Cube Cuboid and Dice

Cube, Cuboid, and Dice: Easy Study Material with Emojis and Practice Questions 🎲📐

1. Cube Basics 🧊

• A cube is a 3D shape with all sides equal: length = width = height = s.
• It has:
  • 6 faces (front, back, right, left, top, bottom)
  • 8 vertices (corners)
  • 12 edges (lines connecting vertices)

2. Surface Area of Cube 📏

• Total Surface Area (TSA) = 6 × (side × side) = $6s^2$
• Lateral Surface Area (LSA) = 4 × (side × side) = $4s^2$ (Sum of 4 side faces, excluding top and bottom)

3. Cuboid Basics 📦

• A cuboid is a 3D shape with length (l), width (w), height (h) all possibly different.
• It has:
  • 6 rectangular faces
  • 8 vertices
  • 12 edges
• Total Surface Area (TSA) = Sum of areas of all 6 faces = $2(lw + lh + wh)$
• Lateral Surface Area (LSA) = Sum of 4 side faces (excluding top and bottom)

4. Unit Cubes and Painted Surfaces 🎨

• When a cube is divided into smaller cubes, these are called unit cubes.
• Example: Dividing a cube into 3 rows × 3 columns × 3 layers = 27 unit cubes.

5. Painted Unit Cubes: How Many Have How Many Painted Faces? 🎨

For a cube divided into $n \times n \times n$ unit cubes:

6. Practice Questions: Painted Cubes 🧩

A cube of side 10 cm is painted red with a 2 cm wide green strip along all sides on all faces. It is divided into 125 smaller cubes.

1. How many cubes have 3 green faces?
   • Answer: 8 (corner cubes) 🟩🟩🟩
2. How many cubes have 1 red face and 1 adjacent green face?
   • Answer: 0 (green cubes have at least 2 faces painted) ❌
3. How many cubes have at least one face coloured?
   • Total cubes = 125
   • Cubes with no paint = 27
   • Painted cubes = 125 - 27 = 98 ✅
4. How many cubes have at least 2 green faces?
   • Painted cubes - cubes with only red paint - cubes with no paint = 44 ✅

7. Dice Basics 🎲

• A dice is a cube with numbers 1 to 6 on its faces.
• Opposite faces always add up to 7:
  • 1 opposite 6
  • 2 opposite 5
  • 3 opposite 4

8. Types of Dice 🎲

• Standard Dice: When two dice are rolled, numbers on faces do not match.
• Ordinary Dice: One or more numbers match between two dice.

9. Rules for Dice Problems 🎲🔄

• Rule 1: If two dice share a common number on the same face, remaining faces are opposite.
• Rule 2: If two dice share two numbers anywhere, the third numbers are opposite.
• Rule 3: If one number is common but in different positions, rotate dice clockwise to find opposite faces.

10. Open Dice (Flattened Dice) 🟥⬜

• Opposite faces never touch each other.
• Common opposite pairs vary, e.g.:
  • 1 opposite 6
  • 2 opposite 5
  • 3 opposite 4

11. Practice Questions: Dice 🎲

1. Possible combinations of dice?
   • Only option 3 is possible based on adjacency and opposition rules.
2. Opposite side to face with 3 dots?
   • Answer: 6 (since 4,5,2 are adjacent to 3).
3. Choose dice similar to the given open die:
   • Answer: Figures 1 and 3.
4. Dots opposite to face with 4 dots?
   • Answer: 2.
5. If 4 is at the bottom, what number is on top?
   • Answer: 1.
6. If 1 is adjacent to 2,3,5, which is necessarily true?
   • Answer: 4 is adjacent to 6.

Summary Table: Cube & Dice Quick Facts 📋

Visualize & Practice! 🎯

• Draw cubes and mark painted faces to understand unit cubes with different painted sides.
• Practice dice problems by sketching dice nets (open dice) and folding them mentally.
• Use the rules to check opposites and adjacency on dice.

Happy Learning! 🎉 Master these concepts with practice to ace your SSC CHSL and other exams!