🧊 Solid Geometry Topics#
🎯 Core Subtopics#
Basic Solids#
- Cube: $V = s^3$, $SA = 6s^2$
- Rectangular Prism: $V = lwh$, $SA = 2(lw + lh + wh)$
- Micro-example: Cube with side 4 has volume $4^3 = 64$ and surface area $6 \cdot 4^2 = 96$
- Trap: Confusing volume and surface area formulas
Circular Solids#
- Cylinder: $V = \pi r^2 h$, $SA = 2\pi r^2 + 2\pi rh$
- Sphere: $V = \frac{4}{3}\pi r^3$, $SA = 4\pi r^2$
- Cone: $V = \frac{1}{3}\pi r^2 h$, $SA = \pi r^2 + \pi rl$
- Micro-example: Cylinder with radius 3 and height 5 has volume $\pi \cdot 3^2 \cdot 5 = 45\pi$
- Trap: Forgetting the $\frac{1}{3}$ factor in cone volume
Pyramids and Prisms#
- Pyramid: $V = \frac{1}{3}Bh$ where $B$ is base area
- Triangular Prism: $V = Bh$ where $B$ is triangular base area
- Micro-example: Square pyramid with base side 6 and height 4 has volume $\frac{1}{3} \cdot 6^2 \cdot 4 = 48$
- Trap: Using wrong height (slant height vs. perpendicular height)
Cross-Sections#
- Horizontal Cross-Sections: Parallel to base
- Vertical Cross-Sections: Perpendicular to base
- Micro-example: Horizontal cross-section of cylinder is always a circle
- Trap: Assuming cross-section shape without considering direction
Surface Area Components#
- Base Areas: Top and bottom faces
- Lateral Surface: Side faces (not including bases)
- Total Surface Area: All faces combined
- Micro-example: Cylinder has 2 circular bases + 1 rectangular lateral surface
- Trap: Forgetting to include all faces in surface area
Volume by Decomposition#
- Breaking Complex Shapes: Into simpler solids
- Adding/Subtracting Volumes: For composite shapes
- Micro-example: L-shaped prism = large rectangular prism - small rectangular prism
- Trap: Double-counting or missing parts in decomposition
🚨 Common Traps#
- Formula Confusion: Mixing up volume and surface area formulas
- Missing Factors: Forgetting $\frac{1}{3}$ in pyramid/cone volume
- Wrong Height: Using slant height instead of perpendicular height
- Unit Errors: Mixing different units in calculations
- Missing Faces: Incomplete surface area calculations
💡 Quick Tips#
- Volume: Always check if you need $\frac{1}{3}$ factor
- Surface Area: Count all faces systematically
- Cross-Sections: Consider the direction of the cut
- Decomposition: Draw a sketch to visualize parts
- Units: Keep track of units throughout calculation