🧊 Solid Geometry Reference
🎯 Key Concepts
Volume
The amount of space a 3D object occupies, measured in cubic units.
Surface Area
The total area of all faces of a 3D object, measured in square units.
Cross-Section
A 2D slice of a 3D object, often used to find volumes.
Net
A 2D representation that can be folded to form a 3D object.
📐 Basic Solids
Cube
- Volume: $V = s^3$ where $s$ is side length
- Surface Area: $SA = 6s^2$
- Faces: 6 square faces
- Edges: 12 edges of length $s$
Rectangular Prism
- Volume: $V = lwh$ where $l$, $w$, $h$ are length, width, height
- Surface Area: $SA = 2(lw + lh + wh)$
- Faces: 6 rectangular faces
- Edges: 12 edges
Cylinder
- Volume: $V = \pi r^2 h$ where $r$ is radius, $h$ is height
- Surface Area: $SA = 2\pi r^2 + 2\pi rh$
- Bases: 2 circular bases
- Lateral Surface: Curved surface area = $2\pi rh$
Sphere
- Volume: $V = \frac{4}{3}\pi r^3$ where $r$ is radius
- Surface Area: $SA = 4\pi r^2$
- No edges or vertices
- Perfectly round
Cone
- Volume: $V = \frac{1}{3}\pi r^2 h$ where $r$ is radius, $h$ is height
- Surface Area: $SA = \pi r^2 + \pi rl$ where $l$ is slant height
- Base: 1 circular base
- Lateral Surface: Curved surface area = $\pi rl$
Pyramid
- Volume: $V = \frac{1}{3}Bh$ where $B$ is base area, $h$ is height
- Surface Area: Base area + lateral faces
- Base: 1 polygonal base
- Lateral Faces: Triangular faces meeting at apex
🔍 Spatial Reasoning
Visualizing 3D Objects
- Isometric Drawings: Show 3D objects in 2D
- Orthographic Views: Front, top, side views
- Cross-Sections: 2D slices of 3D objects
Common Cross-Sections
- Cylinder: Circle (perpendicular to axis), rectangle (parallel to axis)
- Cone: Circle (perpendicular to axis), triangle (through apex)
- Sphere: Circle (any direction)
- Cube: Square, rectangle, triangle, hexagon
💡 Problem-Solving Strategies
- Identify the solid and its dimensions
- Choose the right formula for volume or surface area
- Substitute carefully and watch units
- Use decomposition for complex shapes
- Check your answer with estimation
⚠️ Common Mistakes
- Mixing up formulas for different solids
- Forgetting units in calculations
- Confusing radius and diameter in circular solids
- Missing faces in surface area calculations
- Wrong cross-section identification