Overview

3D geometry on AMC/AIME often reduces to pyramid volume or distance formulas.

Key Ideas

• Pyramid volume: $V = \frac{1}{3}Bh$.
• Tetrahedron volume uses the same formula with a triangular base.

Core Skills

Identify the Base and Height

The height is perpendicular to the base. Slant heights do not work in volume formulas.

Decompose into Pyramids

Split complex solids into pyramids or prisms with known volumes.

Use Coordinate Distances

If coordinates are given, use 3D distance and dot products to find heights.

Worked Example

A pyramid has base area $36$ and height $10$. Find its volume.

$V = \frac{1}{3} \cdot 36 \cdot 10 = 120$.

More Examples

Example 1: Rectangular Prism Diagonal

A box has side lengths $3,4,12$. Find the space diagonal.

$\sqrt{3^2+4^2+12^2}=13$.

Example 2: Tetrahedron Volume

If a tetrahedron has base area $12$ and height $9$, find its volume.

$V=\frac{1}{3}\cdot 12\cdot 9=36$.

Example 3: Stack Volumes

A pyramid is cut by a plane parallel to its base creating a smaller similar pyramid with linear scale factor $1/2$. What fraction of the volume is removed?

Volume scales by $(1/2)^3=1/8$, so $1/8$ is removed.

Strategy Checklist

• Identify the perpendicular height.
• Decompose into known solids.
• Use similarity for volume scaling.

Common Pitfalls

• Forgetting the factor $1/3$.
• Mixing slant height with perpendicular height.
• Scaling volumes linearly instead of cubically.

Practice Problems