| { |
| "name": "SAT Math Formula Sheet", |
| "description": "The 24 formulas and concepts the SAT does not give you on its reference sheet. Each entry has the formula as text, the point it tests, and a link to a full explanation with a solved SAT question.", |
| "source": "https://sigmaprep.io/blog/sat-math-formulas", |
| "publisher": "Sigma Prep", |
| "license": "CC-BY-4.0", |
| "licenseUrl": "https://creativecommons.org/licenses/by/4.0/", |
| "count": 24, |
| "domains": [ |
| "Algebra", |
| "Advanced Math", |
| "Problem-Solving and Data Analysis", |
| "Geometry and Trigonometry" |
| ], |
| "formulas": [ |
| { |
| "n": 1, |
| "slug": "slope-intercept-form-interpretation", |
| "title": "Slope-Intercept Form Interpretation", |
| "domain": "Algebra", |
| "skill": "Linear functions", |
| "formula": [ |
| "y = mx + b" |
| ], |
| "note": "m is the slope · b is the y-intercept", |
| "keyPoint": "Slope is always attached to the variable, and the keywords are per and each. Slope is the change in y for ONE change in x.", |
| "diagram": "figures/slope-intercept-form-interpretation.jpg", |
| "url": "https://sigmaprep.io/formulas/slope-intercept-form-interpretation" |
| }, |
| { |
| "n": 2, |
| "slug": "parallel-and-perpendicular", |
| "title": "Parallel and Perpendicular", |
| "domain": "Algebra", |
| "skill": "Linear equations in two variables", |
| "formula": [ |
| "parallel: m₂ = m₁", |
| "perpendicular: m₂ = −1/m₁" |
| ], |
| "note": "y = 3/5x + 4 → parallel 3/5, perpendicular −5/3", |
| "keyPoint": "You can only read the slope in slope-intercept form, so isolate y first. Perpendicular is opposite reciprocal: flip the sign AND flip the fraction.", |
| "diagram": "figures/parallel-and-perpendicular.jpg", |
| "url": "https://sigmaprep.io/formulas/parallel-and-perpendicular" |
| }, |
| { |
| "n": 3, |
| "slug": "no-solution-and-infinite-solutions", |
| "title": "No Solution and Infinite Solutions", |
| "domain": "Algebra", |
| "skill": "Linear equations in one variable · Systems of two linear equations", |
| "formula": [ |
| "no solution: 3x + 4 = 3x + 2", |
| "infinitely many: 5x + 1 = 5x + 1" |
| ], |
| "note": "Same slope with a different intercept, against the same line twice", |
| "keyPoint": "No solution means parallel lines: same slope, different y-intercept. Infinitely many solutions means the same line: same slope and the same y-intercept. Get both sides into slope-intercept form, then match the slopes (and, for infinite, the intercepts) and solve for the constant.", |
| "diagram": "figures/no-solution-and-infinite-solutions.jpg", |
| "url": "https://sigmaprep.io/formulas/no-solution-and-infinite-solutions" |
| }, |
| { |
| "n": 4, |
| "slug": "mixture-problem", |
| "title": "Mixture Problem", |
| "domain": "Algebra", |
| "skill": "Systems of two linear equations", |
| "formula": [ |
| "(percent × amount) + (percent × amount) = (percent × amount)" |
| ], |
| "note": "If one amount is x, the other is the total minus x", |
| "keyPoint": "Two cups going into one cup. Every cup gets a percent and an amount, and the equation is (percent)(amount) + (percent)(amount) = (percent)(amount). If one amount is x, the other one is the total minus x.", |
| "diagram": "figures/mixture-problem.jpg", |
| "url": "https://sigmaprep.io/formulas/mixture-problem" |
| }, |
| { |
| "n": 5, |
| "slug": "exponent-rules", |
| "title": "Exponent Rules", |
| "domain": "Advanced Math", |
| "skill": "Equivalent expressions", |
| "formula": [ |
| "x^a · x^b = x^a+b", |
| "x^a ÷ x^b = x^a−b", |
| "(x^a)^b = x^ab" |
| ], |
| "note": "Multiply adds, divide subtracts, a power of a power multiplies", |
| "keyPoint": "Three rules, in order. Multiplying means add the exponents, dividing means subtract the exponents, and an exponent raised to another exponent means you multiply the exponents.", |
| "diagram": "figures/exponent-rules.jpg", |
| "url": "https://sigmaprep.io/formulas/exponent-rules" |
| }, |
| { |
| "n": 6, |
| "slug": "exponential-to-radical", |
| "title": "Exponential to Radical", |
| "domain": "Advanced Math", |
| "skill": "Equivalent expressions", |
| "formula": [ |
| "x^power/root = ^root√(x^power)" |
| ], |
| "note": "Power on top, root on the bottom", |
| "keyPoint": "Never leave a radical as a radical. Root on the outside, power on the inside, and as an exponent you write power divided by root. Also remember that a square root has a root of 2 even though nobody writes a 2 on the outside.", |
| "diagram": "figures/exponential-to-radical.jpg", |
| "url": "https://sigmaprep.io/formulas/exponential-to-radical" |
| }, |
| { |
| "n": 7, |
| "slug": "factoring", |
| "title": "Factoring", |
| "domain": "Advanced Math", |
| "skill": "Equivalent expressions", |
| "formula": [ |
| "x² + 2x − 15 = (x − 3)(x + 5)" |
| ], |
| "note": "First times first, last times last", |
| "keyPoint": "When comparing factored form to expanded form, notice that in FOIL first times first gives the first term in expanded form, and last times last gives the last term.", |
| "diagram": "figures/factoring.jpg", |
| "url": "https://sigmaprep.io/formulas/factoring" |
| }, |
| { |
| "n": 8, |
| "slug": "quadratics", |
| "title": "Quadratics", |
| "domain": "Advanced Math", |
| "skill": "Nonlinear functions", |
| "formula": [ |
| "standard: y = ax² + bx + c", |
| "vertex: y = a(x − h)² + k", |
| "factored: y = a(x − r₁)(x − r₂)" |
| ], |
| "note": "Each form hands you one thing: the y-intercept, the vertex, the roots", |
| "keyPoint": "Know the 3 quadratic forms and what information each one gives you. Know the common questions and what they represent in context. Understand symmetry through the vertex of a quadratic.", |
| "diagram": "figures/quadratics.jpg", |
| "url": "https://sigmaprep.io/formulas/quadratics" |
| }, |
| { |
| "n": 9, |
| "slug": "quadratic-interpretations", |
| "title": "Quadratic Interpretations", |
| "domain": "Advanced Math", |
| "skill": "Nonlinear functions", |
| "formula": [], |
| "note": null, |
| "keyPoint": "Three points they ask about: the y-intercept is the \"starting value,\" the x-intercept is when it \"hits the ground,\" the vertex is the \"max or min.\" Those are the kinds of words to listen for; they are not always word for word.", |
| "diagram": "figures/quadratic-interpretations.jpg", |
| "url": "https://sigmaprep.io/formulas/quadratic-interpretations" |
| }, |
| { |
| "n": 10, |
| "slug": "discriminant", |
| "title": "Discriminant", |
| "domain": "Advanced Math", |
| "skill": "Nonlinear equations in one variable and systems of equations in two variables", |
| "formula": [ |
| "b² − 4ac > 0 → 2 solutions", |
| "b² − 4ac = 0 → 1 solution", |
| "b² − 4ac < 0 → no real solution" |
| ], |
| "note": "a, b and c come from ax² + bx + c = 0", |
| "keyPoint": "Two things to spot: you need a quadratic, and a sentence telling you how many solutions. That is a discriminant problem. Know the 3 cases: greater than 0 for 2 solutions, equal to 0 for 1 solution, and less than 0 for no solution. Then let Desmos solve it.", |
| "diagram": "figures/discriminant.jpg", |
| "url": "https://sigmaprep.io/formulas/discriminant" |
| }, |
| { |
| "n": 11, |
| "slug": "exponential-equation", |
| "title": "Exponential Equation", |
| "domain": "Advanced Math", |
| "skill": "Nonlinear functions", |
| "formula": [ |
| "y = a(1 ± r)^x" |
| ], |
| "note": "a is the initial value, r the rate as a decimal", |
| "keyPoint": "Two things to know: the number in front is the initial value, and the number inside the parentheses, with the exponent, represents the increase or decrease (usually in terms of percent).", |
| "diagram": "figures/exponential-equation.jpg", |
| "url": "https://sigmaprep.io/formulas/exponential-equation" |
| }, |
| { |
| "n": 12, |
| "slug": "transformations", |
| "title": "Transformations", |
| "domain": "Advanced Math", |
| "skill": "Nonlinear functions", |
| "formula": [ |
| "f(x) + k → up k", |
| "f(x − h) → right h" |
| ], |
| "note": "Outside is vertical, inside is horizontal and reversed", |
| "keyPoint": "They only really test you on shifts. Vertical, think outside. Horizontal, think inside. Vertical is intuitive: f(x) + 5 means up 5. Horizontal is counterintuitive: f(x + 5) means left 5, not right 5.", |
| "diagram": "figures/transformations.jpg", |
| "url": "https://sigmaprep.io/formulas/transformations" |
| }, |
| { |
| "n": 13, |
| "slug": "unit-conversions", |
| "title": "Unit Conversions", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "Ratios, rates, proportional relationships, and units", |
| "formula": [ |
| "5 ft/min × (1 yd / 3 ft) × (60 min / 1 hr) = 100 yd/hr" |
| ], |
| "note": "Line the units up so each one cancels", |
| "keyPoint": "Per makes the fraction. Line the units up so each one cancels: whatever is on the bottom of one fraction goes on top of the next. If a unit is squared, square the whole conversion factor, or the units will not cancel.", |
| "diagram": "figures/unit-conversions.jpg", |
| "url": "https://sigmaprep.io/formulas/unit-conversions" |
| }, |
| { |
| "n": 14, |
| "slug": "percent-change", |
| "title": "Percent Change", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "Percentages", |
| "formula": [ |
| "final = original × (1 ± r)" |
| ], |
| "note": "Solve for whichever of the three the question asks for", |
| "keyPoint": "Use this equation in 3 different ways: to find the final amount, the original amount, or the percent. Make whatever the question asks for the unknown, x, plug in numbers for the other two, and let Desmos solve it for you. Very useful and versatile.", |
| "diagram": "figures/percent-change.jpg", |
| "url": "https://sigmaprep.io/formulas/percent-change" |
| }, |
| { |
| "n": 15, |
| "slug": "statistics", |
| "title": "Statistics", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "One-variable data: distributions and measures of center and spread", |
| "formula": [ |
| "mean = sum ÷ count", |
| "median = middle number of an ordered list", |
| "range = largest − smallest", |
| "mode = most common number", |
| "standard deviation = spread from the mean" |
| ], |
| "note": "Mean, median and standard deviation can be done in Desmos. Range and mode cannot.", |
| "keyPoint": "Know these 5 calculations for statistics. Mean, median, and standard deviation can be computed in Desmos. Range and mode MUST be memorized, because Desmos will not do them for you.", |
| "diagram": "figures/statistics.jpg", |
| "url": "https://sigmaprep.io/formulas/statistics" |
| }, |
| { |
| "n": 16, |
| "slug": "standard-deviation", |
| "title": "Standard Deviation", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "One-variable data: distributions and measures of center and spread", |
| "formula": [], |
| "note": null, |
| "keyPoint": "For standard deviation, simply think of spread. Just remember that one word. In the examples above, B has more spread than A, so B has the larger standard deviation. C and D always go up by 1, so their spreads are the same.", |
| "diagram": "figures/standard-deviation.jpg", |
| "url": "https://sigmaprep.io/formulas/standard-deviation" |
| }, |
| { |
| "n": 17, |
| "slug": "probability", |
| "title": "Probability", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "Probability and conditional probability", |
| "formula": [ |
| "probability = favorable / total" |
| ], |
| "note": "The words \"given that\" narrow the total", |
| "keyPoint": "Probability just wants you to create a fraction. When you read these problems you need to know what the denominator is (the total) and what the numerator is (the favorable). The phrase \"given that\" swaps the total around.", |
| "diagram": "figures/probability.jpg", |
| "url": "https://sigmaprep.io/formulas/probability" |
| }, |
| { |
| "n": 18, |
| "slug": "statistical-inference", |
| "title": "Statistical Inference", |
| "domain": "Problem-Solving and Data Analysis", |
| "skill": "Inference from sample statistics and margin of error", |
| "formula": [ |
| "true value = estimate ± margin of error" |
| ], |
| "note": "53% with a margin of 2% means 51% to 55%", |
| "keyPoint": "Margin of error is a plus or minus around the reported number: the true value is most likely between the estimate minus the margin and the estimate plus the margin. If they give you the range instead, the margin is the distance from the middle to one end, not the whole width. For sample-to-population questions, set up the proportion and scale it up to the population.", |
| "diagram": "figures/statistical-inference.jpg", |
| "url": "https://sigmaprep.io/formulas/statistical-inference" |
| }, |
| { |
| "n": 19, |
| "slug": "angle-relationships", |
| "title": "Angle Relationships", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Lines, angles, and triangles", |
| "formula": [ |
| "a + b + c = 180°", |
| "linear pair: a + b = 180°", |
| "vertical angles are equal" |
| ], |
| "note": "These three need no parallel lines", |
| "keyPoint": "When they say parallel, you are using one of the parallel-line relationships: alternate interior, alternate exterior, corresponding, or same-side interior angles. The other relationships shown here do not require parallel lines and are always true: vertical angles, linear pair, and triangle sum.", |
| "diagram": "figures/angle-relationships.jpg", |
| "url": "https://sigmaprep.io/formulas/angle-relationships" |
| }, |
| { |
| "n": 20, |
| "slug": "similar-figures", |
| "title": "Similar Figures", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Lines, angles, and triangles", |
| "formula": [], |
| "note": null, |
| "keyPoint": "Similar figures are very common on the test, and it is super important that you always remember the first two properties. The third property, the one about trig, is not always used, but it is nice to know.", |
| "diagram": "figures/similar-figures.jpg", |
| "url": "https://sigmaprep.io/formulas/similar-figures" |
| }, |
| { |
| "n": 21, |
| "slug": "proportionality", |
| "title": "Proportionality", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Area and volume", |
| "formula": [ |
| "sides ×k → area ×k² → volume ×k³" |
| ], |
| "note": "Sides three times bigger gives area nine times, volume twenty-seven", |
| "keyPoint": "If the sides are k times bigger, the area is k squared times bigger and the volume is k cubed times bigger. Area is units squared, volume is units cubed. One calculation, not a whole recomputation.", |
| "diagram": "figures/proportionality.jpg", |
| "url": "https://sigmaprep.io/formulas/proportionality" |
| }, |
| { |
| "n": 22, |
| "slug": "volume-and-surface-area", |
| "title": "Volume and Surface Area", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Area and volume", |
| "formula": [ |
| "rectangular prism: V = LWH, SA = 2LW + 2WH + 2LH", |
| "square prism: V = x²H, SA = 2x² + 4xH", |
| "cube: V = x³, SA = 6x²" |
| ], |
| "note": "The SAT gives you the prism volume on its reference sheet but not surface area", |
| "keyPoint": "The SAT gives you the rectangular prism on its reference sheet, but it does not give you surface area. You need to be able to derive surface area by looking at the figure, or memorize it from here. The square prism and the cube are variations of a rectangular prism, with two sides the same and all three sides the same, respectively.", |
| "diagram": "figures/volume-and-surface-area.jpg", |
| "url": "https://sigmaprep.io/formulas/volume-and-surface-area" |
| }, |
| { |
| "n": 23, |
| "slug": "trigonometry", |
| "title": "Trigonometry", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Right triangles and trigonometry", |
| "formula": [ |
| "a² + b² = c²", |
| "sin = opposite", |
| "cos = adjacent", |
| "tan = opposite" |
| ], |
| "note": "Right triangles only", |
| "keyPoint": "SOH CAH TOA and the Pythagorean theorem ONLY apply to right triangles. Use the Pythagorean theorem to find a missing side. Trig questions just want a ratio.", |
| "diagram": "figures/trigonometry.jpg", |
| "url": "https://sigmaprep.io/formulas/trigonometry" |
| }, |
| { |
| "n": 24, |
| "slug": "circles", |
| "title": "Circles", |
| "domain": "Geometry and Trigonometry", |
| "skill": "Circles", |
| "formula": [ |
| "(x − h)² + (y − k)² = r²" |
| ], |
| "note": "Center (h, k) with the signs flipped, radius squared on the right", |
| "keyPoint": "The right side of (x - h)^2 + (y - k)^2 = r^2 is the radius SQUARED. A radius of 3m makes the right side 9m^2, not 3m^2, and a right side of 49k^2 means the radius is 7k. The center goes in with the signs flipped: center (5, 12) is (x - 5) and (y - 12). Two traps in one question, and both wrong answers are sitting in the choices.", |
| "diagram": "figures/circles.jpg", |
| "url": "https://sigmaprep.io/formulas/circles" |
| } |
| ] |
| } |
|
|
