Volume & Surface Area

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15. Volume & Surface Area Formulas

Volume measures the amount of space occupied by a three-dimensional object.

Surface Area measures the total area of the outer surfaces of a three-dimensional object.

The provided formula sheet covers the following solids:

  1. Cube
  2. Cuboid
  3. Cylinder
  4. Cone
  5. Frustum
  6. Sphere
  7. Hollow Sphere
  8. Hemisphere
  9. Prism
  10. Pyramid

1. Important Units

Measurement Common Unit
Length cm, m, km
Area cm², m²
Volume cm³, m³

Remember: Surface area is always measured in square units, while volume is measured in cubic units.

2. Cube

a a a d

A cube has six equal square faces. Every edge has the same length, represented by a.

Volume

Volume = a³

Lateral Surface Area (LSA)

LSA = 4a²

Total Surface Area (TSA)

TSA = 6a²

Example

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

Volume:

V = a³ = 5³ = 125 cm³

LSA:

LSA = 4a² = 4 × 25 = 100 cm²

TSA:

TSA = 6a² = 6 × 25 = 150 cm²

3. Cuboid

L H B

A cuboid has length L, breadth B, and height H.

Volume

V = L × B × H

Lateral Surface Area

LSA = 2(L + B)H

Total Surface Area

TSA = 2(LB + BH + HL)

Example

A cuboid has:

  • L = 10 cm
  • B = 6 cm
  • H = 4 cm

Volume:

V = 10 × 6 × 4 = 240 cm³

LSA:

LSA = 2(10 + 6) × 4
LSA = 128 cm²

TSA:

TSA = 2(60 + 24 + 40)
TSA = 248 cm²

4. Cylinder

r h

A cylinder has radius r and height h.

Volume

V = πr²h

Curved Surface Area (CSA)

CSA = 2πrh

Total Surface Area (TSA)

TSA = 2πr(r + h)

Example

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 × 49 × 10

V = 1540 cm³

5. Cone

h r l

A cone has radius r, height h, and slant height l.

Volume

V = ⅓πr²h

Curved Surface Area

CSA = πrl

Total Surface Area

TSA = πr(r + l)

Example

A cone has radius 3 cm and height 4 cm. Find its volume.

V = ⅓πr²h

= ⅓ × π × 3² × 4

= ⅓ × π × 9 × 4

V = 12π cm³

6. Frustum

r R h l

A frustum has:

  • R = larger radius
  • r = smaller radius
  • h = height
  • l = slant height

Volume

V = ⅓π(R² + r² + Rr)h

Curved Surface Area

CSA = π(R + r)l

Total Surface Area

TSA = π(R + r)l + π(R² + r²)

Example

A frustum has:

  • R = 5 cm
  • r = 3 cm
  • h = 4 cm

Find its volume.

V = ⅓π(R² + r² + Rr)h

= ⅓π(25 + 9 + 15) × 4

= ⅓π × 49 × 4

V = 196π/3 cm³

7. Sphere

r

Volume

V = ⁴⁄₃πr³

Surface Area

Surface Area = 4πr²

Example

Find the volume of a sphere of radius 3 cm.

V = ⁴⁄₃πr³

= ⁴⁄₃ × π × 3³

= ⁴⁄₃ × π × 27

V = 36π cm³

8. Hollow Sphere

R r

A hollow sphere has an outer radius R and inner radius r.

Volume

V = ⁴⁄₃π(R³ − r³)

Total Surface Area

TSA = 4πR²

The hollow-sphere formulas above are reproduced according to the provided formula sheet.

Example

A hollow sphere has outer radius 5 cm and inner radius 3 cm. Find its volume.

V = ⁴⁄₃π(R³ − r³)

= ⁴⁄₃π(5³ − 3³)

= ⁴⁄₃π(125 − 27)

= ⁴⁄₃π × 98

V = 392π/3 cm³

9. Hemisphere

r

A hemisphere is half of a sphere.

Volume

V = ⅔πr³

Curved Surface Area

CSA = 2πr²

Total Surface Area

TSA = 3πr²

Example

Find the volume of a hemisphere of radius 6 cm.

V = ⅔πr³

= ⅔ × π × 6³

= ⅔ × π × 216

V = 144π cm³

10. Prism

a b c h

For a prism, the formula sheet uses the general terms Base Area, Base Perimeter, and Height.

Volume

Volume = Base Area × Height

Lateral Surface Area

LSA = Base Perimeter × Height

Total Surface Area

TSA = LSA + 2 × Base Area

Example

A prism has a base area of 20 cm², base perimeter of 18 cm, and height of 10 cm.

Volume:

V = 20 × 10
V = 200 cm³

LSA:

LSA = 18 × 10 = 180 cm²

TSA:

TSA = 180 + 2(20)
TSA = 220 cm²

11. Pyramid

h l a b

For a pyramid, the formula sheet uses:

  • Base Area
  • Base Perimeter
  • Height
  • Slant Height

Volume

V = ⅓ × Base Area × Height

Lateral Surface Area

LSA = ½ × Base Perimeter × Slant Height

Total Surface Area

TSA = LSA + Base Area

Example

A pyramid has a base area of 30 cm², height 9 cm, base perimeter 24 cm, and slant height 10 cm.

Volume:

V = ⅓ × 30 × 9
V = 90 cm³

LSA:

LSA = ½ × 24 × 10
LSA = 120 cm²

TSA:

TSA = 120 + 30
TSA = 150 cm²

12. Complete Volume & Surface Area Formula Chart

Solid Volume LSA / CSA TSA / Surface Area
Cube a³ 4a² 6a²
Cuboid LBH 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)
Sphere ⁴⁄₃πr³ — 4πr²
Hollow Sphere ⁴⁄₃π(R³−r³) — 4πR²
Hemisphere ⅔πr³ 2πr² 3πr²
Prism Base Area × Height Base Perimeter × Height LSA + 2 × Base Area
Pyramid ⅓ × Base Area × Height ½ × Base Perimeter × Slant Height LSA + Base Area

13. How to Choose the Correct Formula

If the Question Gives... Use...
Edge of a cube Cube formulas
Length, breadth and height Cuboid formulas
Radius and height of cylinder Cylinder formulas
Radius, height and slant height of cone Cone formulas
Two radii and height of frustum Frustum formulas
Radius of a sphere Sphere formulas
Outer and inner radius Hollow sphere formulas
Radius of a hemisphere Hemisphere formulas
Base area/perimeter and height Prism formulas
Base area/perimeter and height/slant height Pyramid formulas

14. Important Differences to Remember

LSA vs TSA

LSA means Lateral Surface Area. It generally covers the side surfaces of the solid.

TSA means Total Surface Area. It includes the complete outer surface as specified by the formula.

CSA vs TSA

For curved solids such as a cylinder, cone and hemisphere:

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

15. Mixed Solved Example

Example 1 — Cube

Side = 4 cm.

V = 4³ = 64 cm³

Example 2 — Cuboid

L = 8 cm, B = 5 cm, H = 3 cm.

V = 8 × 5 × 3 = 120 cm³

Example 3 — Cylinder

r = 7 cm, h = 5 cm. Use π = 22/7.

V = πr²h
V = 22/7 × 49 × 5
V = 770 cm³

Example 4 — Cone

r = 3 cm, h = 6 cm.

V = ⅓π × 3² × 6
V = 18π cm³

Example 5 — Sphere

r = 2 cm.

V = ⁴⁄₃π × 2³
V = 32π/3 cm³

16. Common Mistakes

1. Confusing area and volume

Area is measured in square units, while volume is measured in cubic units.

2. Forgetting the 1/3 in cone and pyramid

Cone Volume = ⅓πr²h
Pyramid Volume = ⅓ × Base Area × Height

3. Using diameter instead of radius

If diameter d is given:

r = d/2

4. Confusing height and slant height

In a cone, h is vertical height and l is slant height.

5. Forgetting the two bases

For a cylinder:

TSA = 2πr(r+h)

The total surface area includes both circular bases.

17. Competitive Exam Shortcuts

Memorise these high-frequency formulas:

Cube: V = a³
Cuboid: V = LBH
Cylinder: V = πr²h
Cone: V = ⅓πr²h
Sphere: V = ⁴⁄₃πr³
Hemisphere: V = ⅔πr³
Prism: V = Base Area × Height
Pyramid: V = ⅓ × Base Area × Height

18. Practice Questions

Question 1: Find the volume and TSA of a cube whose side is 6 cm.

Question 2: A cuboid has length 12 cm, breadth 8 cm and height 5 cm. Find its volume.

Question 3: Find the volume of a cylinder with radius 7 cm and height 10 cm. Use π = 22/7.

Question 4: Find the volume of a cone with radius 6 cm and height 7 cm.

Question 5: Find the volume of a sphere whose radius is 6 cm.

Question 6: Find the volume of a hemisphere whose radius is 7 cm.

Question 7: A hollow sphere has outer radius 8 cm and inner radius 5 cm. Find its volume.

Question 8: A prism has base area 40 cm² and height 12 cm. Find its volume.

Question 9: A pyramid has base area 60 cm² and height 15 cm. Find its volume.

Question 10: A cylinder has radius 5 cm and height 12 cm. Find its CSA.

19. Quick Revision

Cube → V = a³
Cuboid → V = LBH
Cylinder → V = πr²h
Cone → V = ⅓πr²h
Frustum → V = ⅓π(R²+r²+Rr)h
Sphere → V = ⁴⁄₃πr³
Hollow Sphere → V = ⁴⁄₃π(R³−r³)
Hemisphere → V = ⅔πr³
Prism → V = Base Area × Height
Pyramid → V = ⅓ × Base Area × Height

20. Final Exam Tip

Before solving any volume or surface-area question, identify the solid first.

  1. Cube? → use cube formulas.
  2. Cuboid? → use L, B and H.
  3. Cylinder? → identify r and h.
  4. Cone? → identify r, h and l.
  5. Frustum? → identify R, r, h and l.
  6. Sphere? → identify r.
  7. Hollow sphere? → identify R and r.
  8. Hemisphere? → remember it is half a sphere.
  9. Prism? → Base Area × Height.
  10. Pyramid? → ⅓ × Base Area × Height.

Always check the final unit: cm² for surface area and cm³ for volume.

Formula Source:

This Part 15 follows the final two pages of the provided mensuration formula PDF. Page 10 contains the formulas for Cube, Cuboid, Cylinder, Cone and Frustum, while page 11 contains Sphere, Hollow Sphere, Hemisphere, Prism and Pyramid. :contentReference[oaicite:1]{index=1}

Next Part:

The provided PDF ends with the Volume & Surface Area formulas. If you want to continue the website series, the next part can be prepared as a separate Mensuration Practice Questions & Solved Problems section based on these formulas.