📏 Essential Formulas Bank
Quick Reference Guide
Complete reference for all essential AMC geometry formulas with usage notes and micro-examples. Master these formulas for contest success!
🗂️ Table of Contents
- 🔺 Triangle Formulas
- ⭕ Circle Formulas
- 📐 Coordinate Formulas
- 🔄 Transformation Formulas
- 📦 3D Formulas
- 🎯 Special Formulas
- 💡 Quick Reference Tips
🔺 Triangle Formulas
🎯 Contest Strategy
Triangle problems are extremely common in AMC contests. Master these area and length formulas!
Area Formulas
⚠️ Critical for AMC
These area formulas appear in 90% of AMC geometry problems!
| Formula | Usage | Example | When to Use |
|---|---|---|---|
| $A = \frac{1}{2}bh$ | Basic triangle area | Triangle with base 6, height 4: $A = \frac{1}{2} \cdot 6 \cdot 4 = 12$ | When you have base and height |
| $A = \frac{1}{2}ab\sin C$ | Triangle with two sides and included angle | Sides 5, 7, angle $60°$: $A = \frac{1}{2} \cdot 5 \cdot 7 \cdot \sin 60° = \frac{35\sqrt{3}}{4}$ | When you have two sides and included angle |
| $A = \sqrt{s(s-a)(s-b)(s-c)}$ | Heron’s formula (three sides) | Sides 3, 4, 5: $s = 6$, $A = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = 6$ | When you have all three sides |
Length Formulas
Key Formula: $$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| Median length | $m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$ | Sides 3, 4, 5: $m = \frac{1}{2}\sqrt{2 \cdot 16 + 2 \cdot 9 - 25} = \frac{5}{2}$ | Connects vertex to midpoint of opposite side |
| Altitude length | $h_a = \frac{2A}{a}$ | Area 6, base 3: $h = \frac{2 \cdot 6}{3} = 4$ | Perpendicular from vertex to opposite side |
| Angle bisector | $t_a = \frac{2bc}{b+c}\cos\frac{A}{2}$ | Sides 3, 4, angle $60°$: $t = \frac{2 \cdot 3 \cdot 4}{7} \cdot \cos 30° = \frac{24\sqrt{3}}{14}$ | Divides angle into two equal parts |
Center Formulas
📐 Triangle Centers
These centers are crucial for coordinate geometry and advanced triangle problems!
| Formula | Usage | Example | Center Type |
|---|---|---|---|
| Centroid | $G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$ | Vertices $(0,0)$, $(4,0)$, $(2,3)$: $G = (2,1)$ | Center of mass |
| Inradius | $r = \frac{A}{s}$ | Area 6, semi-perimeter 6: $r = 1$ | Radius of inscribed circle |
| Circumradius | $R = \frac{abc}{4A}$ | Sides 3, 4, 5, area 6: $R = \frac{3 \cdot 4 \cdot 5}{4 \cdot 6} = \frac{5}{2}$ | Radius of circumscribed circle |
⭕ Circle Formulas
🎯 Circle Mastery
Circle problems are high-frequency in AMC contests. Master both basic properties and power theorems!
Basic Properties
Core Circle Formulas:
- Area: $A = \pi r^2$
- Circumference: $C = 2\pi r$
- Diameter: $d = 2r$
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| $A = \pi r^2$ | Circle area | Radius 5: $A = 25\pi$ | Most common area formula |
| $C = 2\pi r$ | Circumference | Radius 5: $C = 10\pi$ | Perimeter of circle |
| $d = 2r$ | Diameter | Radius 5: $d = 10$ | Longest chord through center |
Power of a Point
⚡ Power Theorems
These power theorems are essential for advanced circle problems and often appear in AMC 12!
| Formula | Usage | Example | When to Use |
|---|---|---|---|
| $PA^2 = PB \cdot PC$ | Tangent-secant | Tangent 6, secant segment 4: $6^2 = 4 \cdot PC$, so $PC = 9$ | When you have tangent and secant |
| $PA \cdot PB = PC \cdot PD$ | Two secants/chords | Secant segments 3, 4 and 2, 6: $3 \cdot 4 = 2 \cdot 6 = 12$ | When you have two intersecting chords/secants |
Chord and Arc
Key Formulas:
- Chord length: $L = 2\sqrt{r^2 - d^2}$
- Arc length: $s = r\theta$
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| $L = 2\sqrt{r^2 - d^2}$ | Chord length | Radius 5, distance 3: $L = 2\sqrt{25-9} = 8$ | Distance from center to chord |
| $s = r\theta$ | Arc length | Radius 5, angle $\frac{\pi}{3}$: $s = \frac{5\pi}{3}$ | Angle must be in radians |
📐 Coordinate Formulas
📊 Coordinate Mastery
Coordinate geometry problems are very common in AMC contests. Master these fundamental formulas!
Basic Operations
Essential Coordinate Formulas:
- Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
- Slope: $m = \frac{y_2-y_1}{x_2-x_1}$
- Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ | Distance between points | $(0,0)$ to $(3,4)$: $d = \sqrt{9+16} = 5$ | Pythagorean theorem in 2D |
| $m = \frac{y_2-y_1}{x_2-x_1}$ | Slope | $(1,2)$ to $(3,6)$: $m = \frac{6-2}{3-1} = 2$ | Rise over run |
| Midpoint | $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ | $(0,0)$ to $(4,6)$: $(2,3)$ | Average of coordinates |
Area and Lines
📏 Line and Circle Equations
These equations are fundamental for coordinate geometry problems!
| Formula | Usage | Example | When to Use |
|---|---|---|---|
| $y = mx + b$ | Line equation | Slope 2, y-intercept 3: $y = 2x + 3$ | When you have slope and y-intercept |
| $(x-h)^2 + (y-k)^2 = r^2$ | Circle equation | Center $(2,3)$, radius 5: $(x-2)^2 + (y-3)^2 = 25$ | When you have center and radius |
🔄 Transformation Formulas
Basic Transformations
| Formula | Usage | Example |
|---|---|---|
| $(x,y) \rightarrow (y,x)$ | Reflection across $y = x$ | $(3,4) \rightarrow (4,3)$ |
| $(x,y) \rightarrow (-y,x)$ | Rotation by $90°$ | $(3,4) \rightarrow (-4,3)$ |
| $(x,y) \rightarrow (x+a,y+b)$ | Translation by $(a,b)$ | $(3,4)$ by $(1,2)$: $(4,6)$ |
Scaling
| Formula | Usage | Example |
|---|---|---|
| $(x,y) \rightarrow (kx,ky)$ | Homothety by factor $k$ | $(3,4)$ by factor 2: $(6,8)$ |
📦 3D Formulas
Volume
| Formula | Usage | Example |
|---|---|---|
| $V = s^3$ | Cube volume | Side 4: $V = 64$ |
| $V = lwh$ | Rectangular prism | Length 3, width 4, height 5: $V = 60$ |
| $V = \pi r^2 h$ | Cylinder volume | Radius 3, height 5: $V = 45\pi$ |
| $V = \frac{1}{3}\pi r^2 h$ | Cone volume | Radius 3, height 5: $V = 15\pi$ |
| $V = \frac{4}{3}\pi r^3$ | Sphere volume | Radius 3: $V = 36\pi$ |
Surface Area
| Formula | Usage | Example |
|---|---|---|
| $A = 6s^2$ | Cube surface area | Side 4: $A = 96$ |
| $A = 2(lw + lh + wh)$ | Rectangular prism | Length 3, width 4, height 5: $A = 94$ |
| $A = 2\pi r^2 + 2\pi rh$ | Cylinder surface area | Radius 3, height 5: $A = 48\pi$ |
| $A = 4\pi r^2$ | Sphere surface area | Radius 3: $A = 36\pi$ |
🎯 Special Formulas
🎯 Advanced Theorems
These advanced theorems are crucial for AMC 12 and challenging AMC 10 problems!
Ceva and Menelaus
Ceva’s Theorem: $$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$$
Menelaus’s Theorem: $$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$$
| Theorem | Usage | Example | When to Use |
|---|---|---|---|
| Ceva’s | Check concurrency of cevians | Three cevians meet at a point | Proving concurrency |
| Menelaus’s | Check collinearity of points | Three points lie on a line | Proving collinearity |
Ptolemy’s Theorem
📐 Cyclic Quadrilaterals
Ptolemy’s theorem is essential for problems involving cyclic quadrilaterals!
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| $AC \cdot BD = AB \cdot CD + BC \cdot AD$ | Cyclic quadrilateral | Sides 3, 4, 5, 6: $AC \cdot BD = 3 \cdot 5 + 4 \cdot 6 = 39$ | Product of diagonals = sum of products of opposite sides |
Angle Bisector Theorem
Angle Bisector Theorem: $$\frac{BD}{DC} = \frac{AB}{AC}$$
| Formula | Usage | Example | Key Insight |
|---|---|---|---|
| $\frac{BD}{DC} = \frac{AB}{AC}$ | Angle bisector | Sides 6, 8, bisector divides opposite side: $\frac{BD}{DC} = \frac{6}{8} = \frac{3}{4}$ | Bisector divides opposite side proportionally |
Law of Sines and Cosines
🔺 Triangle Laws
These laws are essential for solving triangles with given information!
| Formula | Usage | Example | When to Use |
|---|---|---|---|
| $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ | Law of sines | Triangle with angles 30°, 60°, 90° and side opposite 30° = 3: $a = 3$, $b = 3\sqrt{3}$, $c = 6$ | When you have angles and sides |
| $a^2 = b^2 + c^2 - 2bc\cos A$ | Law of cosines | Sides 3, 4, angle 60°: $a^2 = 9 + 16 - 24 \cdot \frac{1}{2} = 13$, so $a = \sqrt{13}$ | When you have two sides and included angle |
Shoelace Area Formula
Shoelace Formula: $$A = \frac{1}{2}\left|\sum_{i=1}^n x_iy_{i+1} - \sum_{i=1}^n y_ix_{i+1}\right|$$
Usage: Find area of any polygon given coordinates of vertices
Example: Triangle with vertices $(0,0)$, $(3,0)$, $(0,4)$: $A = \frac{1}{2}|0 \cdot 0 + 3 \cdot 4 + 0 \cdot 0 - (0 \cdot 0 + 0 \cdot 0 + 4 \cdot 3)| = 6$
Common Mistakes
- ⚠️ Wrong units — Make sure units match (degrees vs radians, etc.)
- 📐 Wrong order — Check formula order carefully, especially in coordinate geometry
- 🔢 Missing factors — Don’t forget $\frac{1}{2}$ or other factors in area formulas
- 📏 Sign errors — Be careful with signs in coordinate geometry and transformations
🎉 You're Ready!
You now have a comprehensive geometry formula reference! Practice regularly and use this as your go-to resource during contests.
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