Volume & Surface Area

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20. Volume & Surface Area Formulas

Volume tells us how much three-dimensional space an object occupies.

Surface Area tells us the total area covered by the outer surfaces of a solid.

This section covers the important formulas for:

  • Cube
  • Cuboid
  • Cylinder
  • Cone
  • Frustum

For surface area, the PDF uses:

  • LSA = Lateral Surface Area
  • CSA = Curved Surface Area
  • TSA = Total Surface Area

1. Volume and Surface Area — Basic Idea

Volume: Measures the space occupied by a solid.

Lateral Surface Area (LSA): Measures the area of the side surfaces of a solid.

Curved Surface Area (CSA): Measures the curved portion of a solid.

Total Surface Area (TSA): Measures all the surfaces of the solid.

2. Cube

A cube is a solid having six equal square faces. If each edge has length a, all the edges are equal.

a a a d

Cube Formulas

Volume = a³
LSA = 4a²
TSA = 6a²

Example 1: Cube

A cube has an edge of 5 cm. Find its volume, LSA and TSA.

Volume:

V = a³
V = 5³ = 125 cm³

LSA:

LSA = 4a²
= 4 × 5² = 100 cm²

TSA:

TSA = 6a²
= 6 × 5² = 150 cm²

Answer: Volume = 125 cm³, LSA = 100 cm², TSA = 150 cm².

3. Cuboid

A cuboid is a rectangular solid having length, breadth and height.

Let:

  • L = Length
  • B = Breadth
  • H = Height
L B H d

Cuboid Formulas

Volume = L × B × H
LSA = 2(L + B)H
TSA = 2(LB + BH + HL)

Example 2: Cuboid

A cuboid has length 10 cm, breadth 6 cm and height 4 cm. Find its volume, LSA and TSA.

Volume:

V = L × B × H
= 10 × 6 × 4
V = 240 cm³

LSA:

LSA = 2(L+B)H
= 2(10+6) × 4
= 2 × 16 × 4
LSA = 128 cm²

TSA:

TSA = 2(LB+BH+HL)
= 2(10×6 + 6×4 + 4×10)
= 2(60+24+40)
TSA = 248 cm²

4. Cylinder

A cylinder is a solid with two circular bases and one curved surface.

Let:

  • r = Radius
  • h = Height
r h

Cylinder Formulas

Volume = πr²h
CSA = 2πrh
TSA = 2πr(r + h)

Example 3: Cylinder

A cylinder has radius 7 cm and height 10 cm. Find its volume. Use π = 22/7.

V = πr²h
= 22/7 × 7² × 10
= 22 × 7 × 10
V = 1540 cm³

Answer: 1540 cm³

Example 4: Cylinder CSA

Find the curved surface area of a cylinder with radius 7 cm and height 10 cm. Use π = 22/7.

CSA = 2πrh
= 2 × 22/7 × 7 × 10
CSA = 440 cm²

5. Cone

A cone has a circular base and a curved surface that meets at a single vertex.

Let:

  • r = Radius
  • h = Height
  • l = Slant height
h r l

Cone Formulas

Volume = ⅓πr²h
CSA = πrl
TSA = πr(r + l)

Example 5: Cone Volume

A cone has radius 7 cm and height 12 cm. Find its volume. Use π = 22/7.

V = ⅓πr²h
= ⅓ × 22/7 × 7² × 12
= ⅓ × 22 × 7 × 12
V = 616 cm³

Example 6: Cone Curved Surface Area

A cone has radius 7 cm and slant height 10 cm. Find its CSA. Use π = 22/7.

CSA = πrl
= 22/7 × 7 × 10
CSA = 220 cm²

6. Frustum of a Cone

A frustum is obtained when the top portion of a cone is cut off by a plane parallel to its base.

Let:

  • R = Radius of larger circular base
  • r = Radius of smaller circular base
  • h = Height
  • l = Slant height
r R h l

Frustum Formulas

Volume = ⅓π(R² + r² + Rr)h
CSA = π(R + r)l
TSA = π(R + r)l + π(R² + r²)

Example 7: Frustum Volume

A frustum has:

  • Large radius R = 7 cm
  • Small radius r = 3 cm
  • Height h = 10 cm

Find its volume. Use π = 22/7.

V = ⅓π(R²+r²+Rr)h
= ⅓ × 22/7 × (7² + 3² + 7×3) × 10
= ⅓ × 22/7 × (49 + 9 + 21) × 10
= ⅓ × 22/7 × 79 × 10
V = 1738 cm³ approximately

7. Complete Volume & Surface Area Formula Table

Solid Volume LSA / CSA TSA
Cube a³ 4a² 6a²
Cuboid L × B × H 2(L+B)H 2(LB+BH+HL)
Cylinder πr²h 2πrh 2πr(r+h)
Cone ⅓πr²h πrl πr(r+l)
Frustum ⅓π(R²+r²+Rr)h π(R+r)l π(R+r)l + π(R²+r²)

8. Meaning of Symbols

Symbol Meaning
a Edge of a cube
L Length of cuboid
B Breadth of cuboid
H Height
r Radius
R Larger radius of frustum
l Slant height
π Pi, commonly taken as 22/7 or 3.14

9. Which Formula Should You Use?

Cube: If all edges are equal, use cube formulas.

Cuboid: If length, breadth and height are given, use cuboid formulas.

Cylinder: If the solid has two circular bases and a curved side, use cylinder formulas.

Cone: If the solid has one circular base and a vertex, use cone formulas.

Frustum: If a cone is cut parallel to its base, use frustum formulas.

10. Practice Questions

1. Find the volume of a cube whose edge is 6 cm.

2. Find the TSA of a cube whose edge is 8 cm.

3. Find the volume of a cuboid with L = 12 cm, B = 5 cm and H = 4 cm.

4. Find the LSA of a cuboid with L = 10 cm, B = 6 cm and H = 5 cm.

5. Find the volume of a cylinder with radius 7 cm and height 10 cm. Use π = 22/7.

6. Find the CSA of a cylinder with radius 7 cm and height 8 cm. Use π = 22/7.

7. Find the volume of a cone with radius 7 cm and height 9 cm. Use π = 22/7.

8. Find the CSA of a cone with radius 7 cm and slant height 12 cm. Use π = 22/7.

9. A frustum has R = 7 cm, r = 3 cm and h = 5 cm. Find its volume.

11. Answers

Question Answer
1 216 cm³
2 384 cm²
3 240 cm³
4 160 cm²
5 1540 cm³
6 352 cm²
7 462 cm³
8 264 cm²
9 Approximately 869 cm³

12. Important Exam Shortcuts

Shortcut 1: Cube volume is simply the cube of its edge.

V = a³

Shortcut 2: Cube TSA is six times the area of one face.

TSA = 6a²

Shortcut 3: Cuboid volume is the product of length, breadth and height.

V = LBH

Shortcut 4: Cylinder volume is:

V = πr²h

Shortcut 5: Cone volume is one-third of the corresponding cylinder volume.

V = ⅓πr²h

Shortcut 6: Always distinguish between CSA/LSA and TSA.

13. Common Mistakes to Avoid

1. Do not confuse radius r with diameter.

2. Volume is measured in cubic units: cm³, m³, etc.

3. Surface area is measured in square units: cm², m², etc.

4. For a cone, do not confuse height h with slant height l.

5. For a frustum, carefully identify the larger radius R and smaller radius r.

6. Use CSA for curved surfaces and TSA when all surfaces are included.

14. Quick Revision

Cube

V = a³
LSA = 4a²
TSA = 6a²

Cuboid

V = LBH
LSA = 2(L+B)H
TSA = 2(LB+BH+HL)

Cylinder

V = πr²h
CSA = 2πrh
TSA = 2πr(r+h)

Cone

V = ⅓πr²h
CSA = πrl
TSA = πr(r+l)

Frustum

V = ⅓π(R²+r²+Rr)h
CSA = π(R+r)l
TSA = π(R+r)l + π(R²+r²)

Source: Part 20 is based on the “VOLUME & SURFACE AREA” page of the provided Mensuration Formula PDF. The source page contains the formulas for Cube, Cuboid, Cylinder, Cone and Frustum. :contentReference[oaicite:1]{index=1}