Volume and Surface Area are among the most scoring and formula-driven topics in Mensuration. But many students find them confusing because every solid has unique dimensions, different types of areas, and additional parameters such as slant height or radius.

If you understand these three ideas:
✔ Dimensions of the solid
✔ Surface area to be covered
✔ Volume occupied inside

Then you can solve ANY mensuration question easily.

This Volume and Surface Area guide covers all formulas, concepts, shortcuts, cases, diagrams, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Volume & Surface Area Formulas

Shape	Dimensions	Volume	Curved / Lateral Area	Total Surface Area	Diagonal / Slant Height
Cuboid	l, b, h	lbh	–	2(lb + bh + lh)	√(l² + b² + h²)
Cube	a	a³	–	6a²	a√3
Cylinder	r, h	πr²h	2πrh	2πr(h+r)	–
Cone	r, h	(1/3)πr²h	πrl	πr(l+r)	l = √(h² + r²)
Sphere	r	(4/3)πr³	–	4πr²	–
Hemisphere	r	(2/3)πr³	2πr²	3πr²	–

Volume & Surface Area Formulas

1. Fundamentals of Mensuration (Base Concepts)

Before using shape-specific formulas, understand these foundations clearly. Every question relies on these ideas:

• Volume = space inside a solid
• Surface Area = total area covering the outside
• Curved Surface Area (CSA) = outer curved region
• Total Surface Area (TSA) = Curved Area + Base Areas
• All dimensions must be in the same unit

These basics apply to cuboid, cube, cylinder, cone, sphere, and hemisphere.

2. Cuboid Formulas (Length, Breadth, Height Based Solid)

A cuboid has different edge lengths, making each surface a rectangle.

Key Dimensions

• Length = l
• Breadth = b
• Height = h

Volume of a Cuboid

Volume = l×b×h

Meaning: space inside the 3D box.

Total Surface Area (TSA)

TSA = 2(lb+bh+lh)

Reason: A cuboid has 6 rectangular faces.

Diagonal of a Cuboid

Diagonal = √l2 + b2 + h2​

This represents the longest internal distance inside the cuboid.

3. Cube Formulas (Equal Edge Solid)

A cube has all edges equal, so calculations become simpler.

Let edge = a

Volume of Cube

Volume = a3

Total Surface Area (TSA)

TSA = 6a2

Reason: 6 identical square faces.

Diagonal of Cube

Diagonal = a√3

This connects two opposite corners of the cube.

4. Cylinder Formulas (Circular-Based Solid)

A cylinder has two circular bases and a curved surface around them.

Dimensions

• Radius = r
• Height = h

Volume of Cylinder

Volume = πr2h

Curved Surface Area (CSA)

CSA = 2πrh

Represents the curved wrapper around the cylinder.

Total Surface Area (TSA)

TSA = 2πr(h+r)

Includes both circular bases + curved area.

5. Cone Formulas (Tapered Circular Solid)

A cone has a pointed top and a circular base.

Dimensions

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

Where

l = √h2+r2

Volume of Cone

Volume = (⅓)πr2h

Curved Surface Area (CSA)

CSA = πrl

Total Surface Area (TSA)

TSA = πr(l+r)

6. Sphere Formulas (Perfect Round Solid)

A sphere is a perfectly round 3D object.

Radius = r

Volume of Sphere

(4/3)πr3

Surface Area of Sphere

4πr2

A sphere has no edges or vertices.

7. Hemisphere Formulas (Half of a Sphere)

A hemisphere is simply half of a sphere, but area changes because the flat face is exposed.

Radius = r

Volume of Hemisphere

(⅔)πr3

Curved Surface Area (CSA)

2πr2

Total Surface Area (TSA)

3πr2

(TSA = Curved Area + Circular Base)

Smart Tips and Practical Tricks for Solving Volume and Surface Area Problems

Mastering Volume and Surface Area becomes simple once you understand how dimensions, shapes, and units work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding what each measurement represents. This section breaks down the most important ideas into clear, actionable tips so you can solve questions faster and more accurately.

1. Convert All Units Before Using Any Formula

Mensuration questions often mix cm, m, and mm. If you don’t convert them to the same unit, the answer will always be wrong.

Always remember:

• Area uses square units (cm², m²)
• Volume uses cubic units (cm³, m³)

Example:
1 litre = 1000 cm³

This one step prevents most calculation errors.

2. Identify the Exact Surface Area Required

Different solids require different kinds of areas.

• Curved Surface Area (CSA) → only round outer surface
• Total Surface Area (TSA) → curved area + base areas
• Lateral Surface Area → sides only, without top/bottom

Students often confuse CSA with TSA, which changes the answer completely.

Always check the question carefully:
✔ Do they want curved?
✔ Total?
✔ Only side area?

3. Visualize the Solid Before Applying a Formula

Many mistakes happen because students cannot imagine the shape.

Think of:

• A cuboid as a rectangular box
• A cube as a dice
• A cylinder as a tin can
• A cone as an ice-cream cone
• A sphere as a ball
• A hemisphere as half a ball

Visualizing helps you instantly know which surfaces and dimensions matter.

4. Understand Why Formulas Work Instead of Memorizing

Every formula comes from simple shapes:

• Cylinder surface area → rolled rectangle + two circles
• Cube → 6 identical squares
• Cone → slanted triangle + circle
• Sphere → curved surface only

When you understand the structure, you will never forget a formula.

5. Double-Check Formula Selection Before Calculating

Students often use the wrong formula because shapes look similar.

Always confirm:
✔ Cube or cuboid?
✔ Sphere or hemisphere?
✔ CSA or TSA?
✔ Volume or surface area?

A quick 2-second check can save the entire question.

6. Use π Efficiently for Faster Calculations

Most circular solids use π.

Use:

• π = 22/7 when radius is multiple of 7
• π = 3.14 for approximate answers

This helps computations run much faster.

7. Practice Exam-Pattern Questions Regularly

Mensuration problems appear frequently in:

• SSC (CGL, CHSL, GD)
• Banking (IBPS, SBI)
• Railway RRB
• State exams
• Campus aptitude tests

Practicing these shapes and formulas improves accuracy and reduces time during exams.

FAQs About Volume and Surface Area

Q1. Why do some solids have curved surface area and total surface area separately?

Because curved solids like cylinders, cones, and spheres have a rounded surface plus base/flat surfaces. TSA includes both; CSA includes only the curved part.

Q2. Why does the volume of a cube use a³ while a cuboid uses l × b × h?

A cube has all edges equal, so multiplying a × a × a gives a³. A cuboid has different lengths, so all three dimensions remain separate.

Q3. Why is the volume of a cone one-third of a cylinder?

A cone with the same radius and height occupies exactly one-third the space of the cylinder because of its tapering shape.

Q4. Why does a hemisphere have a larger surface area than half a sphere?

Cutting a sphere exposes a new circular face, adding extra area. That is why TSA of hemisphere is 3πr², not 2πr².

Q5. Why is π used in cylinder, cone, sphere, and hemisphere formulas?

Because these shapes are derived from circles, and π is the constant relationship between radius and circular measurements.

Q6. Why do mensuration questions often require unit conversion?

Because lengths may be given in cm, area in m², and volume in litres. Using the same unit ensures accurate results.

Q7. Why is slant height necessary for calculating the surface area of a cone?

The curved surface of a cone is shaped by the slant height, not the vertical height. That is why l = √(h² + r²) is required.

Q8. Why does a cuboid’s surface area use 2(lb + bh + lh)?

Because a cuboid has three pairs of identical rectangles, and multiplying the sum of their areas by 2 gives the total surface area.

Q9. Why is the diagonal of a cuboid calculated using √(l² + b² + h²)?

Because the diagonal is the longest straight line inside the cuboid, connecting opposite corners in 3D space using Pythagoras in three directions.

Q10. Why is practice so important in mensuration?

Formulas are easy, but identifying which one to use quickly is the real challenge. Regular practice improves recognition and reduces mistakes.