Area of a Circle:

\[ A = \pi r^2 \]

where \(r\) = radius

Circumference of a Circle:

\[ C = 2\pi r \]

OR

\[ C = \pi d \]

where \(d\) = diameter = \(2r\)

Area of a Rectangle:

\[ A = lw \]

where \(l\) = length, \(w\) = width

Area of a Triangle:

\[ A = \frac{1}{2}bh \]

where \(b\) = base, \(h\) = height

Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

where \(a\) and \(b\) are legs, \(c\) = hypotenuse (longest side)

45-45-90 Triangle (Isosceles Right Triangle):

Side lengths: \(x\), \(x\), \(x\sqrt{2}\)

Example: \(5\), \(5\), \(5\sqrt{2}\)

30-60-90 Triangle:

Side lengths: \(x\), \(x\sqrt{3}\), \(2x\)

Opposite 30°: \(x\) (shortest)

Opposite 60°: \(x\sqrt{3}\) (middle)

Opposite 90°: \(2x\) (hypotenuse)

Example: \(3\), \(3\sqrt{3}\), \(6\)

Volume of a Rectangular Solid (Box):

\[ V = lwh \]

Volume of a Cylinder:

\[ V = \pi r^2 h \]

Volume of a Sphere:

\[ V = \frac{4}{3}\pi r^3 \]

Volume of a Cone:

\[ V = \frac{1}{3}\pi r^2 h \]

Volume of a Pyramid:

\[ V = \frac{1}{3}lwh \]

• A circle has 360°

• A circle has \(2\pi\) radians

• A triangle has 180°

Slope Formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}} \]

Given two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\)

Slope-Intercept Form:

\[ y = mx + b \]

where \(m\) = slope, \(b\) = y-intercept

⚡ Always rewrite equations into this form to identify slope and intercept!

Point-Slope Form:

\[ y - y_1 = m(x - x_1) \]

Use when you have a point \((x_1, y_1)\) and slope \(m\)

Standard Form:

\[ Ax + By = C \]

Special Slope Cases:

• Horizontal line: slope = 0 (equation: \(y = b\))

• Vertical line: slope = undefined (equation: \(x = a\))

• Parallel lines: equal slopes (\(m_1 = m_2\))

• Perpendicular lines: negative reciprocal slopes (\(m_1 \cdot m_2 = -1\))

Midpoint Formula:

\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Distance Formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

💡 Alternative: Graph points and use Pythagorean Theorem!

Method 1: Substitution

1. Solve one equation for one variable

2. Substitute into the other equation

3. Solve for the remaining variable

4. Back-substitute to find the other variable

Method 2: Elimination (Addition/Subtraction)

1. Multiply equations to align coefficients

2. Add or subtract equations to eliminate a variable

3. Solve for the remaining variable

4. Substitute back to find the other variable

⚡ Fastest method when possible!

Method 3: Graphing

Convert both equations to slope-intercept form and graph. The intersection point is the solution.

🎯 SAT Shortcut:

If asked to find \(x + y\) or \(2x - 3y\) (not individual variables), add/subtract equations directly without solving for each variable!

Solving Inequalities:

Solve like regular equations, BUT:

⚠️ FLIP the inequality sign when multiplying or dividing by a negative number!

Example:

\(-2x > 6\)

Divide by -2 and flip sign:

\(x < -3\)

Compound Inequalities:

\(a < x < b\) means \(x\) is between \(a\) and \(b\)

\(x < a\) OR \(x > b\) means \(x\) is outside the range

🔥 Critical: Rules ONLY work with the SAME BASE!

1. Multiplication Rule (add exponents):

\[ a^b \cdot a^c = a^{b+c} \]

Example: \(x^3 \cdot x^5 = x^8\)

2. Division Rule (subtract exponents):

\[ \frac{a^b}{a^c} = a^{b-c} \]

Example: \(\frac{x^7}{x^2} = x^5\)

3. Power Rule (multiply exponents):

\[ (a^b)^c = a^{bc} \]

Example: \((x^3)^4 = x^{12}\)

4. Zero Exponent:

\[ a^0 = 1 \]

(for any \(a \neq 0\))

5. Negative Exponent:

\[ a^{-b} = \frac{1}{a^b} \]

Example: \(x^{-3} = \frac{1}{x^3}\)

6. Fractional Exponent:

\[ a^{\frac{b}{c}} = \sqrt[c]{a^b} = (\sqrt[c]{a})^b \]

Example: \(x^{\frac{1}{2}} = \sqrt{x}\), \(x^{\frac{2}{3}} = \sqrt[3]{x^2}\)

7. Power of a Product:

\[ (ab)^c = a^c b^c \]

8. Power of a Quotient:

\[ \left(\frac{a}{b}\right)^c = \frac{a^c}{b^c} \]

Standard Form:

\[ f(x) = ax^2 + bx + c \]

• y-intercept = \(c\)

• Vertex x-coordinate = \(-\frac{b}{2a}\)

• Opens up if \(a > 0\), down if \(a < 0\)

Quadratic Formula (MUST MEMORIZE):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

🎵 Memory trick: Sing to "Pop Goes the Weasel" or "Row Your Boat"

Gives you BOTH solutions (use + and - separately)

Vertex Form:

\[ f(x) = a(x - h)^2 + k \]

• Vertex = \((h, k)\)

Factored Form:

\[ f(x) = a(x - m)(x - n) \]

• x-intercepts (roots) = \(m\) and \(n\)

• Vertex x-coordinate = \(\frac{m + n}{2}\)

The Discriminant:

\[ b^2 - 4ac \]

• If positive: 2 real solutions

• If zero: 1 real solution (repeated root)

• If negative: 0 real solutions (2 imaginary)

Sum and Product of Solutions:

For \(ax^2 + bx + c = 0\) with solutions \(r\) and \(s\):

Sum: \(r + s = -\frac{b}{a}\)

Product: \(r \cdot s = \frac{c}{a}\)

Difference of Squares:

\[ a^2 - b^2 = (a + b)(a - b) \]

Example: \(x^2 - 9 = (x + 3)(x - 3)\)

Perfect Square Trinomial:

\[ a^2 + 2ab + b^2 = (a + b)^2 \]

\[ a^2 - 2ab + b^2 = (a - b)^2 \]

FOIL Method (Multiplying Binomials):

\[ (a + b)(c + d) = ac + ad + bc + bd \]

First, Outer, Inner, Last

Squaring a Binomial:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

Exponential Growth/Decay:

\[ y = a(1 + r)^t \]

• \(a\) = initial amount

• \(r\) = rate (as decimal)

• \(t\) = time

Growth: use \((1 + r)\)

Decay: use \((1 - r)\)

Compound Interest:

\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

• \(P\) = principal (initial investment)

• \(r\) = annual interest rate (decimal)

• \(n\) = number of times compounded per year

• \(t\) = time in years

Absolute Value:

\(|x|\) = distance from zero (always non-negative)

\(|x| = a\) means \(x = a\) or \(x = -a\)

Standard Equation of a Circle:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

• Center = \((h, k)\)

• Radius = \(r\)

Example: \((x - 3)^2 + (y + 2)^2 = 25\)

Center: \((3, -2)\), Radius: \(5\)

Arc Length:

\[ L = 2\pi r \left(\frac{\theta}{360}\right) \]

where \(\theta\) = central angle in degrees

Area of a Sector:

\[ A = \pi r^2 \left(\frac{\theta}{360}\right) \]

Part-Whole Relationship:

\[ \frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100} \]

Find x% of n:

\[ n \times \frac{x}{100} \]

Example: 25% of 80 = \(80 \times 0.25 = 20\)

Percent Change:

\[ \text{Percent Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\% \]

Also written as: \(\frac{\text{Difference}}{\text{Original}} \times 100\%\)

Increase by r%:

Multiply by \((1 + r)\) where r is in decimal form

Example: Increase 100 by 20% → \(100 \times 1.20 = 120\)

Decrease by r%:

Multiply by \((1 - r)\) where r is in decimal form

Example: Decrease 100 by 20% → \(100 \times 0.80 = 80\)

Direct Proportion:

\[ \frac{a}{b} = \frac{c}{d} \]

Cross-multiply: \(ad = bc\)

Rate Formula:

\[ \text{Distance} = \text{Rate} \times \text{Time} \]

Also: \(d = rt\)

Mean (Average):

\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

Also: Sum = Mean × Count

Median:

The middle value when data is arranged in order

If ODD number of values:

Median = middle number

If EVEN number of values:

Median = average of two middle numbers

Mode:

The value that appears most frequently

If no repeats → No mode

Range:

\[ \text{Range} = \text{Maximum} - \text{Minimum} \]

📌 Key Insight:

Mean > Median → Data skewed right (high outliers)

Mean < Median → Data skewed left (low outliers)

Basic Probability Formula:

\[ P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Important Facts:

• Probability ranges from 0 to 1 (or 0% to 100%)

• Probability = 0 → Event will never happen

• Probability = 1 → Event is guaranteed

• Probability of complement: \(P(\text{not A}) = 1 - P(A)\)

Independent Events (AND):

Multiply probabilities

\(P(A \text{ and } B) = P(A) \times P(B)\)

Sum of Angles:

All angles in a triangle add to \(180°\)

\[ A + B + C = 180° \]

Pythagorean Triples (Memorize!):

Common right triangle side ratios:

• 3-4-5 (and multiples: 6-8-10, 9-12-15, etc.)

• 5-12-13 (and multiples)

• 8-15-17

• 7-24-25

Exterior Angle Theorem:

An exterior angle equals the sum of the two remote interior angles

Area of Equilateral Triangle:

\[ A = \frac{s^2\sqrt{3}}{4} \]

where \(s\) = side length

Triangle Similarity:

Triangles are similar if:

• All corresponding angles are equal (AA)

• All corresponding sides are proportional (SSS)

• Two sides proportional and included angle equal (SAS)

SOHCAHTOA

Memory trick for trig ratios in RIGHT TRIANGLES

Sine (SOH):

\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]

Cosine (CAH):

\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

Tangent (TOA):

\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]

Complementary Angle Identity:

\[ \sin(\theta) = \cos(90° - \theta) \]

The sine of an angle equals the cosine of its complement

Radian Conversion:

\[ 360° = 2\pi \text{ radians} \]

\(180° = \pi\) radians

\(90° = \frac{\pi}{2}\) radians

Vertical Angles:

Opposite angles formed by intersecting lines are equal

Linear Pair (Supplementary):

Adjacent angles on a straight line add to \(180°\)

Parallel Lines Cut by Transversal:

• Corresponding angles are equal

• Alternate interior angles are equal

• Alternate exterior angles are equal

Sum of Interior Angles of a Polygon:

\[ S = 180°(n - 2) \]

where \(n\) = number of sides

One Interior Angle of Regular Polygon:

\[ \text{Angle} = \frac{180°(n - 2)}{n} \]

Sum of Exterior Angles:

Always \(360°\) for any polygon

Area of Trapezoid:

\[ A = \frac{1}{2}(b_1 + b_2)h \]

where \(b_1\) and \(b_2\) are the parallel bases

Radius ⊥ Tangent:

A radius and tangent line form a \(90°\) angle at the point of tangency

Central vs. Inscribed Angle:

A central angle is twice the inscribed angle that subtends the same arc

Central angle = \(2 \times\) Inscribed angle

✓ Algebra Essentials

Slopes, systems, inequalities, linear equations

✓ Exponents & Radicals

8 rules, fractional exponents, simplification

✓ Quadratics

Factoring, quadratic formula, vertex form

✓ Data Analysis

Mean, median, mode, probability, percentages

✓ Geometry

Triangles, circles, angles, area, volume

✓ Trigonometry

SOHCAHTOA, special triangles, radians