Preparing for the SAT Math section? You’ve come to the right place! This comprehensive formula sheet contains every formula you need to know for SAT Math, organized by topic for easy reference.
Whether you’re tackling algebra, geometry, trigonometry, or statistics, this guide has you covered. We’ve included not just the formulas, but also clear descriptions and tips for when to use them.
Pro Tip: Bookmark this page and review it regularly during your SAT prep. The SAT doesn’t provide all formulas, so memorizing these is crucial for success!
📚 Table of Contents – All 28 Topics
- 1. Algebra I – Linear Equations & Inequalities
- 2. Algebra II – Quadratics & Polynomials
- 3. Exponentials & Logarithms
- 4. Systems of Equations
- 5. Functions
- 6. Polynomials & Rational Expressions
- 7. Radicals & Rational Exponents
- 8. Geometry – Lines & Angles
- 9. Geometry – Triangles
- 10. Geometry – Circles
- 11. Geometry – Quadrilaterals
- 12. Solid Geometry (3D Shapes)
- 13. Trigonometry – Right Triangles
- 14. Trigonometry – Identities
- 15. Special Angles & Unit Circle
- 16. Coordinate Geometry
- 17. Statistics – Central Tendency
- 18. Statistics – Measures of Spread
- 19. Probability
- 20. Sequences & Series
- 21. Complex Numbers (Advanced)
- 22. Matrices (Advanced)
- 23. Ratio, Proportion & Variation
- 24. Percent, Percent Change & Applications
- 25. Rate, Work & Mixture Problems
- 26. Important Constants & Values
- 27. Useful Calculation Shortcuts
- 28. Tips for SAT Math Success
💾 Download This Formula Sheet
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1. Algebra I – Linear Equations & Inequalities
Linear equations form the foundation of algebra. These formulas help you work with straight lines, slopes, and systems of equations. Master these first!
| Topic | Formula | Description |
|---|---|---|
| Slope-Intercept Form | y = mx + b | m = slope, b = y-intercept |
| Point-Slope Form | y – y₁ = m(x – x₁) | Line through point (x₁, y₁) with slope m |
| Standard Form | Ax + By = C | A, B, C are constants |
| Slope Formula | m = (y₂ – y₁)/(x₂ – x₁) | Slope between two points (rise over run) |
| Parallel Lines | m₁ = m₂ | Same slope means parallel |
| Perpendicular Lines | m₁ × m₂ = -1 | Negative reciprocal slopes |
| Distance Between Points | d = √[(x₂ – x₁)² + (y₂ – y₁)²] | Pythagorean theorem in coordinate plane |
| Midpoint Formula | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) | Point halfway between two points |
2. Algebra II – Quadratics & Polynomials
Quadratic equations are tested heavily on the SAT. You’ll need to factor, complete the square, and use the quadratic formula with confidence.
| Topic | Formula | Description |
|---|---|---|
| Standard Form | y = ax² + bx + c | a ≠ 0, opens up if a > 0, down if a < 0 |
| Vertex Form | y = a(x – h)² + k | Vertex at (h, k) |
| Factored Form | y = a(x – r₁)(x – r₂) | r₁, r₂ are x-intercepts (roots) |
| Quadratic Formula | x = [-b ± √(b² – 4ac)]/(2a) | Solutions to ax² + bx + c = 0 |
| Discriminant | Δ = b² – 4ac | Δ > 0: two real roots Δ = 0: one real root Δ < 0: no real roots |
| Vertex (x-coordinate) | x = -b/(2a) | x-coordinate of vertex/axis of symmetry |
| Vertex (y-coordinate) | y = c – b²/(4a) | y-coordinate of vertex |
| Axis of Symmetry | x = -b/(2a) | Vertical line through vertex |
| Sum of Roots | r₁ + r₂ = -b/a | Vieta’s formula |
| Product of Roots | r₁ × r₂ = c/a | Vieta’s formula |
| Completing the Square | x² + bx + (b/2)² = (x + b/2)² | Add (b/2)² to both sides |
| Difference of Squares | a² – b² = (a + b)(a – b) | Important factoring pattern |
| Perfect Square Trinomial | a² ± 2ab + b² = (a ± b)² | Factoring pattern |
3. Exponentials & Logarithms
Exponential functions model growth and decay in real-world scenarios like compound interest, population growth, and radioactive decay.
| Topic | Formula | Description |
|---|---|---|
| Exponential Growth | y = a(1 + r)ᵗ or y = a·bᵗ | a = initial, r = rate, t = time |
| Exponential Decay | y = a(1 – r)ᵗ or y = a·bᵗ | 0 < b < 1 for decay |
| Compound Interest | A = P(1 + r/n)^(nt) | P = principal, r = rate, n = times/year, t = years |
| Continuous Compound | A = Pe^(rt) | e ≈ 2.718 |
| Logarithm Definition | log_a(x) = b ↔ a^b = x | a is base |
| Natural Log | ln(x) = log_e(x) | Base e |
| Common Log | log(x) = log₁₀(x) | Base 10 |
| Product Rule | log_a(xy) = log_a(x) + log_a(y) | Log of product |
| Quotient Rule | log_a(x/y) = log_a(x) – log_a(y) | Log of quotient |
| Power Rule | log_a(x^n) = n·log_a(x) | Log of power |
| Change of Base | log_a(x) = log(x)/log(a) = ln(x)/ln(a) | Convert to any base |
4. Systems of Equations
Systems of equations appear frequently on the SAT, often in word problems. You’ll solve them using substitution or elimination.
| Topic | Formula | Description |
|---|---|---|
| Linear System (2×2) | ax + by = e cx + dy = f | Solve by substitution or elimination |
| Consistent System | One or infinite solutions | Lines intersect or coincide |
| Inconsistent System | No solution | Parallel lines |
| Cramer’s Rule (x) | x = (ed – bf)/(ad – bc) | For 2×2 systems |
| Cramer’s Rule (y) | y = (af – ec)/(ad – bc) | For 2×2 systems |
5. Functions
Understanding function notation and properties is crucial for the SAT. You’ll work with domain, range, and function transformations.
| Topic | Formula | Description |
|---|---|---|
| Function Notation | f(x) = … | Output for input x |
| Domain | All possible x-values | Input values |
| Range | All possible y-values | Output values |
| Composite Functions | (f ∘ g)(x) = f(g(x)) | Function of a function |
| Inverse Function | f⁻¹(f(x)) = x | Reflects over y = x |
| Even Function | f(-x) = f(x) | Symmetric about y-axis |
| Odd Function | f(-x) = -f(x) | Symmetric about origin |
| Absolute Value | |x| = x if x ≥ 0 |x| = -x if x < 0 | Distance from zero |
| Piecewise Function | f(x) = {g(x) if condition 1 h(x) if condition 2} | Different rules for different domains |
6. Polynomials & Rational Expressions
| Topic | Formula | Description |
|---|---|---|
| FOIL Method | (a + b)(c + d) = ac + ad + bc + bd | First, Outer, Inner, Last |
| Difference of Cubes | a³ – b³ = (a – b)(a² + ab + b²) | Factoring pattern |
| Sum of Cubes | a³ + b³ = (a + b)(a² – ab + b²) | Factoring pattern |
| Rational Expression | P(x)/Q(x), Q(x) ≠ 0 | Polynomial over polynomial |
| Vertical Asymptote | x = a where Q(a) = 0 | Denominator = 0 |
| Horizontal Asymptote | y = lim(x→∞) f(x) | Behavior as x → ∞ |
7. Radicals & Rational Exponents
Exponent rules are essential for simplifying expressions. These rules apply to both whole number and fractional exponents.
| Topic | Formula | Description |
|---|---|---|
| Square Root | √(x²) = |x| | Principal square root |
| Rational Exponent | x^(m/n) = ⁿ√(x^m) = (ⁿ√x)^m | Fractional exponent |
| Product Rule | x^a · x^b = x^(a+b) | Same base |
| Quotient Rule | x^a / x^b = x^(a-b) | Same base |
| Power Rule | (x^a)^b = x^(ab) | Power of a power |
| Negative Exponent | x^(-n) = 1/x^n | Reciprocal |
| Zero Exponent | x^0 = 1 | x ≠ 0 |
| Distributive Power | (xy)^n = x^n·y^n | Power of product |
| Quotient Power | (x/y)^n = x^n/y^n | Power of quotient |
| Simplifying Radicals | √(ab) = √a · √b | Product property |
| Rationalizing | 1/√a = √a/a | Multiply by √a/√a |
8. Geometry – Lines & Angles
Angle relationships are fundamental to geometry problems on the SAT. Know these cold!
| Topic | Formula | Description |
|---|---|---|
| Supplementary Angles | α + β = 180° | Angles on a straight line |
| Complementary Angles | α + β = 90° | Angles in right angle |
| Vertical Angles | α = β | Opposite angles are equal |
| Sum of Triangle Angles | α + β + γ = 180° | Interior angles of triangle |
| Exterior Angle | Exterior = sum of remote interior angles | Triangle exterior angle theorem |
| Parallel Lines (Corresponding) | Angles are equal | Same position |
| Parallel Lines (Alternate Interior) | Angles are equal | Z-pattern |
| Parallel Lines (Co-interior) | Angles sum to 180° | C-pattern |
9. Geometry – Triangles
Triangles are heavily tested on the SAT. Pay special attention to the special right triangles!
| Topic | Formula | Description |
|---|---|---|
| Area | A = ½bh | b = base, h = height |
| Area (Heron’s Formula) | A = √[s(s-a)(s-b)(s-c)] | s = (a+b+c)/2 (semi-perimeter) |
| Perimeter | P = a + b + c | Sum of all sides |
| Pythagorean Theorem | a² + b² = c² | Right triangle, c = hypotenuse |
| Pythagorean Triples | 3-4-5, 5-12-13, 8-15-17, 7-24-25 | Common right triangles |
| 45-45-90 Triangle | Sides: x, x, x√2 | Isosceles right triangle |
| 30-60-90 Triangle | Sides: x, x√3, 2x | Special right triangle |
| Isosceles Triangle | Two sides equal, two angles equal | Base angles are equal |
| Equilateral Triangle | All sides equal, all angles = 60° | All sides equal |
| Area (Equilateral) | A = (s²√3)/4 | s = side length |
| Similar Triangles | a₁/a₂ = b₁/b₂ = c₁/c₂ | Corresponding sides proportional |
| Triangle Inequality | a + b > c | Sum of two sides > third side |
- 3-4-5 (and multiples: 6-8-10, 9-12-15, etc.)
- 5-12-13 (and multiples: 10-24-26, etc.)
- 8-15-17
- 7-24-25
10. Geometry – Circles
Circle problems test your understanding of circumference, area, arcs, and the circle equation in coordinate geometry.
| Topic | Formula | Description |
|---|---|---|
| Circumference | C = 2πr = πd | r = radius, d = diameter |
| Area | A = πr² | r = radius |
| Arc Length | s = rθ | θ in radians |
| Arc Length (degrees) | s = (θ/360°) × 2πr | θ in degrees |
| Sector Area | A = ½r²θ | θ in radians |
| Sector Area (degrees) | A = (θ/360°) × πr² | θ in degrees |
| Chord Length | c = 2r sin(θ/2) | θ = central angle |
| Circle Equation (Standard) | (x – h)² + (y – k)² = r² | Center (h, k), radius r |
| Circle Equation (General) | x² + y² + Dx + Ey + F = 0 | Expand to find center/radius |
| Inscribed Angle | θ = ½ × central angle | Angle on circumference |
| Tangent Line | Perpendicular to radius at point | Forms 90° with radius |
11. Geometry – Quadrilaterals
| Topic | Formula | Description |
|---|---|---|
| Rectangle Area | A = lw | l = length, w = width |
| Rectangle Perimeter | P = 2l + 2w | Sum of all sides |
| Square Area | A = s² | s = side length |
| Square Perimeter | P = 4s | Four equal sides |
| Square Diagonal | d = s√2 | Diagonal of square |
| Parallelogram Area | A = bh | b = base, h = height |
| Parallelogram (alt) | A = ab sin(θ) | a, b = sides, θ = angle |
| Trapezoid Area | A = ½(b₁ + b₂)h | b₁, b₂ = parallel sides, h = height |
| Rhombus Area | A = ½d₁d₂ | d₁, d₂ = diagonals |
| Kite Area | A = ½d₁d₂ | d₁, d₂ = diagonals |
| Sum of Interior Angles | (n – 2) × 180° | n = number of sides |
12. Solid Geometry (3D Shapes)
Three-dimensional geometry problems require you to calculate volume and surface area of common solids.
| Topic | Formula | Description |
|---|---|---|
| Rectangular Prism Volume | V = lwh | l = length, w = width, h = height |
| Rectangular Prism SA | SA = 2(lw + lh + wh) | Surface area |
| Cube Volume | V = s³ | s = side length |
| Cube Surface Area | SA = 6s² | Six square faces |
| Cylinder Volume | V = πr²h | r = radius, h = height |
| Cylinder Surface Area | SA = 2πr² + 2πrh = 2πr(r + h) | Two circles + lateral area |
| Cone Volume | V = ⅓πr²h | r = radius, h = height |
| Cone Surface Area | SA = πr² + πrl | l = slant height |
| Sphere Volume | V = (4/3)πr³ | r = radius |
| Sphere Surface Area | SA = 4πr² | r = radius |
| Pyramid Volume | V = ⅓Bh | B = base area, h = height |
13. Trigonometry – Right Triangles
Remember: SOH-CAH-TOA! This mnemonic helps you remember the basic trig ratios.
| Topic | Formula | Description |
|---|---|---|
| Sine | sin(θ) = opposite/hypotenuse | SOH |
| Cosine | cos(θ) = adjacent/hypotenuse | CAH |
| Tangent | tan(θ) = opposite/adjacent | TOA |
| Cosecant | csc(θ) = 1/sin(θ) = hypotenuse/opposite | Reciprocal of sine |
| Secant | sec(θ) = 1/cos(θ) = hypotenuse/adjacent | Reciprocal of cosine |
| Cotangent | cot(θ) = 1/tan(θ) = adjacent/opposite | Reciprocal of tangent |
14. Trigonometry – Identities
| Topic | Formula | Description |
|---|---|---|
| Pythagorean Identity | sin²(θ) + cos²(θ) = 1 | Fundamental identity |
| Pythagorean (tan/sec) | 1 + tan²(θ) = sec²(θ) | Divide by cos²(θ) |
| Pythagorean (cot/csc) | 1 + cot²(θ) = csc²(θ) | Divide by sin²(θ) |
| Quotient Identity (tan) | tan(θ) = sin(θ)/cos(θ) | Definition |
| Quotient Identity (cot) | cot(θ) = cos(θ)/sin(θ) | Definition |
| Co-function (sine/cosine) | sin(90° – θ) = cos(θ) | Complementary angles |
| Co-function (tan/cot) | tan(90° – θ) = cot(θ) | Complementary angles |
15. Special Angles & Unit Circle
Memorize these special angle values – they appear constantly on the SAT!
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 = 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
| Topic | Formula | Description |
|---|---|---|
| Radian Conversion | radians = (π/180°) × degrees | Degree to radian |
| Degree Conversion | degrees = (180°/π) × radians | Radian to degree |
| Common Conversions | π rad = 180°, π/2 = 90°, π/3 = 60°, π/4 = 45°, π/6 = 30° | Key angles |
| Arc Length | s = rθ | θ in radians |
| Unit Circle | x² + y² = 1 | Radius = 1 |
| Coordinates | (cos(θ), sin(θ)) | Point on unit circle |
16. Coordinate Geometry
| Topic | Formula | Description |
|---|---|---|
| Distance Formula | d = √[(x₂ – x₁)² + (y₂ – y₁)²] | Distance between points |
| Midpoint Formula | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) | Midpoint between points |
| Section Formula (Internal) | x = (mx₂ + nx₁)/(m + n) y = (my₂ + ny₁)/(m + n) | Divides segment m:n internally |
| Slope Formula | m = (y₂ – y₁)/(x₂ – x₁) | Rise over run |
| Collinear Points | Slope AB = Slope BC | Three points on same line |
| Area of Triangle | A = ½|x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)| | Using coordinates |
17. Statistics – Measures of Central Tendency
| Topic | Formula | Description |
|---|---|---|
| Mean (Average) | x̄ = (Σx)/n | Sum divided by count |
| Weighted Mean | x̄ = (Σwx)/(Σw) | w = weights |
| Median | Middle value when ordered | 50th percentile |
| Mode | Most frequent value | Can have multiple modes |
| Range | Range = Max – Min | Spread of data |
18. Statistics – Measures of Spread
| Topic | Formula | Description |
|---|---|---|
| Variance (Population) | σ² = Σ(x – μ)²/N | N = population size |
| Variance (Sample) | s² = Σ(x – x̄)²/(n – 1) | n = sample size |
| Standard Deviation (Pop) | σ = √[Σ(x – μ)²/N] | Square root of variance |
| Standard Deviation (Sample) | s = √[Σ(x – x̄)²/(n – 1)] | Square root of variance |
| Interquartile Range | IQR = Q₃ – Q₁ | Middle 50% spread |
| Outlier (Lower) | x < Q₁ - 1.5(IQR) | Below lower fence |
| Outlier (Upper) | x > Q₃ + 1.5(IQR) | Above upper fence |
19. Probability
| Topic | Formula | Description |
|---|---|---|
| Basic Probability | P(A) = (favorable outcomes)/(total outcomes) | 0 ≤ P(A) ≤ 1 |
| Complement Rule | P(A’) = 1 – P(A) | Probability of NOT A |
| Addition Rule (Mutually Exclusive) | P(A or B) = P(A) + P(B) | Events cannot occur together |
| Addition Rule (General) | P(A or B) = P(A) + P(B) – P(A and B) | Avoid double counting |
| Multiplication Rule (Independent) | P(A and B) = P(A) × P(B) | Events don’t affect each other |
| Conditional Probability | P(A|B) = P(A and B)/P(B) | Probability of A given B |
| Permutations | ₙPᵣ = n!/(n – r)! | Order matters |
| Combinations | ₙCᵣ = n!/[r!(n – r)!] | Order doesn’t matter |
| Factorial | n! = n × (n-1) × (n-2) × … × 1 | 0! = 1 by definition |
20. Sequences & Series
| Topic | Formula | Description |
|---|---|---|
| Arithmetic Sequence | aₙ = a₁ + (n – 1)d | d = common difference |
| Arithmetic Sum | Sₙ = n(a₁ + aₙ)/2 | Sum of n terms |
| Arithmetic Sum (alt) | Sₙ = n[2a₁ + (n – 1)d]/2 | Alternative formula |
| Geometric Sequence | aₙ = a₁ · r^(n-1) | r = common ratio |
| Geometric Sum (finite) | Sₙ = a₁(1 – rⁿ)/(1 – r) | r ≠ 1 |
| Geometric Sum (infinite) | S = a₁/(1 – r) | |r| < 1 only |
21. Complex Numbers (Advanced)
| Topic | Formula | Description |
|---|---|---|
| Imaginary Unit | i = √(-1), i² = -1 | Definition |
| Complex Number | z = a + bi | a = real part, b = imaginary part |
| Complex Conjugate | z̄ = a – bi | Conjugate of z |
| Addition | (a + bi) + (c + di) = (a + c) + (b + d)i | Add real and imaginary parts |
| Multiplication | (a + bi)(c + di) = (ac – bd) + (ad + bc)i | FOIL and use i² = -1 |
| Modulus | |z| = √(a² + b²) | Distance from origin |
22. Matrices (Advanced)
| Topic | Formula | Description |
|---|---|---|
| Matrix Addition | [aᵢⱼ] + [bᵢⱼ] = [aᵢⱼ + bᵢⱼ] | Add corresponding elements |
| Scalar Multiplication | k[aᵢⱼ] = [k·aᵢⱼ] | Multiply each element by k |
| Matrix Multiplication | (AB)ᵢⱼ = Σ(aᵢₖ·bₖⱼ) | Row × column |
| Identity Matrix | I = [1 0; 0 1] for 2×2 | AI = IA = A |
| Determinant (2×2) | det(A) = ad – bc | For A = [a b; c d] |
23. Ratio, Proportion & Variation
| Topic | Formula | Description |
|---|---|---|
| Ratio | a:b or a/b | Comparison of quantities |
| Proportion | a/b = c/d or a:b = c:d | Equal ratios |
| Cross Multiplication | If a/b = c/d, then ad = bc | Solving proportions |
| Direct Variation | y = kx or y/x = k | k = constant of variation |
| Inverse Variation | y = k/x or xy = k | Product is constant |
| Joint Variation | z = kxy | z varies with both x and y |
| Combined Variation | z = kx/y | Direct with x, inverse with y |
| Scale Factor | New/Original = k | For similar figures |
24. Percent, Percent Change & Applications
| Topic | Formula | Description |
|---|---|---|
| Percent | Percent = (Part/Whole) × 100% | Convert to percentage |
| Percent of a Number | Part = (Percent/100) × Whole | Find the part |
| Percent Change | % Change = [(New – Old)/Old] × 100% | Increase or decrease |
| Percent Increase | New = Old × (1 + r/100) | r = percent increase |
| Percent Decrease | New = Old × (1 – r/100) | r = percent decrease |
| Simple Interest | I = Prt | P = principal, r = rate, t = time |
| Total with Interest | A = P(1 + rt) | Simple interest |
| Markup | Selling Price = Cost × (1 + markup%) | Retail pricing |
| Discount | Sale Price = Original × (1 – discount%) | Reduced price |
25. Rate, Work & Mixture Problems
| Topic | Formula | Description |
|---|---|---|
| Distance = Rate × Time | d = rt | Uniform motion |
| Average Speed | Average = Total Distance/Total Time | NOT average of speeds |
| Work Rate | Work = Rate × Time | W = rt |
| Combined Work | 1/t = 1/t₁ + 1/t₂ | Two workers together |
| Mixture Formula | Amount × Concentration = Pure quantity | For solutions |
26. Important Constants & Values
| Constant | Value | Description |
|---|---|---|
| π (Pi) | ≈ 3.14159 or 22/7 | Circle constant |
| e (Euler’s number) | ≈ 2.71828 | Natural exponential base |
| √2 | ≈ 1.414 | Square root of 2 |
| √3 | ≈ 1.732 | Square root of 3 |
| Golden Ratio (φ) | ≈ 1.618 | (1 + √5)/2 |
27. Useful Calculation Shortcuts
| Topic | Formula | Description |
|---|---|---|
| Sum of First n Integers | 1 + 2 + 3 + … + n = n(n + 1)/2 | Quick sum |
| Sum of First n Odd | 1 + 3 + 5 + … + (2n-1) = n² | Odd numbers |
| Sum of First n Even | 2 + 4 + 6 + … + 2n = n(n + 1) | Even numbers |
| Sum of Squares | 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 | Squared integers |
| Difference of Squares | a² – b² = (a + b)(a – b) | Quick factoring |
🎯 Top 10 Tips for SAT Math Success
- Memorize special right triangles: The 30-60-90 and 45-45-90 triangles appear frequently
- Know Pythagorean triples: Recognize 3-4-5, 5-12-13, 8-15-17 instantly
- Master the quadratic formula: It’s NOT given on the test
- Practice unit circle values: Know sin, cos, tan for 0°, 30°, 45°, 60°, 90°
- Understand slope concepts: Parallel lines (same slope), perpendicular lines (negative reciprocal)
- Master percent change: Growth/decay problems are common
- Know circle equations: Both standard (x-h)²+(y-k)²=r² and general form
- Be comfortable with functions: Function notation, composition, transformations
- Review exponent rules: These appear in multiple contexts
- Practice word problems: Learn to translate English to equations efficiently
Final Thoughts
This comprehensive formula sheet covers all the mathematical concepts tested on the SAT. While memorizing formulas is important, understanding when and how to apply them is even more crucial.
Here are some final study tips:
- Practice regularly: Review these formulas daily, not just the night before the test
- Create flashcards: Make cards for formulas you struggle with most
- Apply them: Work through practice problems to see these formulas in action
- Understand, don’t just memorize: Know WHY formulas work, not just WHAT they are
- Take timed practice tests: Speed and accuracy both matter on the SAT
Remember, the SAT provides some formulas at the beginning of each math section (like area formulas for basic shapes), but most of these formulas you’ll need to know by heart. Focus especially on the quadratic formula, special right triangles, and trigonometric ratios.
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