Volume & Surface Areas
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Important Formulas & Concepts
Study MaterialVolume & Surface Areas
Volume and Surface Areas is one of the most important chapters in Quantitative Aptitude and Mensuration. Questions from this chapter are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, Defence, and various competitive examinations.
This chapter mainly deals with:
- Volume of solids
- Surface area calculations
- Cube and cuboid problems
- Cylinder and cone formulas
- Sphere and hemisphere concepts
- Pipe and tank problems
- Melting and recasting solids
- Mensuration-based aptitude questions
Strong understanding of formulas, visualization of 3D objects, and geometry concepts helps candidates solve mensuration problems quickly and accurately.
What is Volume?
The space occupied by a solid object is called its Volume.
Volume is measured in cubic units like:
- cm3
- m3
- km3
Volume = Length × Breadth × Height
Example:
If length = 9 cm, breadth = 4 cm, and height = 2 cm:
Volume = 9 × 4 × 2 = 72 cm3
Volume Concept Diagram
Illustration showing volume of a cuboid.
What is Surface Area?
The total area covered by all outer surfaces of a solid object is called its Surface Area.
Surface area is measured in square units like:
- cm2
- m2
- km2
Types of Surface Areas:
- Curved Surface Area (CSA)
- Total Surface Area (TSA)
Surface Area = Sum of Areas of All Surfaces
Surface Area Diagram
Illustration showing the surface area of a cube.
Difference Between Volume and Surface Area
| Concept | Meaning | Units |
|---|---|---|
| Volume | Space occupied by object | Cubic Units |
| Surface Area | Total outer area | Square Units |
Part – I : Cube
A cube is a solid figure in which all edges are equal.
Cube Diagram
3D cube showing edges, faces, and dimensions.
Let side of cube = a
Important Formulas of Cube
| Concept | Formula |
|---|---|
| Volume | a3 |
| Total Surface Area | 6a2 |
| Lateral Surface Area | 4a2 |
| Diagonal | a√3 |
Important Properties of Cube
- Cube has 6 square faces.
- Cube has 12 equal edges.
- Cube has 8 vertices.
- All angles are 90°.
Part – II : Cuboid
A cuboid is a rectangular solid figure having length, breadth, and height.
Cuboid Diagram
Cuboid showing length, breadth, height, and 3D structure.
Let:
- Length = l
- Breadth = b
- Height = h
Important Formulas of Cuboid
| Concept | Formula |
|---|---|
| Volume | l × b × h |
| Total Surface Area | 2(lb + bh + hl) |
| Lateral Surface Area | 2h(l+b) |
| Diagonal | √(l2 + b2 + h2) |
Important Properties of Cuboid
- Opposite faces are equal.
- All angles are right angles.
- Diagonal connects opposite vertices.
Part – III : Cylinder
A cylinder is a solid having two circular bases connected by a curved surface.
Cylinder Diagram
Cylinder showing radius, height, circular bases, and curved surface area.
Let:
- Radius = r
- Height = h
Important Formulas of Cylinder
| Concept | Formula |
|---|---|
| Volume | πr2h |
| Curved Surface Area | 2Ï€rh |
| Total Surface Area | 2Ï€r(h+r) |
Important Cylinder Concepts
- Cylinder has two circular bases.
- Radius remains constant.
- Curved surface joins both bases.
Part – IV : Sphere and Hemisphere
A sphere is a perfectly round solid figure.
Sphere Diagram
Sphere showing radius, diameter, center point, and curved surface.
Let radius = r
Important Formulas of Sphere
| Concept | Formula |
|---|---|
| Volume | (4/3)Ï€r3 |
| Surface Area | 4Ï€r2 |
Hemisphere Diagram
Hemisphere showing radius, curved surface, and flat circular base.
Important Formulas of Hemisphere
| Concept | Formula |
|---|---|
| Volume | (2/3)Ï€r3 |
| Curved Surface Area | 2Ï€r2 |
| Total Surface Area | 3Ï€r2 |
Part – V : Cone
A cone is a solid having one circular base and one vertex.
Cone Diagram
Cone showing radius, height, slant height, circular base, and vertex.
Let:
- Radius = r
- Height = h
- Slant Height = l
Important Formulas of Cone
| Concept | Formula |
|---|---|
| Slant Height | √(h2 + r2) |
| Volume | (1/3)Ï€r2h |
| Curved Surface Area | πrl |
| Total Surface Area | πr(l+r) |
Important Mensuration Concepts
1. Melting and Recasting
When a solid is melted and recast into another shape, volume remains constant.
2. Hollow Cylinder Formula
Volume = πh(R2 − r2)
where:
- R = External Radius
- r = Internal Radius
3. Water Tank Formula
Volume = Base Area × Height
4. Wire and Pipe Problems
Most wire problems use cylinder formulas.
Important Unit Conversion Table
| Conversion | Value |
|---|---|
| 1 m3 | 1,000,000 cm3 |
| 1 litre | 1000 cm3 |
| 1 m2 | 10,000 cm2 |
Important Geometry Properties
- Sphere has no edges or vertices.
- Cylinder has curved surface.
- Cone has one vertex.
- Cube has equal edges.
- Cuboid has rectangular faces.
Common Mistakes in Volume & Surface Areas
- Confusing CSA and TSA formulas.
- Using diameter instead of radius.
- Ignoring unit conversions.
- Using wrong π value.
- Calculation mistakes in cube roots and square roots.
Important Exam Tips
- Memorize all standard formulas.
- Practice 3D visualization regularly.
- Use π = 22/7 whenever possible.
- Draw diagrams for clarity.
- Practice melting and recasting problems.
- Verify units carefully.
- Learn difference between CSA and TSA.
Quick Revision Formula Table
| Solid | Volume | Surface Area |
|---|---|---|
| Cube | a3 | 6a2 |
| Cuboid | lbh | 2(lb+bh+hl) |
| Cylinder | πr2h | 2πr(h+r) |
| Sphere | (4/3)Ï€r3 | 4Ï€r2 |
| Cone | (1/3)πr2h | πr(l+r) |
| Hemisphere | (2/3)Ï€r3 | 3Ï€r2 |
Volume and Surface Areas form one of the most important parts of Mensuration and Geometry. Strong understanding of formulas, properties, and visualization techniques helps candidates solve aptitude problems quickly and accurately in competitive examinations.