Pyramid Formulas

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25. Pyramid Formulas

A pyramid is a three-dimensional solid with a polygonal base and triangular faces that meet at a common point called the vertex.

The perpendicular distance from the vertex to the base is called the height of the pyramid.

The source Mensuration Formula PDF gives three important formulas for a pyramid:

  • Volume
  • Lateral Surface Area (LSA)
  • Total Surface Area (TSA)

These formulas appear in the final page of the source PDF. :contentReference[oaicite:1]{index=1}

1. Pyramid Diagram

h l Base Vertex Pyramid

2. Important Parts of a Pyramid

  • Base: The polygonal bottom face of the pyramid.
  • Vertex: The common point where the triangular faces meet.
  • Height (h): The perpendicular distance from the vertex to the base.
  • Slant Height (l): The height of a triangular lateral face.
  • Base Perimeter: The perimeter of the polygonal base.
  • Base Area: The area of the base.

3. Volume of a Pyramid

Volume = ⅓ × Base Area × Height

The source PDF gives the volume formula as one-third of the product of the base area and perpendicular height. :contentReference[oaicite:2]{index=2}

V = ⅓Bh

Where:

  • B = Base Area
  • h = Perpendicular Height

4. Example 1 — Volume of a Pyramid

A pyramid has a base area of 60 cm² and a perpendicular height of 9 cm. Find its volume.

Step 1: Formula

V = ⅓Bh

Step 2: Substitute the values

V = ⅓ × 60 × 9

Step 3: Calculate

V = 20 × 9
V = 180 cm³

Answer: 180 cm³.

5. Lateral Surface Area of a Pyramid

LSA = ½ × Base Perimeter × Slant Height

The lateral surface area includes the triangular side faces of the pyramid but does not include the base.

LSA = ½Pl

Where:

  • P = Perimeter of the base
  • l = Slant Height

This is the LSA formula shown in the source PDF. :contentReference[oaicite:3]{index=3}

6. Example 2 — Lateral Surface Area

A pyramid has a base perimeter of 40 cm and a slant height of 12 cm. Find its LSA.

Step 1: Formula

LSA = ½Pl

Step 2: Substitute

LSA = ½ × 40 × 12

Step 3: Calculate

LSA = 20 × 12
LSA = 240 cm²

Answer: 240 cm².

7. Total Surface Area of a Pyramid

TSA = LSA + Base Area

The total surface area includes:

  • The lateral triangular faces
  • The base
TSA = ½Pl + B

This follows directly from the source formula: TSA = LSA + Base Area. :contentReference[oaicite:4]{index=4}

8. Example 3 — Total Surface Area

A pyramid has an LSA of 240 cm² and a base area of 100 cm². Find its TSA.

TSA = LSA + Base Area
TSA = 240 + 100
TSA = 340 cm²

Answer: 340 cm².

9. Complete Pyramid Formula Sheet

Quantity Formula
Volume ⅓ × Base Area × Height
Volume V = ⅓Bh
LSA ½ × Base Perimeter × Slant Height
LSA LSA = ½Pl
TSA LSA + Base Area
TSA TSA = ½Pl + B

10. Height vs Slant Height

Height (h)

The height is the perpendicular distance from the vertex to the base.

Used in Volume = ⅓Bh

Slant Height (l)

The slant height is measured along the triangular lateral face from the vertex toward the base.

Used in LSA = ½Pl

11. Do Not Confuse h and l

Symbol Name Used In
h Perpendicular Height Volume
l Slant Height LSA
B Base Area Volume and TSA
P Base Perimeter LSA

12. Example 4 — Complete Calculation

A pyramid has:

  • Base area = 80 cm²
  • Base perimeter = 36 cm
  • Height = 10 cm
  • Slant height = 12 cm

Volume

V = ⅓ × 80 × 10
V = 266.67 cm³ approximately

LSA

LSA = ½ × 36 × 12
LSA = 216 cm²

TSA

TSA = 216 + 80
TSA = 296 cm²

13. Common Types of Pyramids

Type Base Shape
Triangular Pyramid Triangle
Square Pyramid Square
Rectangular Pyramid Rectangle
Pentagonal Pyramid Pentagon
Hexagonal Pyramid Hexagon

14. Example 5 — Square Pyramid

A square pyramid has a base side of 10 cm and a perpendicular height of 12 cm. Find its volume.

Step 1: Find base area

B = a²
B = 10² = 100 cm²

Step 2: Apply pyramid volume formula

V = ⅓Bh
V = ⅓ × 100 × 12
V = 400 cm³

Answer: 400 cm³.

15. LSA vs TSA of Pyramid

Surface Includes Formula
LSA Only triangular side faces ½Pl
TSA Triangular side faces + base LSA + B

16. Practice Questions

1. A pyramid has a base area of 90 cm² and height 12 cm. Find its volume.

2. A pyramid has a base perimeter of 50 cm and slant height 14 cm. Find its LSA.

3. A pyramid has LSA 300 cm² and base area 120 cm². Find its TSA.

4. A square pyramid has a base side of 8 cm and height 15 cm. Find its volume.

5. A pyramid has base area 150 cm² and height 18 cm. Find its volume.

6. A pyramid has base perimeter 48 cm and slant height 10 cm. Find its LSA.

17. Answers

Question Answer
1 360 cm³
2 350 cm²
3 420 cm²
4 320 cm³
5 900 cm³
6 240 cm²

18. Common Mistakes to Avoid

1. Do not use slant height in the volume formula.

2. Do not use perpendicular height directly in the LSA formula.

3. Remember that TSA includes the base.

4. LSA does not include the base.

5. Always calculate the correct area and perimeter of the base before applying the formula.

6. Volume is measured in cubic units.

7. Surface area is measured in square units.

19. One-Minute Revision

Volume = ⅓ × Base Area × Height
V = ⅓Bh
LSA = ½ × Base Perimeter × Slant Height
LSA = ½Pl
TSA = LSA + Base Area

20. Competitive Exam Shortcut

When you see a PYRAMID question, remember:

V = ⅓Bh
LSA = ½Pl
TSA = LSA + B

Memory trick:

Volume → Base Area × Height ÷ 3

LSA → Base Perimeter × Slant Height ÷ 2

TSA → LSA + Base Area

Source: Complete Mensuration Formulas PDF, Page 11 — Surface Area & Volume. The source lists Pyramid under item (x) and gives the Volume, LSA and TSA formulas used in this article. :contentReference[oaicite:5]{index=5}

This completes the topic-wise formula series from the uploaded PDF.