SAT® math formulas: what to memorize

None of these appear on the reference sheet, and they carry more Algebra questions than anything else on this page.None of these appear on the reference sheet, and they carry more Algebra questions than anything else on this page.

• Slope from two points

  Memorize

  m = y₂ - y₁x₂ - x₁m = (y2 - y1)/(x2 - x1)

  The change in y over the change in x, in that order.The change in y over the change in x, in that order.

• Slope-intercept form

  Memorize

  y = mx + by = mx + b

  m is the slope; b is the value of y where the line crosses the y-axis.m is the slope; b is the value of y where the line crosses the y-axis.

• Point-slope form

  Memorize

  y - y₁ = m(x - x₁)y - y1 = m(x - x1)

  Fastest way to write a line when you have one point and the slope.Fastest way to write a line when you have one point and the slope.

• Parallel and perpendicular

  Memorize

  parallel: m₁ = m₂ perpendicular: m₁ x m₂ = -1parallel: m1 = m2 perpendicular: m1 x m2 = -1

  Perpendicular slopes are negative reciprocals of each other.Perpendicular slopes are negative reciprocals of each other.

• Midpoint

  Memorize

  (x₁ + x₂2, y₁ + y₂2)((x1 + x2)/2, (y1 + y2)/2)

  Average each coordinate separately.Average each coordinate separately.

• Distance between two points

  Memorize

  d = √(x₂ - x₁)² + (y₂ - y₁)²d = √((x2 - x1)^2 + (y2 - y1)^2)

  The Pythagorean theorem applied to the coordinate plane.The Pythagorean theorem applied to the coordinate plane.

The forms matter as much as the formulas: each one hands you a different fact for free.The forms matter as much as the formulas: each one hands you a different fact for free.

• Quadratic formula

  Memorize

  x = -b ± √b² - 4ac2ax = (-b ± √(b^2 - 4ac))/(2a)

  For ax² + bx + c = 0. Try factoring first; most SAT quadratics factor.For ax^2 + bx + c = 0. Try factoring first; most SAT quadratics factor.

• Discriminant

  Memorize

  b² - 4acb^2 - 4ac

  Positive gives two real solutions, zero gives one, negative gives none.Positive gives two real solutions, zero gives one, negative gives none.

• Vertex form

  Memorize

  y = a(x - h)² + ky = a(x - h)^2 + k

  The vertex is (h, k). Note the subtraction: y = (x - 4)² has its vertex at x = 4.The vertex is (h, k). Note the subtraction: y = (x - 4)^2 has its vertex at x = 4.

• Axis of symmetry

  Memorize

  x = -b / (2a)x = -b / (2a)

  From standard form, this is the x-coordinate of the vertex.From standard form, this is the x-coordinate of the vertex.

• Difference of squares

  Memorize

  a² - b² = (a - b)(a + b)a^2 - b^2 = (a - b)(a + b)

  The single most useful factoring pattern on the test.The single most useful factoring pattern on the test.

• Sum and product of roots

  Memorize

  sum = -b / a product = c / asum = -b / a product = c / a

  Lets you answer questions about the roots without solving for them.Lets you answer questions about the roots without solving for them.

Rules, not formulas. Every one of them requires the same base to apply.Rules, not formulas. Every one of them requires the same base to apply.

• Product rule

  Memorize

  x^a x^b = x^{a + b}x^a x^b = x^(a + b)

  Only when the bases match. 2³ x 3² does not simplify this way.Only when the bases match. 2^3 x 3^2 does not simplify this way.

• Quotient rule

  Memorize

  x^a / x^b = x^{a - b}x^a / x^b = x^(a - b)

  Same base requirement as the product rule.Same base requirement as the product rule.

• Power of a power

  Memorize

  (x^a)^b = x^ab(x^a)^b = x^(ab)

  The exponents multiply rather than add.The exponents multiply rather than add.

• Negative exponent

  Memorize

  x^-a = 1 / x^ax^(-a) = 1 / x^a

  A negative exponent moves the term across the fraction bar.A negative exponent moves the term across the fraction bar.

• Fractional exponent

  Memorize

  x^1/n = the nth root of xx^(1/n) = the nth root of x

  So x¹² is the square root and x²³ is the cube root of x squared.So x^(1/2) is the square root and x^(2/3) is the cube root of x squared.

• Zero exponent

  Memorize

  x⁰ = 1x^0 = 1

  True for every nonzero x.True for every nonzero x.

The arithmetic here is easy; the errors come from applying a percent to the wrong amount.The arithmetic here is easy; the errors come from applying a percent to the wrong amount.

• Percent change

  Memorize

  (new - old) / old x 100(new - old) / old x 100

  Always divide by the original amount, never the new one.Always divide by the original amount, never the new one.

• Percent increase, applied

  Memorize

  new = old x (1 + rate)new = old x (1 + rate)

  A 25 percent rise multiplies by 1.25.A 25 percent rise multiplies by 1.25.

• Reversing a percent

  Memorize

  old = new / (1 + rate)old = new / (1 + rate)

  Divide to undo an increase. Subtracting the percent from the new amount gives a different, wrong answer.Divide to undo an increase. Subtracting the percent from the new amount gives a different, wrong answer.

• Mean

  Memorize

  mean = sum / countmean = sum / count

  Rearranged as sum = mean x count, this solves every changing-mean question.Rearranged as sum = mean x count, this solves every changing-mean question.

• Probability

  Memorize

  P = favorable outcomes / total outcomesP = favorable outcomes / total outcomes

  Read carefully which group forms the denominator.Read carefully which group forms the denominator.

• Rate

  Memorize

  distance = rate x timedistance = rate x time

  Same shape as work = rate x time, which is why combined-rate questions add rates rather than times.Same shape as work = rate x time, which is why combined-rate questions add rates rather than times.

Most of this group is supplied during the test. Knowing that is worth more than memorizing it: use the reference and spend your time on setup.Most of this group is supplied during the test. Knowing that is worth more than memorizing it: use the reference and spend your time on setup.

• Area of a circle

  Given

  A = π r²A = π r^2

  Halve a given diameter before substituting.Halve a given diameter before substituting.

• Circumference

  Given

  C = 2π rC = 2π r

  Equivalently π times the diameter.Equivalently π times the diameter.

• Area of a rectangle

  Given

  A = lwA = lw

  Perimeter is 2l + 2w.Perimeter is 2l + 2w.

• Area of a triangle

  Given

  A = (12) b hA = (1/2) b h

  The height must be perpendicular to the chosen base.The height must be perpendicular to the chosen base.

• Area of a trapezoid

  Memorize

  A = (12)(b₁ + b₂) hA = (1/2)(b1 + b2) h

  The average of the two bases times the height.The average of the two bases times the height.

• Volume of a rectangular solid

  Given

  V = lwhV = lwh

  A cube is the case where all three are equal.A cube is the case where all three are equal.

• Volume of a cylinder

  Given

  V = π r² hV = π r^2 h

  The circular area times the height.The circular area times the height.

• Volume of a cone

  Given

  V = (13) π r² hV = (1/3) π r^2 h

  One third of the cylinder with the same base and height.One third of the cylinder with the same base and height.

• Volume of a sphere

  Given

  V = (43) π r³V = (4/3) π r^3

  The r³ is why doubling a radius multiplies volume by eight.The r^3 is why doubling a radius multiplies volume by eight.

• Scaling

  Memorize

  lengths x k, areas x k², volumes x k³lengths x k, areas x k^2, volumes x k^3

  Not on any reference sheet, and the most reliable trap in geometry.Not on any reference sheet, and the most reliable trap in geometry.

The special triangles are supplied. The angle relationships and the ratios are not.The special triangles are supplied. The angle relationships and the ratios are not.

• Pythagorean theorem

  Given

  a² + b² = c²a^2 + b^2 = c^2

  c is the hypotenuse, always the side opposite the right angle.c is the hypotenuse, always the side opposite the right angle.

• Special right triangles

  Given

  30-60-90: x, x √3, 2x 45-45-90: x, x, x √230-60-90: x, x √(3), 2x 45-45-90: x, x, x √(2)

  Recognizing these skips the theorem entirely.Recognizing these skips the theorem entirely.

• Triangle angle sum

  Memorize

  the three angles total 180°the three angles total 180°

  A polygon with n sides totals (n - 2) x 180°.A polygon with n sides totals (n - 2) x 180°.

• Sine, cosine, tangent

  Memorize

  sin = opp / hyp cos = adj / hyp tan = opp / adjsin = opp / hyp cos = adj / hyp tan = opp / adj

  Tangent is the only one that does not use the hypotenuse.Tangent is the only one that does not use the hypotenuse.

• Complementary angles

  Memorize

  sin(x) = cos(90 - x)sin(x) = cos(90 - x)

  Why a question can give you a sine and ask for a cosine.Why a question can give you a sine and ask for a cosine.

• Similar figures

  Memorize

  corresponding sides share one scale factorcorresponding sides share one scale factor

  Find the factor once, then apply it to the side you want.Find the factor once, then apply it to the side you want.

• Angles on parallel lines

  Memorize

  corresponding and alternate angles are equalcorresponding and alternate angles are equal

  Same-side interior angles add to 180°.Same-side interior angles add to 180°.

• Degrees and radians

  Given

  180° = π radians180° = π radians

  Multiply by π/180 to convert degrees to radians.Multiply by π/180 to convert degrees to radians.