📦 3D Projections & Sections

3D geometry problems often involve projections and cross-sections. Master these techniques for AMC 12 success.

🎯 Recognition Cues

Key Triggers

  • 3D figures - Look for cubes, prisms, pyramids, or spheres
  • Projections - Look for problems involving projecting 3D figures onto 2D planes
  • Cross-sections - Look for problems involving slicing 3D figures
  • Areas/lengths - Look for problems asking for areas or lengths in 3D

Common Setups

  • Cubes with cross-sections
  • Prisms with projections
  • Pyramids with sections
  • Spheres with projections

🧩 Solution Template

Step 1: Identify the 3D Figure

  • Determine the type of 3D figure
  • Identify key dimensions and properties
  • Mark given information

Step 2: Find the Projection or Section

  • Determine what 2D figure results from projection or section
  • Calculate dimensions of the 2D figure
  • Use 2D geometry to find areas or lengths

Step 3: Apply 3D Properties

  • Use 3D formulas for volumes or surface areas
  • Apply projection formulas
  • Use similar triangles or other 2D techniques

Step 4: Verify

  • Check that the 2D figure makes sense
  • Ensure all calculations are correct
  • Verify the final answer

🔍 Worked Example

Problem: A cube with side length 6 is cut by a plane that passes through the midpoints of three edges meeting at a vertex. Find the area of the cross-section.

Solution: Step 1: Identify 3D figure

  • Cube with side length 6
  • Plane passes through midpoints of three edges

Step 2: Find the cross-section

  • The cross-section is a triangle
  • Vertices are at midpoints of three edges
  • Each edge of the triangle is the diagonal of a face

Step 3: Calculate dimensions

  • Each edge of the triangle is $\sqrt{6^2 + 6^2} = 6\sqrt{2}$
  • This is an equilateral triangle with side length $6\sqrt{2}$
  • Area = $\frac{\sqrt{3}}{4} \cdot (6\sqrt{2})^2 = \frac{\sqrt{3}}{4} \cdot 72 = 18\sqrt{3}$

Step 4: Verify

  • Check that cross-section is triangle ✓
  • Calculate area correctly ✓
  • Answer makes sense ✓

Answer: $18\sqrt{3}$

⚠️ Common Pitfalls

Pitfall: Wrong cross-section identification

  • Fix: Carefully determine what 2D figure results from the cut

Pitfall: Forgetting about 3D properties

  • Fix: Use 3D formulas when appropriate

Pitfall: Wrong projection setup

  • Fix: Make sure you understand the projection direction

Pitfall: Forgetting about similar triangles

  • Fix: Look for similar triangles in 3D problems

🔗 Related Patterns

💡 Quick Reference

3D Projection Properties

  • Orthogonal projection: Project perpendicular to plane
  • Cross-sections: 2D slices of 3D figures
  • Areas: Use 2D formulas for cross-sections

Common 3D Figures

  • Cube: All faces are squares
  • Prism: Two parallel bases connected by rectangles
  • Pyramid: Base connected to apex
  • Sphere: All points equidistant from center

Solution Strategy

  • Identify: Look for 3D figures with projections or sections
  • Project: Find the 2D projection or section
  • Calculate: Use 2D geometry to find areas or lengths
  • Verify: Check that answer makes geometric sense

Next: Geometric Probability Patterns → | Prev: Mass Points Templates → | Back to: Geometry Mastery Guide →