🎯 Solid Geometry Problem Types

📊 Problem Pattern Catalog

Type 1: Basic Volume Calculations

Pattern: Find volume of a basic solid given dimensions Key Formula: Use appropriate volume formula for the solid

Worked Example:

Find the volume of a cylinder with radius 4 cm and height 10 cm.

Solution: $V = \pi r^2 h = \pi \cdot 4^2 \cdot 10 = \pi \cdot 16 \cdot 10 = 160\pi$ cm³

Type 2: Surface Area Calculations

Pattern: Find total surface area of a 3D object Key Formula: Sum of all face areas

Worked Example:

Find the surface area of a cube with side length 5.

Solution: $SA = 6s^2 = 6 \cdot 5^2 = 6 \cdot 25 = 150$ square units

Type 3: Volume by Decomposition

Pattern: Find volume of complex shape by breaking into simpler parts Key Concept: Add or subtract volumes of component solids

Worked Example:

Find the volume of an L-shaped prism with dimensions as shown.

Solution: Break into two rectangular prisms:

  • Large prism: $8 \times 6 \times 4 = 192$
  • Small prism: $4 \times 2 \times 4 = 32$
  • Total volume: $192 + 32 = 224$ cubic units

Type 4: Cross-Section Problems

Pattern: Find area or perimeter of cross-section Key Concept: Identify the shape of the cross-section

Worked Example:

A cylinder is cut by a plane perpendicular to its axis. What is the area of the cross-section?

Solution: Cross-section is a circle with same radius as cylinder. If radius is $r$, then area = $\pi r^2$

Type 5: Scaling and Similarity

Pattern: Find volume/surface area when dimensions are scaled Key Concept: Volume scales by cube of factor, surface area by square of factor

Worked Example:

A cube has side length 3. If all dimensions are doubled, what is the new volume?

Solution: Original volume: $3^3 = 27$ Scaling factor: 2 New volume: $27 \times 2^3 = 27 \times 8 = 216$

🔍 Problem-Solving Strategy

  1. Identify the solid and its type
  2. List given dimensions and what you need to find
  3. Choose appropriate formula for volume or surface area
  4. Substitute values carefully
  5. Check units and simplify

⚠️ Common Mistakes

  • Wrong formula for the solid type
  • Missing the $\frac{1}{3}$ factor in pyramid/cone volume
  • Incomplete surface area calculations
  • Unit conversion errors
  • Wrong cross-section identification