📏 Essential Formulas

Master these essential formulas for AMC geometry success. Each formula includes usage notes and micro-examples for quick reference.

🎯 How to Use This Reference

For Practice

1. Study each formula - Understand what it calculates
2. Practice applications - Work through the micro-examples
3. Build recognition - Learn when to use each formula

For Contests

1. Quick lookup - Find the right formula fast
2. Apply correctly - Use the formula with proper values
3. Verify results - Check that your answer makes sense

For Review

1. Identify gaps - Which formulas need more practice
2. Focus study - Work on weak formula applications
3. Build speed - Practice quick formula recall

📚 Formula Categories

🔺 Triangle Formulas

• Area: $A = \frac{1}{2}bh$, $A = \frac{1}{2}ab\sin C$, Heron’s formula
• Lengths: Medians, altitudes, angle bisectors
• Centers: Centroid, incenter, circumcenter, orthocenter

⭕ Circle Formulas

• Area: $A = \pi r^2$
• Circumference: $C = 2\pi r$
• Power of a Point: $PA \cdot PB = PC \cdot PD$
• Chord length: $2\sqrt{r^2 - d^2}$

📐 Coordinate Formulas

• Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
• Shoelace: $A = \frac{1}{2}|\sum_{i=1}^n x_iy_{i+1} - \sum_{i=1}^n y_ix_{i+1}|$

🔄 Transformation Formulas

• Reflection: Across $y = x$: $(x,y) \rightarrow (y,x)$
• Rotation: By $90°$: $(x,y) \rightarrow (-y,x)$
• Translation: By $(a,b)$: $(x,y) \rightarrow (x+a,y+b)$

📦 3D Formulas

• Volume: Cube $s^3$, Cylinder $\pi r^2 h$, Sphere $\frac{4}{3}\pi r^3$
• Surface Area: Cube $6s^2$, Cylinder $2\pi r^2 + 2\pi rh$, Sphere $4\pi r^2$

💡 Pro Tips

• Memorize the basics - Area, perimeter, distance formulas
• Practice applications - Work through examples for each formula
• Build recognition - Learn when to use each formula
• Check units - Make sure units match in your calculations

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