The SAT Math Formulas You Actually Need to Memorize

By Dr. Brink, Ph.D., 32 years teaching mathematics

The short answer: the SAT gives you a reference sheet at the start of every Math section, so the common geometry formulas (areas, volumes, the Pythagorean theorem, the two special right triangles) are already printed for you. You do not need to memorize those. What you do need by heart are the algebra, coordinate-geometry, and data formulas that are not on the sheet: slope and the three forms of a line, the quadratic formula and discriminant, vertex form, the distance and midpoint formulas, the equation of a circle, percent change, averages, the exponent rules, exponential growth and decay, and right-triangle trig. Those are the ones worth putting on a flashcard.

Most students waste study time memorizing formulas the test is going to hand them anyway, and then walk in without the handful that actually matter. So before the list, know the split.

What the SAT already gives you

ON THE REFERENCE SHEET

These sit at the top of every Math section. Do not spend flashcards on them.

Area of a circle, $A = \pi r^{2}$, and circumference, $C = 2\pi r$
Area of a rectangle, $A = lw$, and area of a triangle, $A = \dfrac{1}{2}bh$
The Pythagorean theorem, $a^{2} + b^{2} = c^{2}$
The two special right triangles: the 45°-45°-90° with sides $x,\ x,\ x\sqrt{2}$, and the 30°-60°-90° with sides $x,\ x\sqrt{3},\ 2x$
Volumes of a box, cylinder, sphere, cone, and pyramid
A full circle is 360°, or $2\pi$ radians, and the angles of a triangle add to 180°

That is real ground covered for free. The catch is that the sheet stops at geometry. Everything below is not on it, and that is where the points are won or lost.

START HERE

The ten to memorize first

If you learn nothing else, learn these. They cover the largest share of the test.

Slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form: $y = mx + b$
The quadratic formula: $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
The discriminant: $b^{2} - 4ac$
Vertex form: $y = a(x - h)^{2} + k$, with vertex $(h, k)$
Distance: $d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$
Midpoint: $\left( \dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2} \right)$
Equation of a circle: $(x - h)^{2} + (y - k)^{2} = r^{2}$
Percent change: $\dfrac{\text{new} - \text{old}}{\text{old}} \times 100$
Right-triangle trig: SOH-CAH-TOA

Lines

Slope $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form $y = mx + b$
Point-slope form $y - y_1 = m(x - x_1)$
Parallel and perpendicular lines Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

Quadratics

Quadratic formula $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
Discriminant (how many real solutions) $b^{2} - 4ac$ Positive gives two real solutions, zero gives one, negative gives none.
Vertex form $y = a(x - h)^{2} + k$, vertex at $(h, k)$
Axis of symmetry $x = -\dfrac{b}{2a}$
Factored form $y = a(x - r_1)(x - r_2)$, roots at $r_1$ and $r_2$

Coordinate geometry

Distance between two points $d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$
Midpoint $\left( \dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2} \right)$
Equation of a circle $(x - h)^{2} + (y - k)^{2} = r^{2}$, center $(h, k)$, radius $r$

Percents, averages, and probability

Percent change $\dfrac{\text{new} - \text{old}}{\text{old}} \times 100$
Average (arithmetic mean) $\text{average} = \dfrac{\text{sum}}{\text{count}}$ The version that solves SAT problems: $\text{sum} = \text{average} \times \text{count}$.
Probability $P = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$

Exponents and growth

Exponent rules $x^{a} \cdot x^{b} = x^{a+b}, \quad \dfrac{x^{a}}{x^{b}} = x^{a-b}, \quad (x^{a})^{b} = x^{ab}$ $x^{-a} = \dfrac{1}{x^{a}}, \quad x^{1/n} = \sqrt[n]{x}$
Exponential growth and decay Growth: $y = a(1 + r)^{t}$ Decay: $y = a(1 - r)^{t}$

Right-triangle trig

SOH-CAH-TOA $\sin = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos = \dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan = \dfrac{\text{opposite}}{\text{adjacent}}$
Complementary angle relationship $\sin(x) = \cos(90 - x)$, with the angles measured in degrees The SAT tests this one directly, so it is worth knowing.

Circles: the angle-proportion idea

The occasional arc-length or sector-area question is just a fraction of the whole circle. The slice you want is the same fraction of the circumference or area as the central angle is of the full 360°.

Arc length and sector area $\dfrac{\text{arc length}}{\text{circumference}} = \dfrac{\text{sector area}}{\text{circle area}} = \dfrac{\text{central angle}}{360}$

How to actually use this list

Do not sit and memorize all of them cold. Put the ten in the box above on flashcards, then take a full practice test and watch which formulas you reach for and freeze on. Those are the ones your memory is missing, and they are worth far more than the ones you already recall. Learn them in the context of real questions, not in a vacuum.

See which formulas you actually reach for under time. Every question you miss on a free practice test here comes with a full worked solution.

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