Prism Formulas

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24. Prism Formulas

A prism is a three-dimensional solid having two identical and parallel bases connected by rectangular or parallelogram-shaped lateral faces.

The two bases of a prism have the same shape and size.

Depending on the shape of its base, a prism may be called a triangular prism, rectangular prism, pentagonal prism, etc.

1. Prism Diagram

Height Base Prism

2. Parts of a Prism

  • Base: The identical faces at the two ends of the prism.
  • Height: The perpendicular distance between the two bases.
  • Lateral Faces: The faces connecting the two bases.
  • Base Area: Area of one base.
  • Base Perimeter: Perimeter of one base.

3. Volume of a Prism

Volume = Base Area × Height

The volume of a prism is obtained by multiplying the area of one base by the perpendicular height of the prism.

V = B × h

Where:

  • B = Area of the base
  • h = Height of the prism

4. Example 1 — Volume of a Prism

The area of the base of a prism is 40 cm² and its height is 12 cm. Find the volume.

Step 1: Formula

V = Base Area × Height

Step 2: Substitute

V = 40 × 12

Step 3: Calculate

V = 480 cm³

Answer: 480 cm³.

5. Lateral Surface Area of a Prism

LSA = Base Perimeter × Height

The lateral surface area includes only the side faces of the prism. The two bases are not included.

LSA = P × h

Where:

  • P = Perimeter of the base
  • h = Height of the prism

6. Example 2 — Lateral Surface Area

A prism has a base perimeter of 30 cm and a height of 10 cm. Find its lateral surface area.

LSA = Base Perimeter × Height
LSA = 30 × 10
LSA = 300 cm²

Answer: 300 cm².

7. Total Surface Area of a Prism

TSA = LSA + 2 × Base Area

A prism has two identical bases. Therefore, the total surface area includes:

  • Lateral Surface Area
  • Area of the first base
  • Area of the second base
TSA = LSA + Base Area + Base Area
TSA = LSA + 2B

8. Example 3 — Total Surface Area

A prism has a base area of 25 cm² and a lateral surface area of 200 cm². Find its total surface area.

TSA = LSA + 2 × Base Area
TSA = 200 + 2(25)
TSA = 200 + 50
TSA = 250 cm²

Answer: 250 cm².

9. Complete Prism Formula Sheet

Quantity Formula
Volume Base Area × Height
Volume V = B × h
Lateral Surface Area LSA = Base Perimeter × Height
Total Surface Area TSA = LSA + 2 × Base Area

10. Example — Triangular Prism

Suppose the base of a prism is a triangle.

If the triangle has base b and perpendicular height h₁, then:

Triangle Area = ½ × b × h₁

If the prism length is L, then:

Volume = ½ × b × h₁ × L

11. Example 4 — Triangular Prism

The triangular base of a prism has base 8 cm and height 6 cm. The length of the prism is 10 cm. Find its volume.

Step 1: Find area of triangular base

Base Area = ½ × 8 × 6
Base Area = 24 cm²

Step 2: Find volume

V = Base Area × Length
V = 24 × 10
V = 240 cm³

Answer: 240 cm³.

12. Rectangular Prism

A rectangular prism is commonly represented by a cuboid. If its length, breadth and height are L, B and H, then its volume is:

V = L × B × H

This is a special case of the general prism formula:

V = Base Area × Height

For a rectangular base:

Base Area = L × B

13. LSA vs TSA

Surface What Is Included? Formula
LSA Only lateral/side faces P × h
TSA Side faces + two bases LSA + 2B

14. Example 5 — Finding TSA from LSA

A prism has base area 35 cm² and LSA 280 cm². Find the TSA.

TSA = LSA + 2B
TSA = 280 + 2(35)
TSA = 280 + 70
TSA = 350 cm²

Answer: 350 cm².

15. Important Points to Remember

1. A prism has two equal and parallel bases.

2. Volume is found using Base Area × Height.

3. LSA includes only the lateral faces.

4. TSA includes the lateral faces and both bases.

5. Always identify the shape of the base before calculating its area.

6. Volume is expressed in cubic units such as cm³ and m³.

7. Surface area is expressed in square units such as cm² and m².

16. Practice Questions

1. A prism has a base area of 50 cm² and height 12 cm. Find its volume.

2. A prism has base perimeter 36 cm and height 8 cm. Find its LSA.

3. A prism has base area 40 cm² and LSA 250 cm². Find its TSA.

4. A triangular prism has a triangular base of base 10 cm and height 6 cm. Its length is 15 cm. Find its volume.

5. A prism has base area 75 cm² and height 20 cm. Find its volume.

6. The base perimeter of a prism is 44 cm and its height is 12 cm. Find its LSA.

17. Answers

Question Answer
1 600 cm³
2 288 cm²
3 330 cm²
4 450 cm³
5 1500 cm³
6 528 cm²

18. One-Minute Revision

Volume = Base Area × Height
V = B × h
LSA = Base Perimeter × Height
LSA = P × h
TSA = LSA + 2 × Base Area

19. Competitive Exam Tip

When you see the word PRISM, remember:

V = B × h
LSA = P × h
TSA = LSA + 2B

The most important step is to correctly calculate the area and perimeter of the base.

Source: Your uploaded Mensuration Formulas PDF includes Prism in the final Surface Area & Volume section.

This page explains the Prism formulas with diagrams, worked examples and practice questions for your Trunesthub website.