Math Formula Reference — Everything Student

A quick place to check a formula instead of digging through old notes. It covers the formulas that come up most in algebra, geometry, trigonometry, and intro statistics, not every formula there is.

Algebra

Quadratic formula, for solving $\vA{a}x^2 + \vB{b}x + \vC{c} = 0$ (where $\vA{a} \ne 0$ ):

x = \frac{-\vB{b} \pm \sqrt{\vB{b}^2 - 4\vA{a}\vC{c}}}{2\vA{a}}

The part under the square root, $\vB{b}^2 - 4\vA{a}\vC{c}$ , is the discriminant: positive means two real solutions, zero means one, negative means none that are real.

Slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$ :

m = \frac{y_2 - y_1}{x_2 - x_1}

Slope-intercept and point-slope form of a line:

\begin{aligned} y &= mx + b \\[0.6em] y - y_1 &= m(x - x_1) \end{aligned}

Distance between two points, and their midpoint:

\begin{aligned} d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\[0.6em] M &= \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \end{aligned}

Exponents and Logarithms

Rule	Formula
Product	$a^m \cdot a^n = a^{m+n}$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$
Zero and negative exponents	$a^0 = 1, \quad a^{-n} = \dfrac{1}{a^n}$
Fractional exponents	$a^{1/n} = \sqrt[n]{a}$
Definition of a log	$\log_b x = y \iff b^y = x$
Log of a product	$\log_b (xy) = \log_b x + \log_b y$
Log of a quotient	$\log_b \dfrac{x}{y} = \log_b x - \log_b y$
Log of a power	$\log_b x^n = n \log_b x$
Change of base	$\log_b x = \dfrac{\ln x}{\ln b}$

Geometry

Shape	Area	Perimeter
Rectangle	$lw$	$2(l + w)$
Triangle	$\tfrac{1}{2}bh$	sum of the three sides
Circle	$\pi r^2$	$2\pi r$ (circumference)
Trapezoid	$\tfrac{1}{2}(b_1 + b_2)h$	sum of the four sides
Parallelogram	$bh$	$2(a + b)$

Solid	Volume	Surface area
Rectangular prism	$lwh$	$2(lw + lh + wh)$
Cylinder	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$
Cone	$\tfrac{1}{3}\pi r^2 h$	$\pi r^2 + \pi r s$ ( $s$ = slant height)
Sphere	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$

Pythagorean theorem, for a right triangle with legs $\vA{a}$ and $\vB{b}$ and hypotenuse $\vC{c}$ :

\vA{a}^2 + \vB{b}^2 = \vC{c}^2

Arc length and sector area for an angle $\theta$ in radians:

\begin{aligned} s &= r\theta \\[0.6em] A &= \tfrac{1}{2} r^2 \theta \end{aligned}

Trigonometry

For an angle $\theta$ in a right triangle (SOH-CAH-TOA):

\begin{aligned} \sin\theta &= \frac{\text{opposite}}{\text{hypotenuse}} \\[0.6em] \cos\theta &= \frac{\text{adjacent}}{\text{hypotenuse}} \\[0.6em] \tan\theta &= \frac{\text{opposite}}{\text{adjacent}} \end{aligned}

The Pythagorean identity, true for every angle:

\sin^2\theta + \cos^2\theta = 1

For any triangle with sides $a, b, c$ opposite angles $A, B, C$ , the law of sines and the law of cosines:

\begin{aligned} \frac{a}{\sin A} &= \frac{b}{\sin B} = \frac{c}{\sin C} \\[0.6em] c^2 &= a^2 + b^2 - 2ab\cos C \end{aligned}

To convert between degrees and radians: $\pi \text{ rad} = 180^\circ$ .

Statistics and Probability

Mean of $n$ values, and the sample standard deviation:

\begin{aligned} \bar{x} &= \frac{\sum x_i}{n} \\[0.6em] s &= \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \end{aligned}

z-score: how many standard deviations a value sits from the mean.

z = \frac{x - \mu}{\sigma}

Percent error between a measured and an accepted value:

\text{percent error} = \frac{\lvert \text{measured} - \text{accepted} \rvert}{\lvert \text{accepted} \rvert} \times 100\%

Probability basics: the complement, and “and” for independent events.

\begin{aligned} P(\text{not } A) &= 1 - P(A) \\[0.6em] P(A \text{ and } B) &= P(A) \cdot P(B) \end{aligned}

Common Mistakes

Dropping the ± in the quadratic formula. Most quadratics have two solutions; find both.
Using degrees where radians are expected. Arc length, sector area, and most calculus formulas assume radians. Check your calculator’s mode.
Dividing by $n$ instead of $n - 1$ . The sample standard deviation divides by $n - 1$ ; dividing by $n$ gives the population version.
Mixing up area and perimeter units. Area is in square units (cm²); perimeter is in plain units (cm).

Frequently Asked Questions

Is this every math formula I could need?

No. It covers the formulas that come up most in algebra, geometry, trigonometry, and intro statistics. For calculus, linear algebra, or anything more specialized, your textbook or course notes are the complete source.

Why does the quadratic formula have a ± in it?

Because a quadratic can have two solutions. Work it out once adding the square root and once subtracting it. If the discriminant is zero there is only one solution, and if it is negative there are no real ones.

Should my calculator be in degrees or radians?

Use whatever unit the problem gives the angle in. Formulas like arc length and sector area on this page need radians, and so does most of calculus. If a trig answer looks wildly off, check the mode first.

What is the difference between sample and population standard deviation?

The sample version divides by n - 1 and is what you use when your data is a sample of a larger group, which is the usual case in class. The population version divides by n and is only for data that covers the entire group.

What are the colored letters in some formulas?

They follow one quantity through a formula, so you can see where each value goes. In the quadratic formula, for example, a, b, and c keep the same color in the equation and in the answer. The color is only a guide; the letters mean the same thing without it.

Algebra

Exponents and Logarithms

Geometry

Trigonometry

Statistics and Probability

Common Mistakes

Frequently Asked Questions

Related Guides

Unit Conversion Chart

Roman Numerals & Greek Alphabet

Periodic Table Reference

Sources