Surface Area and Volume Formulas Summary

Class 9Class 10Surface Areas and Volumes

This is a comprehensive reference of all surface area and volume formulas for 3D shapes covered in Class 9 and Class 10 NCERT Mathematics.

Use this as a quick revision sheet before exams. Each formula includes the shape, the formula, and what each variable represents.

What is Surface Area and Volume Formulas Summary?

Key terms:

LSA / CSA: Lateral Surface Area / Curved Surface Area — the area of the curved or side surfaces only (excludes bases).
TSA: Total Surface Area — the area of ALL surfaces including bases.
Volume: The space enclosed by the solid, measured in cubic units.

Surface Area and Volume Formulas Summary Formula

CUBOID (l × b × h):

LSA = 2h(l + b)
TSA = 2(lb + bh + hl)
Volume = l × b × h
Diagonal = √(l² + b² + h²)

CUBE (side a):

LSA = 4a²
TSA = 6a²
Volume = a³
Diagonal = a√3

CYLINDER (radius r, height h):

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

CONE (radius r, height h, slant height l):

l = √(r² + h²)
CSA = πrl
TSA = πr(l + r)
Volume = (1/3)πr²h

SPHERE (radius r):

SA = 4πr²
Volume = (4/3)πr³

HEMISPHERE (radius r):

CSA = 2πr²
TSA = 3πr²
Volume = (2/3)πr³

FRUSTUM OF A CONE (radii R, r; height h; slant height l):

l = √[h² + (R−r)²]
CSA = π(R + r)l
TSA = π(R + r)l + πR² + πr²
Volume = (1/3)πh(R² + r² + Rr)

Types and Properties

Relationship between shapes:

Volume of cone = (1/3) × Volume of cylinder (same base, same height)
Volume of hemisphere = (2/3) × Volume of cylinder (same base, height = radius)
Volume of sphere = (2/3) × Volume of circumscribing cylinder
If a cone, hemisphere, and cylinder have same radius and height = radius: their volumes are in ratio 1 : 2 : 3

Solved Examples

Example 1: Example 1: Cuboid

Problem: Find TSA and volume of a cuboid 12 × 8 × 5 cm.

Solution:

TSA = 2(12×8 + 8×5 + 12×5) = 2(96 + 40 + 60) = 2(196) = 392 cm²
Volume = 12 × 8 × 5 = 480 cm³

Answer: TSA = 392 cm², Volume = 480 cm³.

Example 2: Example 2: Cylinder

Problem: A cylinder has radius 7 cm and height 10 cm. Find CSA, TSA, and volume.

Solution:

CSA = 2πrh = 2 × 22/7 × 7 × 10 = 440 cm²
TSA = 2πr(r+h) = 2 × 22/7 × 7 × 17 = 748 cm²
Volume = πr²h = 22/7 × 49 × 10 = 1540 cm³

Answer: CSA = 440 cm², TSA = 748 cm², Volume = 1540 cm³.

Example 3: Example 3: Cone

Problem: A cone has radius 5 cm and height 12 cm. Find slant height, CSA, and volume.

Solution:

l = √(25 + 144) = √169 = 13 cm
CSA = πrl = π × 5 × 13 = 65π ≈ 204.29 cm²
Volume = (1/3)πr²h = (1/3) × π × 25 × 12 = 100π ≈ 314.29 cm³

Answer: l = 13 cm, CSA ≈ 204.29 cm², V ≈ 314.29 cm³.

Example 4: Example 4: Sphere

Problem: Find SA and volume of a sphere of radius 14 cm.

Solution:

SA = 4πr² = 4 × 22/7 × 196 = 2464 cm²
Volume = (4/3)πr³ = (4/3) × 22/7 × 2744 = 11498.67 cm³

Answer: SA = 2464 cm², V ≈ 11498.67 cm³.

Example 5: Example 5: Hemisphere

Problem: Find CSA, TSA, and volume of a hemisphere of radius 3.5 cm.

Solution:

CSA = 2πr² = 2 × 22/7 × 12.25 = 77 cm²
TSA = 3πr² = 3 × 22/7 × 12.25 = 115.5 cm²
Volume = (2/3)πr³ = (2/3) × 22/7 × 42.875 = 89.83 cm³

Answer: CSA = 77 cm², TSA = 115.5 cm², V ≈ 89.83 cm³.

Example 6: Example 6: Volume ratio — cone, hemisphere, cylinder

Problem: A cone, hemisphere, and cylinder all have radius 7 cm and height 7 cm. Find the ratio of their volumes.

Solution:

V(cone) = (1/3)πr²h = (1/3)π(49)(7) = 343π/3
V(hemisphere) = (2/3)πr³ = (2/3)π(343) = 686π/3
V(cylinder) = πr²h = π(49)(7) = 343π
Ratio = 343π/3 : 686π/3 : 343π = 1 : 2 : 3

Answer: Ratio = 1 : 2 : 3.

Example 7: Example 7: Frustum

Problem: A frustum has top radius 3 cm, bottom radius 7 cm, and height 6 cm. Find volume and slant height.

Solution:

l = √[36 + 16] = √52 ≈ 7.21 cm
V = (1/3)πh(R² + r² + Rr) = (1/3)π(6)(49 + 9 + 21) = 2π(79) = 158π ≈ 496.37 cm³

Answer: Slant height ≈ 7.21 cm, Volume ≈ 496.37 cm³.

Example 8: Example 8: Quick comparison

Problem: A cube and a sphere both have surface area 600 cm². Which has more volume?

Solution:

Cube: 6a² = 600 → a² = 100 → a = 10. V = 1000 cm³.
Sphere: 4πr² = 600 → r² = 150/π → r ≈ 6.91. V = (4/3)πr³ ≈ 1381.5 cm³.

Answer: The sphere has more volume for the same surface area.

Real-World Applications

When to use which formula:

Painting, wrapping → Surface area (TSA for fully covered, CSA for open containers)
Filling, capacity → Volume
Material for curved part only → CSA
Melting and recasting → Equate volumes

Key Points to Remember

Memorise all formulas — they are foundational for Class 10 and competitive exams.
Cone volume = (1/3) of cylinder volume (same base and height).
Hemisphere volume = (2/3) of cylinder volume (height = radius).
For combined solids: add volumes, but for surface area, exclude hidden areas.
1 litre = 1000 cm³ = 0.001 m³.
Slant height of cone: l = √(r² + h²).
Slant height of frustum: l = √[h² + (R−r)²].
A sphere has the maximum volume for a given surface area among all 3D shapes.

Practice Problems

Find TSA and volume of a cube with side 8 cm.
A cone has slant height 15 cm and radius 9 cm. Find its height, CSA, and volume.
Find the volume of a hemisphere of diameter 21 cm.
A cuboid has volume 1200 cm³, length 15 cm, breadth 10 cm. Find its height and TSA.
A cylinder and cone have the same base (r = 6 cm) and height (14 cm). Find the ratio of their volumes.
A frustum has radii 5 cm and 10 cm with height 8 cm. Find its volume.

Frequently Asked Questions

Q1. What is the difference between CSA and TSA?

CSA includes only the curved (lateral) surface. TSA includes CSA plus the area of all flat bases/faces.

Q2. Why is cone volume one-third of cylinder volume?

Experimentally, if you fill a cone with water and pour into a cylinder of same base and height, it takes exactly 3 cone-fills. Mathematically, it is proved using integral calculus.

Q3. What is the ratio 1:2:3 for cone, hemisphere, cylinder?

When all three have the same radius and height equal to the radius, their volumes are in the ratio 1:2:3.

Q4. How do you find slant height?

For cone: l = √(r² + h²). For frustum: l = √[h² + (R−r)²]. Use the Pythagoras theorem.

Q5. Which shape has maximum volume for given surface area?

The sphere. Among all 3D shapes with the same surface area, the sphere encloses the maximum volume.

Q6. How do you convert volume to capacity?

Volume in cm³ = capacity in ml. 1000 cm³ = 1 litre. 1 m³ = 1000 litres.

Surface Area and Volume Formulas Summary

What is Surface Area and Volume Formulas Summary?

Surface Area and Volume Formulas Summary Formula

Types and Properties

Solved Examples

Example 1: Example 1: Cuboid

Example 2: Example 2: Cylinder

Example 3: Example 3: Cone

Example 4: Example 4: Sphere

Example 5: Example 5: Hemisphere

Example 6: Example 6: Volume ratio — cone, hemisphere, cylinder

Example 7: Example 7: Frustum

Example 8: Example 8: Quick comparison

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the difference between CSA and TSA?

Q2. Why is cone volume one-third of cylinder volume?

Q3. What is the ratio 1:2:3 for cone, hemisphere, cylinder?

Q4. How do you find slant height?

Q5. Which shape has maximum volume for given surface area?

Q6. How do you convert volume to capacity?

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