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3D Mensuration — Cubes & Cuboids

Introduction

3D Mensuration deals with solid shapes and their measurements - such as volume, surface area, and diagonal lengths. Cubes and cuboids form the foundation of all 3D geometry problems because they appear in various real-life applications like boxes, containers, and storage spaces.

This pattern helps you calculate dimensions and areas of cubical and rectangular solids efficiently using direct formulas.

Pattern: 3D Mensuration — Cubes & Cuboids

Pattern

Key concept: Use formulas to calculate volume, total surface area (TSA), lateral surface area (LSA), and diagonal length of cubes and cuboids.

Formulas to remember:
• For a cube (side = a):
  Volume = a³, TSA = 6a², LSA = 4a², Diagonal = a√3.

• For a cuboid (length = l, breadth = b, height = h):
  Volume = l × b × h, TSA = 2(lb + bh + hl), LSA = 2h(l + b), Diagonal = √(l² + b² + h²).

Step-by-Step Example

Question

A cuboid has dimensions 10 cm × 8 cm × 6 cm. Find its volume, total surface area (TSA), and diagonal length.

Solution

  1. Step 1: Find the Volume.

    Volume = l × b × h = 10 × 8 × 6 = 480 cm³.

  2. Step 2: Find the Total Surface Area (TSA).

    TSA = 2(lb + bh + hl) = 2(10×8 + 8×6 + 6×10) = 2(80 + 48 + 60) = 2 × 188 = 376 cm².

  3. Step 3: Find the Diagonal.

    Diagonal = √(l² + b² + h²) = √(10² + 8² + 6²) = √(100 + 64 + 36) = √200 = 10√2 cm.

  4. Final Answers:

    • Volume = 480 cm³
    • TSA = 376 cm²
    • Diagonal = 10√2 cm

  5. Quick Check:

    Substitute the values into each formula again - all consistent ✅

Quick Variations

1. If side is given, use cube formulas directly.

2. Given volume, find one dimension using cube root or simple division.

3. Find ratio of areas or volumes between two cubes.

4. Practical questions - painting surface area, melting and recasting metal cubes, etc.

Trick to Always Use

  • Step 1 → Always identify whether shape is a cube (all sides equal) or cuboid (different dimensions).
  • Step 2 → For area problems, apply TSA or LSA depending on what’s asked.
  • Step 3 → For diagonal, square and add all sides, then take the square root.

Summary

Summary

In the 3D Mensuration - Cubes & Cuboids pattern:

  • Cube → Volume = a³, TSA = 6a², Diagonal = a√3.
  • Cuboid → Volume = l×b×h, TSA = 2(lb + bh + hl), Diagonal = √(l² + b² + h²).
  • Diagonal represents the longest straight line inside the solid.
  • Quick check: For equal edges → cube; for unequal → cuboid.

Practice

(1/5)
1. Find the volume of a cube with side 5 cm.
easy
A. 125 cm³
B. 100 cm³
C. 150 cm³
D. 225 cm³

Solution

  1. Step 1: Apply cube volume formula

    Use formula for volume of a cube → Volume = a³.

  2. Step 2: Substitute the side value

    Substitute a = 5 → 5³ = 125.

  3. Final Answer:

    125 cm³ → Option A.

  4. Quick Check:

    5 × 5 × 5 = 125 ✅
Hint: For cubes, cube the side directly to get volume.
Common Mistakes: Multiplying only two sides instead of three.
2. A cuboid has dimensions 8 cm × 6 cm × 4 cm. Find its volume.
easy
A. 180 cm³
B. 192 cm³
C. 200 cm³
D. 210 cm³

Solution

  1. Step 1: Use cuboid volume formula

    Use formula → Volume = l × b × h.

  2. Step 2: Multiply all dimensions

    Substitute values → 8 × 6 × 4 = 192.

  3. Final Answer:

    192 cm³ → Option B.

  4. Quick Check:

    8 × 6 = 48, 48 × 4 = 192 ✅
Hint: Multiply all three dimensions directly.
Common Mistakes: Adding instead of multiplying sides.
3. Find the total surface area (TSA) of a cube with side 10 cm.
easy
A. 400 cm²
B. 500 cm²
C. 600 cm²
D. 700 cm²

Solution

  1. Step 1: Apply TSA formula for cubes

    Formula → TSA = 6a².

  2. Step 2: Substitute side value

    Substitute a = 10 → TSA = 6 × 10² = 6 × 100 = 600.

  3. Final Answer:

    600 cm² → Option C.

  4. Quick Check:

    Each face = 100 cm² × 6 faces = 600 ✅
Hint: Multiply one face area by 6 for cubes.
Common Mistakes: Using formula for volume instead of TSA.
4. Find the diagonal length of a cuboid with l = 9 cm, b = 12 cm, h = 20 cm.
medium
A. 22 cm
B. 24 cm
C. 26 cm
D. 25 cm

Solution

  1. Step 1: Write the diagonal formula

    Use diagonal formula → √(l² + b² + h²).

  2. Step 2: Substitute values and compute sum

    Substitute → √(9² + 12² + 20²) = √(81 + 144 + 400) = √625.

  3. Step 3: Extract the square root

    √625 = 25 cm.

  4. Final Answer:

    25 cm → Option D.

  5. Quick Check:

    Perfect square 625 → 25 ✅
Hint: Use Pythagoras in 3D: √(l² + b² + h²).
Common Mistakes: Forgetting to square all sides before adding.
5. A cube has a total surface area of 150 cm². Find its side length.
medium
A. 5 cm
B. 4 cm
C. 6 cm
D. 7 cm

Solution

  1. Step 1: Write TSA formula

    Formula → TSA = 6a².

  2. Step 2: Isolate a²

    Substitute → 150 = 6a² ⇒ a² = 150 ÷ 6 = 25.

  3. Step 3: Take square root

    a = √25 = 5 cm.

  4. Final Answer:

    5 cm → Option A.

  5. Quick Check:

    6 × 5² = 6 × 25 = 150 ✅
Hint: a = √(TSA ÷ 6).
Common Mistakes: Dividing by wrong number of faces or forgetting to take square root.

Mock Test

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