to their base shape.

1] Rectangular Prism

A Rectangular Prism has 2 parallel rectangular bases and 4 rectangular faces.

• Base Area = b×l
• Surface area = 2× (bl+lh+hb)
• Volume = l×b×h

b	base length
l	base width
h	height

2] Triangular Prism

A triangular prism has 3 rectangular faces and 2 parallel triangular bases.

• Base area = \(\frac{1}{2}ab\)
• Surface area=ab+3bh
• Volume=\(\frac{1}{2}abh\)

3] Pentagonal Prism

A pentagonal prism has 5 rectangular faces and 2 parallel pentagonal bases.

• Base Area= \(\frac{5}{2}ab\)
• Surface area=5ab+5bh
• Volume =\(\frac{5}{2}abh\)

4] Hexagonal Prism

A hexagonal prism has six rectangular faces and two parallel hexagonal bases.

• Base area=3ab
• Surface area=6ab+6bh
• Volume=3abh

b	base length
a	apothem length
h	height

Solved Examples

Q 1: What will be the surface area of a triangular prism if the apothem length, base length, and height are 5 cm, 10 cm, and 18 cm respectively?

Solution: Given,

a = 5 cm; b = 10 cm; h = 18 cm

The surface area of a triangular prism

= ab + 3bh

= (5 cm × 10 cm) + (3 × 10 cm × 18 cm)

= 50 cm² + 540 cm²

= 590 cm²

Q2. The base of a triangular prism is ΔABC, where AB = 6 cm, BC = 8 cm and ∠B = 90. The height of the prism is 20 cm. Find lateral surface area and total surface area.

Solution: By Pythagorean theorem

c ²= a²+ b² = 6²+ 8² 

c²= 100 c = 10 cm

Perimeter  = a + b + c

= 6 cm + 8 cm + 10 cm = 24 cm

∴ Area of Δ ABC = \(\frac{1}{2} × 6 × 8 = 24 cm\)

Lateral surface area = perimeter × height = 24 × 20 = 480 cm²

Total surface area = 2 × area of triangle + Lateral surface are

= 2 × 24 + 480 = 48 +480

∴ Total surface area = 520 cm²