How to Solve: Volume and Surface Area (3D Figures)

How to Solve: Volume and Surface Area (3D Figures)

For SSC / Bank / Railway Exams

Introduction

"Mastering volume and surface area can add 5–8 marks to your SSC, Bank, or Railway exam—enough to push you from ‘just passing’ to ‘top rank.’ These questions appear in every paper, and if you follow this exact method, you’ll solve them in under 60 seconds."

What You Need To Know First

1. Basic 2D shapes: Know the area formulas for squares, rectangles, triangles, and circles.
2. Units: Understand how to convert between cm³, m³, liters, and cm², m².
3. Pythagoras’ theorem: Needed for diagonal problems in cubes/cuboids.

Key Vocabulary

Formulas To Know

1. Cuboid (Rectangular Box)

• Volume (V) = length × width × height V = l × w × h MEMORISE THIS (l, w, h = dimensions in same units)

• Total Surface Area (TSA) = 2(lw + lh + wh) MEMORISE THIS

• Lateral Surface Area (LSA) = 2h(l + w) Given on exam sheet (but memorise to save time)

2. Cube (Special Cuboid)

• Volume (V) = side³ V = a³ (a = side length) MEMORISE THIS

• Total Surface Area (TSA) = 6 × side² TSA = 6a² MEMORISE THIS

• Lateral Surface Area (LSA) = 4 × side² LSA = 4a² Given on exam sheet

3. Cylinder

• Volume (V) = π × radius² × height V = πr²h MEMORISE THIS (r = radius, h = height)

• Total Surface Area (TSA) = 2πr(r + h) MEMORISE THIS

• Lateral Surface Area (LSA) = 2πrh Given on exam sheet

4. Cone

• Volume (V) = (1/3) × π × radius² × height V = (1/3)πr²h MEMORISE THIS

• Total Surface Area (TSA) = πr(r + l) l = slant height (use Pythagoras: l = √(r² + h²)) MEMORISE THIS

• Lateral Surface Area (LSA) = πrl Given on exam sheet

5. Sphere

• Volume (V) = (4/3)πr³ MEMORISE THIS

• Surface Area (SA) = 4πr² MEMORISE THIS

Step-by-Step Method

Follow these 5 steps for every problem:

1. Identify the shape → Is it a cube, cylinder, cone, or sphere?
2. List given values → Write down all numbers (e.g., r = 7 cm, h = 10 cm).
3. Check units → Convert all to the same unit (e.g., cm → m if needed).
4. Pick the right formula → Volume or surface area? Total or lateral?
5. Plug in numbers & solve → Show every step to avoid mistakes.

Worked Examples

Example 1 – Basic (Cuboid Volume)

Question: A box has length 5 cm, width 3 cm, and height 2 cm. Find its volume.

Solution: 1. Shape: Cuboid. 2. Given: l = 5 cm, w = 3 cm, h = 2 cm. 3. Units: All in cm → no conversion needed. 4. Formula: Volume = l × w × h. 5. Calculation: 5 × 3 × 2 = 30 cm³.

What we did and why: We multiplied the three dimensions because volume measures how much space the box occupies.

Example 2 – Medium (Cylinder Surface Area)

Question: A cylindrical can has radius 7 cm and height 10 cm. Find its total surface area.

Solution: 1. Shape: Cylinder. 2. Given: r = 7 cm, h = 10 cm. 3. Units: All in cm → no conversion. 4. Formula: TSA = 2πr(r + h). 5. Calculation:
- First, r + h = 7 + 10 = 17.
- Then, 2 × π × 7 × 17 = 2 × 22/7 × 7 × 17 = 2 × 22 × 17 = 748 cm².

What we did and why: We used the TSA formula because the question asked for the entire outer area (including top and bottom circles).

Example 3 – Exam-Style (Cone Volume)

Question: A cone has a base diameter of 14 cm and height 24 cm. Find its volume. (Take π = 22/7)

Solution: 1. Shape: Cone. 2. Given: Diameter = 14 cm → radius (r) = 7 cm, h = 24 cm. 3. Units: All in cm → no conversion. 4. Formula: Volume = (1/3)πr²h. 5. Calculation:
- r² = 7² = 49.
- (1/3) × (22/7) × 49 × 24 = (1/3) × 22 × 7 × 24 = 22 × 7 × 8 = 1232 cm³.

What we did and why: We halved the diameter to get the radius, then plugged into the cone volume formula. The (1/3) factor is crucial—missing it is a common mistake!

Common Mistakes

Exam Traps

1-Minute Recap

"Night before the exam? Here’s what to remember: 1. Volume = space inside (cube: a³, cylinder: πr²h, cone: (1/3)πr²h). 2. Surface area = outer covering (cube: 6a², cylinder: 2πr(r + h)). 3. Radius vs. diameter – halve the diameter to get radius. 4. Units matter – convert cm to m if needed. 5. Read the question – is it total or lateral surface area? Write down the formula, plug in numbers, and solve step by step. You’ve got this!