SAT® math formulas: what to memorise

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

• Slope from two points

  Memorise
  m = (y2 - y1) / (x2 - x1)

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

• Slope-intercept form

  Memorise
  y = mx + b

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

• Point-slope form

  Memorise
  y - y1 = m(x - x1)

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

• Parallel and perpendicular

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

  Perpendicular slopes are negative reciprocals of each other.

• Midpoint

  Memorise
  ((x1 + x2) / 2, (y1 + y2) / 2)

  Average each coordinate separately.

• Distance between two points

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

  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.

• Quadratic formula

  Memorise
  x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

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

• Discriminant

  Memorise
  b^2 - 4ac

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

• Vertex form

  Memorise
  y = a(x - h)^2 + k

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

• Axis of symmetry

  Memorise
  x = -b / (2a)

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

• Difference of squares

  Memorise
  a^2 - b^2 = (a - b)(a + b)

  The single most useful factoring pattern on the test.

• Sum and product of roots

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

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

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

• Product rule

  Memorise
  x^a x^b = x^(a + b)

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

• Quotient rule

  Memorise
  x^a / x^b = x^(a - b)

  Same base requirement as the product rule.

• Power of a power

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

  The exponents multiply rather than add.

• Negative exponent

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

  A negative exponent moves the term across the fraction bar.

• Fractional exponent

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

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

• Zero exponent

  Memorise
  x^0 = 1

  True for every nonzero x.

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

• Percent change

  Memorise
  (new - old) / old x 100

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

• Percent increase, applied

  Memorise
  new = old x (1 + rate)

  A 25 percent rise multiplies by 1.25.

• Reversing a percent

  Memorise
  old = new / (1 + rate)

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

• Mean

  Memorise
  mean = sum / count

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

• Probability

  Memorise
  P = favourable outcomes / total outcomes

  Read carefully which group forms the denominator.

• Rate

  Memorise
  distance = rate x time

  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 memorising it: use the reference and spend your time on setup.

• Area of a circle

  Given
  A = pi r^2

  Halve a given diameter before substituting.

• Circumference

  Given
  C = 2 pi r

  Equivalently pi times the diameter.

• Area of a rectangle

  Given
  A = lw

  Perimeter is 2l + 2w.

• Area of a triangle

  Given
  A = (1/2) b h

  The height must be perpendicular to the chosen base.

• Area of a trapezoid

  Memorise
  A = (1/2)(b1 + b2) h

  The average of the two bases times the height.

• Volume of a rectangular solid

  Given
  V = lwh

  A cube is the case where all three are equal.

• Volume of a cylinder

  Given
  V = pi r^2 h

  The circular area times the height.

• Volume of a cone

  Given
  V = (1/3) pi r^2 h

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

• Volume of a sphere

  Given
  V = (4/3) pi r^3

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

• Scaling

  Memorise
  lengths x k, areas x k^2, volumes x k^3

  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.

• Pythagorean theorem

  Given
  a^2 + b^2 = c^2

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

• Special right triangles

  Given
  30-60-90: x, x sqrt(3), 2x 45-45-90: x, x, x sqrt(2)

  Recognising these skips the theorem entirely.

• Triangle angle sum

  Memorise
  the three angles total 180 degrees

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

• Sine, cosine, tangent

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

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

• Complementary angles

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

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

• Similar figures

  Memorise
  corresponding sides share one scale factor

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

• Angles on parallel lines

  Memorise
  corresponding and alternate angles are equal

  Same-side interior angles add to 180 degrees.

• Degrees and radians

  Given
  180 degrees = pi radians

  Multiply by pi/180 to convert degrees to radians.