SAT Math reference sheet

This sheet follows the six units of the course in order. Each unit starts with the big idea in a few sentences, then gives the definitions, formulas and rules you will look up most often, with the conditions they need. When you want the full explanation or more examples, follow the lesson links under each topic.

A reference sheet of common formulas is available during the test, but it cannot tell you which formula a question needs or what its letters mean there. This page collects the rules the lessons rely on, whether or not they appear on the test’s own sheet, so learn the ones you use often well enough to recognize them quickly.

Unless a question says otherwise, every variable and expression on the SAT is a real number, a function’s inputs must give real outputs, and figures lie in a plane and are drawn to scale. On this page angles are in degrees unless they are marked in radians, and a formula holds wherever both sides are defined unless a condition says otherwise.

Essential Skills

Nearly every SAT question uses fractions, signs, ratios, percents or units somewhere along the way. This unit makes those steps reliable, then reads expressions precisely and turns a situation described in words into an equation.

Fractions, signs and order of operations

Work in this order. Grouping symbols first, then powers and roots, then multiplication and division from left to right, and last addition and subtraction from left to right. A fraction bar groups its whole numerator and its whole denominator. An exponent applies only to the symbol or group just before it, so $(-3)^2 = 9$ while $-3^2 = -9$.

Division by zero is undefined, while $\frac{0}{b} = 0$ for $b \ne 0$. A product or quotient of two numbers with the same sign is positive, and with opposite signs it is negative. Subtracting a negative number adds its size, as in $5 - (-2) = 7$. For example

Lessons Fractions, signs and order of operations

Ratios, proportions and rates

A ratio $a : b$ compares two quantities, and multiplying both parts by the same positive number gives an equivalent ratio. Read whether a ratio compares part to part or part to whole. If boys and girls in a class are in the ratio $3 : 5$, then the boys are $\frac{3}{8}$ of the class, not $\frac{3}{5}$. With $40$ students that means $15$ boys and $25$ girls.

A proportion states that two ratios are equal. For $b \ne 0$ and $d \ne 0$,

So $\frac{3}{8} = \frac{x}{56}$ gives $8x = 168$ and $x = 21$. Two quantities are directly proportional when $y = kx$ for a constant $k \ne 0$, so the quotient $\frac{y}{x}$ stays equal to $k$ for every $x \ne 0$ and the graph is a line through the origin. A unit rate is the amount per one unit of the second quantity, such as $12 for $4$ pounds, which is $3 per pound.

Lessons Ratios and proportions

Percents and percent change

A percent is a number out of $100$, so $p\%$ of a quantity $Q$ is $\frac{p}{100} \cdot Q$. The percent change from an original value $a \ne 0$ to a new value $b$ is

always divided by the original value. A change of $r$ percent is a multiplication. An increase by $r\%$ multiplies by $1 + \frac{r}{100}$, and a decrease by $r\%$ multiplies by $1 - \frac{r}{100}$.

Successive percent changes multiply, so equal increases and decreases do not cancel. To undo a change, divide by its multiplier rather than subtracting the same percent. A percentage point is a difference of two percentages. A rise from $40\%$ to $46\%$ is $6$ percentage points, but a relative increase of $15\%$.

Lessons Percentages and percent change

Units and conversions

Multiply by conversion factors written as fractions equal to $1$, arranged so the unwanted units cancel. For area square the length factor, and for volume cube it.

The denominator of a rate names the reference unit. With $1$ mile $= 5280$ feet and $1$ hour $= 3600$ seconds,

Keep exact values until the last step and round only as the question asks.

Lessons Units, rates and conversions

Expressions and word problems

An expression such as $3x - 7$ has no equals sign and is simplified or evaluated, while an equation such as $3x - 7 = 11$ makes a claim that is solved. Terms are joined by addition or subtraction, factors by multiplication, and a coefficient is the number multiplying a variable. When you substitute, put the value in parentheses, so $x = -2$ in $x^2 - 3x$ gives $(-2)^2 - 3(-2) = 10$.

To write an equation from words, name each unknown with its unit, then translate whole relationships rather than single words.

For example, a gym charges a $25 fee plus $12 per month, so $m$ months cost $25 + 12m$ dollars, and a budget of at most $150 means $25 + 12m \le 150$, so $m \le \frac{125}{12} \approx 10.4$ and at most $10$ whole months. Check that an answer fits the context, since a count must be a whole number and a length cannot be negative.

Lessons Expressions and substitution, Word problems into equations

Algebra

Linear equations and linear functions are the largest single part of SAT Math. A linear relationship changes by the same amount for every unit step in the input, so its graph is a line, and almost every question here comes down to slope, an intercept, or a point that satisfies several conditions at once.

Linear equations in one variable

Adding or subtracting the same quantity on both sides, and multiplying or dividing both sides by the same nonzero number, give an equivalent equation with the same solutions. Distribute first, collect the variable terms on one side, and check the answer in the original equation. For example

Read what the question asks for. If $3x + 2 = 11$, the value of $6x + 4$ is $2 \cdot 11 = 22$ without finding $x$ first.

The equation $ax + b = cx + d$ has

So $2(x + 3) = 2x + 6$ holds for every real $x$, while $2x + 5 = 2x + 1$ has no solution. When the variable cancels, the answer is never automatically $x = 0$. A question asking for a constant that makes an equation have no solution is asking for equal coefficients of $x$ and unequal constants.

Lessons Solving linear equations, No or infinitely many solutions

Rearranging formulas

To make a variable the subject, undo the operations applied to it in reverse order. Dividing by an expression requires that expression to be nonzero, and the zero case is a separate question. For example, solving $A = \frac{1}{2}bh$ for $h$ gives

and in geometry $b > 0$ anyway, since a length is positive. Solving $y = mx + b$ for $x$ gives $x = \frac{y - b}{m}$ only when $m \ne 0$.

Lessons Rearranging formulas

Functions and graphs

A function $f$ assigns exactly one output $f(x)$ to each allowed input $x$. The symbol $f(x)$ is a value, not $f$ times $x$, and evaluating at an expression substitutes the whole expression, so for $f(x) = x^2 + 1$ the value $f(x + 1)$ is $(x + 1)^2 + 1$. The point $(a, b)$ lies on the graph of $f$ exactly when $f(a) = b$. The $y$-intercept is $(0, f(0))$, and the $x$-intercepts are the points $(r, 0)$ with $f(r) = 0$. A question asking for “the value of $x$ when $f(x) = 7$” reads the graph or table backward, from output to input.

Lessons Coordinates, functions and graphs

Slope and the forms of a line

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 \ne x_2$, is

the change in $y$ for each increase of $1$ in $x$, in units of $y$ per unit of $x$. A positive slope rises from left to right, a horizontal line has slope $0$, and a vertical line has undefined slope.

In standard form with $A \ne 0$ and $B \ne 0$, the $x$-intercept is $\left(\frac{C}{A}, 0\right)$ and the $y$-intercept is $\left(0, \frac{C}{B}\right)$. For example the points $(1, 2)$ and $(4, 11)$ give $m = \frac{9}{3} = 3$, and then $2 = 3 + b$ gives $y = 3x - 1$.

In a linear model $y = mx + b$ the slope is the rate of change and $b$ is the starting value, the value of $y$ when $x = 0$. If a plumber charges $C = 45h + 60$ dollars for $h$ hours, then $45 is the charge per hour and $60 is the fixed charge. A model is valid only on the inputs that make sense in the situation.

Lessons Slope and rate of change, Intercepts and linear models, Writing the equation of a line

Parallel and perpendicular lines

Two distinct nonvertical lines are parallel exactly when their slopes are equal. Two lines with slopes $m_1$ and $m_2$ are perpendicular exactly when

which needs both slopes defined and nonzero. A horizontal line and a vertical line are also perpendicular. Lines with equal slopes and equal intercepts are the same line, not two parallel lines. For example, the line through $(4, 1)$ perpendicular to $y = 2x + 1$ has slope $-\frac{1}{2}$, so it is $y = -\frac{1}{2}x + 3$.

Lessons Parallel and perpendicular lines

Systems of linear equations

A solution of a system is an ordered pair that satisfies every equation, and on a graph it is a point where the lines meet. Substitution solves one equation for a variable and puts that expression into the other. Elimination adds a multiple of one whole equation, both sides, to the other so that a variable cancels. For example

adds to $2x = 14$, so $x = 7$ and $y = 3$, the solution $(7, 3)$. Sometimes the question asks for a combination such as $x + y$, which one addition or subtraction of the equations may give directly.

For the system $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$,

For example $kx + 3y = 6$ and $2x + y = 5$ have no solution exactly when $k = 6$, because then the first equation is $6x + 3y = 6$ while three times the second is $6x + 3y = 15$. Equal coefficient ratios alone do not separate the last two cases, so compare the constants too, and avoid ratio tests that divide by a zero coefficient.

Lessons Graphs and substitution, Elimination, One, no or infinite solutions

Linear inequalities

Adding the same number to both sides of an inequality, or multiplying or dividing both sides by a positive number, keeps its direction. Multiplying or dividing by a negative number reverses it. So $-3x + 5 > 11$ gives $-3x > 6$ and $x < -2$. A strict inequality ($<$ or $>$) excludes its boundary, drawn as a hollow dot or a dashed line, and an inclusive one ($\le$ or $\ge$) includes it, drawn as a filled dot or a solid line.

The graph of $y > mx + b$ is the half-plane above the line $y = mx + b$, and $y < mx + b$ is the half-plane below it. When the inequality is not solved for $y$, test a point not on the boundary line, such as $(0, 0)$. The solution set of a system of inequalities is the region where every inequality holds at the same time, and a point on a dashed boundary is not in it.

Lessons Solving linear inequalities, Graphing inequalities

Advanced Math

Advanced Math is the nonlinear algebra on the test. The key skill is choosing the form of an expression that answers the question, such as factored form for zeros, vertex form for a maximum or minimum, and the exponential form $a \cdot b^t$ for a growth rate, while keeping track of every value a denominator or a root rules out.

Exponent rules

For a positive integer $n$ the power $a^n$ is the product of $n$ factors of $a$. A negative exponent means a reciprocal, never a negative number.

These rules hold for all real exponents when the bases are positive, and for integer exponents with any nonzero bases. They apply to products, never to sums, since $(a + b)^2 \ne a^2 + b^2$ in general. For example $2^3 \cdot 2^{-5} = 2^{-2} = \frac{1}{4}$, and $\frac{x^7}{x^3} = x^4$ for $x \ne 0$, where the restriction stays after canceling.

Lessons Exponent rules

Radicals and rational exponents

For $a \ge 0$ the principal square root $\sqrt{a}$ is the number $b \ge 0$ with $b^2 = a$. An even root needs a nonnegative radicand and is never negative, while an odd root accepts every real number, as in $\sqrt[3]{-8} = -2$.

For example $\sqrt{50} = \sqrt{25}\,\sqrt{2} = 5\sqrt{2}$, $8^{2/3} = 4$ and $27^{-1/3} = \frac{1}{3}$. The equation $x^2 = 9$ has the two solutions $\pm 3$, but $\sqrt{9} = 3$ only.

Lessons Radicals and rational exponents

Polynomials and factoring

To multiply polynomials, multiply every term of one by every term of the other, then combine like terms. For all real $a$ and $b$,

A sum of squares $x^2 + c^2$ with $c \ne 0$ does not factor over the real numbers. To factor, take out the greatest common factor first, then look for these patterns. A trinomial $x^2 + bx + c$ factors as $(x + p)(x + q)$ when $p + q = b$ and $pq = c$, so $x^2 - 5x + 6 = (x - 2)(x - 3)$. Grouping handles four terms, as in $x^3 + 2x^2 - 9x - 18 = (x + 2)(x - 3)(x + 3)$. Look for structure too, since $x^4 - 16$ is a difference of squares in $x^2$ and factors as $(x^2 + 4)(x - 2)(x + 2)$. Two polynomials are equal for every $x$ exactly when their coefficients of each power match, which is how questions with unknown constants such as $(2x + k)^2 = 4x^2 + 12x + 9$ are solved, here with $k = 3$. Expand a factored answer to check it.

Lessons Expanding polynomials, Factoring

Solving quadratic equations

A quadratic equation can be written as $ax^2 + bx + c = 0$ with $a \ne 0$. The zero product property says that $AB = 0$ exactly when $A = 0$ or $B = 0$, so move everything to one side before factoring. For example $x^2 = 5x$ gives $x(x - 5) = 0$, so $x = 0$ or $x = 5$, and dividing both sides by $x$ would lose the solution $0$. The square root method gives $z^2 = c$ the two solutions $z = \pm\sqrt{c}$ when $c > 0$, one when $c = 0$, and none when $c < 0$.

The quadratic formula gives the real solutions of $ax^2 + bx + c = 0$, for $a \ne 0$ and $b^2 - 4ac \ge 0$,

with the whole numerator divided by $2a$. The discriminant $D = b^2 - 4ac$ tells how many real solutions there are.

For example $2x^2 - 4x - 3 = 0$ has $D = 16 + 24 = 40$, so

Completing the square rewrites $x^2 + bx$ as $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$. With a leading coefficient other than $1$, factor it out of the $x$ terms first.

Lessons Solving by factoring, Completing the square, The quadratic formula

Quadratic graphs and their forms

The graph of $y = ax^2 + bx + c$ with $a \ne 0$ is a parabola that opens up when $a > 0$ and down when $a < 0$. Its three common forms describe the same graph.

The vertex has

Watch the sign inside the square, since $(x + 3)^2$ means $h = -3$. When the zeros are real, the axis of symmetry lies halfway between them, at $x = \frac{r_1 + r_2}{2}$. If $a > 0$ the minimum value is $k$, and if $a < 0$ the maximum value is $k$, both attained at $x = h$. On a restricted domain, such as a time interval, check the endpoints as well. For example

has vertex $(3, -4)$, minimum value $-4$, $y$-intercept $(0, 5)$ and $x$-intercepts $(1, 0)$ and $(5, 0)$.

Lessons Quadratic graphs

Sum and product of the roots

If $a \ne 0$ and $r_1$ and $r_2$ are the solutions of $ax^2 + bx + c = 0$, counted twice when repeated, then

Read $a$, $b$ and $c$ from the standard form with $0$ on one side. For example $3x^2 - 12x + 7 = 0$ has two real roots, since $D = 60 > 0$, with sum $4$ and product $\frac{7}{3}$.

A parameter in a quadratic usually enters through the discriminant. The equation $x^2 + kx + 9 = 0$ has exactly one real solution when $k^2 - 36 = 0$, that is when $k = 6$ or $k = -6$. If the parameter multiplies $x^2$, check separately the value that makes the leading coefficient $0$, because the equation is then linear and none of these formulas apply.

Lessons Parameters and root relationships

Polynomial zeros and graphs

For a polynomial $p$, the number $c$ is a zero exactly when $x - c$ is a factor, and dividing $p(x)$ by $x - c$ leaves the remainder $p(c)$. So $p(x) = x^3 - 2x^2 + x - 5$ leaves the remainder $p(2) = -3$, and $x - 2$ is not a factor. A real zero is an $x$-intercept of the graph. A zero $c$ of multiplicity $m$, meaning the factor $x - c$ appears exactly $m$ times, is crossed by the graph when $m$ is odd and touched when $m$ is even. For example $y = (x + 2)(x - 1)^2$ crosses the $x$-axis at $-2$ and touches it at $1$. The zeros alone do not fix the whole graph, since multiplying by a nonzero constant keeps them.

Lessons Polynomial zeros and graph features

Rational and radical equations

A rational expression is undefined wherever its original denominator is $0$, and those values stay excluded after any simplification. Cancel common factors, never terms. For example

so the graph is the line $y = x + 3$ with a hole at $(3, 6)$.

To solve a rational equation, list the excluded values, multiply both sides by the common denominator, solve, and reject any candidate that makes an original denominator $0$. For example

leads to $x = 2 + 3(x - 2)$ and the single candidate $x = 2$, which is excluded, so the equation has no solution.

To solve a radical equation, isolate the radical, square both sides, and check every candidate in the original equation, because squaring can add extraneous solutions. For example $\sqrt{2x + 3} = x$ squares to $x^2 - 2x - 3 = 0$, with candidates $3$ and $-1$. Only $x = 3$ works, since $x = -1$ gives $1$ on the left and $-1$ on the right.

Lessons Rational expressions, Rational equations, Radical equations

Absolute value

The absolute value $\lvert x \rvert$ is the distance from $x$ to $0$, so it is never negative, and $\lvert x - a \rvert$ is the distance between $x$ and $a$. For a number $d$,

For example $\lvert 2x - 3 \rvert = 7$ gives $2x - 3 = 7$ or $2x - 3 = -7$, so $x = 5$ or $x = -2$, and $\lvert x - 4 \rvert < 3$ gives $1 < x < 7$. When the other side contains $x$, check each candidate, because that side must not be negative. The graph of $y = a\lvert x - h \rvert + k$ with $a \ne 0$ is a V with vertex $(h, k)$, opening up when $a > 0$.

Lessons Absolute value equations and graphs

Systems with lines and curves

Solve a system of a line and a curve by substituting the line into the curve, then find both coordinates of each solution. For example $y = x^2$ and $y = 2x + 3$ give $x^2 - 2x - 3 = 0$, so $x = 3$ or $x = -1$, and the solutions are $(3, 9)$ and $(-1, 1)$. When the substitution leads to a quadratic, its discriminant counts the intersections, with $D = 0$ meaning the line touches the curve at exactly one point.

Lessons Systems with lines and curves

Exponential growth and decay

An exponential model has the form

with $a \ne 0$ and a base $b > 0$, $b \ne 1$. In applications $a > 0$ is the initial value $f(0)$ and $t$ counts time intervals. Over each interval the quantity is multiplied by $b$, so it grows when $b > 1$ and decays when $0 < b < 1$.

A percent rate always belongs to a stated time interval. For example $2000 growing by $5\%$ per year is worth $2000 \cdot 1.05^3 = 2315.25$ dollars after $3$ years, and $80$ milligrams with a half-life of $6$ hours leaves $80 \cdot \left(\frac{1}{2}\right)^{18/6} = 10$ milligrams after $18$ hours. Equivalent forms show other rates, since $1.44^t = (1.2^2)^t = 1.2^{2t}$, so growth of $44\%$ every $2$ years is growth of $20\%$ per year. An exponential equation with a common base is solved by comparing exponents, as $2^{x + 1} = 32 = 2^5$ gives $x = 4$.

Linear models add the same amount over equal steps of input, and exponential models multiply by the same factor. In a table with equally spaced inputs, constant differences of outputs point to a linear model and constant ratios, with no output $0$, point to an exponential one. A short table supports a model but cannot prove it.

Lessons Exponential growth and decay, Models from tables

Shifting, reflecting and stretching graphs

Here $h$ and $k$ are real numbers and $c > 0$. The graph of each new function comes from the graph of $y = f(x)$ as follows.

A change inside the parentheses acts on the input, so it moves the graph horizontally and the opposite way to how it looks. The point $(u, f(u))$ lies on $y = f(x)$, and $y = f(x - h)$ reaches the same height at $x = u + h$. For example $y = (x + 2)^2 - 5$ is $y = x^2$ moved $2$ left and $5$ down, with vertex $(-2, -5)$. The new function is defined only where the whole new input lies in the domain of $f$.

Lessons Shifting, reflecting and stretching graphs

Problem-Solving and Data Analysis

This part of the test asks you to read data from tables and graphs, summarize it, and judge what it supports. Most errors come from using the wrong group as the total, so always ask what the denominator of a rate, percent or probability is, and how the data were collected before deciding what they show.

Data displays

A dot plot shows every value, and a histogram groups values into intervals, so it shows how many fall in each interval but not the exact values. A box plot marks the minimum, the first quartile $Q_1$, the median, the third quartile $Q_3$ and the maximum. About half of the data lie between $Q_1$ and $Q_3$, and the interquartile range is

Read every axis label and scale before comparing heights or lengths. Quartile conventions differ, so use the quartiles a question gives.

Lessons Reading data displays

Mean, median and weighted averages

The mean of $n$ values is

so the total of the values equals $n \bar{x}$, which helps when a value is added or removed. The median is the middle value of the ordered data, or the mean of the two middle values when $n$ is even. For the data $3$, $7$, $8$, $12$, $20$, $41$ the median is $\frac{8 + 12}{2} = 10$ and the mean is $\frac{91}{6} \approx 15.2$. An extreme value pulls the mean toward it but barely moves the median, so in data skewed to the right the mean is usually larger than the median.

To combine groups, weight each group mean by its size. A class of $20$ with mean $80$ and a class of $30$ with mean $70$ have the combined mean

not $75$. A frequency table works the same way, with each value weighted by its frequency.

Lessons Mean, median and weighted averages

Spread and standard deviation

The range is the maximum minus the minimum. The standard deviation measures how far the values typically lie from the mean, and it is $0$ only when all values are equal. For a population of $n$ values with mean $\mu$,

and the sample standard deviation $s$ divides by $n - 1$ instead, for $n \ge 2$. SAT questions usually ask you to compare spreads rather than compute them. Data bunched near the mean have a smaller standard deviation than data spread far from it, even with the same mean and range.

For example $2$, $4$, $4$, $4$, $5$, $5$, $7$, $9$ has mean $5$ and population standard deviation $2$. Removing an outlier usually lowers the standard deviation and the range, and it usually changes the mean more than the median.

Lessons Spread, outliers and standard deviation

Scatterplots and models

A scatterplot shows pairs of values. A line of best fit summarizes the trend without passing through every point, and its slope is the predicted change in $y$ for each increase of $1$ in $x$. The residual of a data point is

positive for a point above the model and negative for one below. For example, with the model $y = 2.5x + 10$ the point $(8, 27)$ has predicted value $30$ and residual $-3$. A prediction inside the range of the data is an interpolation, and one outside it is an extrapolation, which is less reliable the farther it goes. An association between two variables does not show that one causes the other.

Lessons Scatterplots, models and predictions

Probability rules

When all outcomes are equally likely,

and every probability lies between $0$ and $1$.

A conditional probability restricts the total to the given group, and $P(A \mid B)$ usually differs from $P(B \mid A)$. In this two-way table of $150$ students,

the probability that a grade 10 student said yes is $\frac{30}{50} = 0.6$, while the probability that a student who said yes is in grade 10 is $\frac{30}{75} = 0.4$. The probability that a student is in grade 10 or said yes is $\frac{50 + 75 - 30}{150} = \frac{19}{30}$. Treat events as independent only when the question says so or the numbers show it.

Lessons Probability and two-way tables, Conditional probability

Samples and margin of error

A sample statistic estimates a population value but is almost never exactly equal to it. Results generalize only to the population that was sampled, and only when the sample was chosen at random from it. A sample that favors some members, such as volunteers or people who are easy to reach, is biased, and a larger sample does not fix that bias.

A margin of error $E$ describes the uncertainty from random sampling, and an estimate with margin $E$ gives the interval

It is plausible that the population value lies in this interval, but it is not guaranteed, and values outside it are less plausible rather than impossible. For example a poll result of $54\%$ with a margin of $3$ percentage points gives $51\%$ to $57\%$. Under comparable conditions a larger random sample gives a smaller margin of error, and under the usual formula the margin shrinks in proportion to $\frac{1}{\sqrt{n}}$, so four times the sample size roughly halves it. A higher confidence level gives a wider interval from the same data.

Lessons Samples, populations and bias, Margin of error

Studies and conclusions

An observational study records what happens without assigning treatments, so a confounding variable can explain an association, and it cannot show cause and effect. Random sampling and random assignment answer different questions.

Lessons Experiments, observations and causal claims

Geometry and Trigonometry

SAT geometry rests on a short list of facts about angles, triangles and circles. Most questions combine two of them, such as similar triangles with a proportion, or a right triangle inside a circle, so it pays to know the facts below well enough to spot them in a figure.

Angles, lines and triangles

When a line crosses two parallel lines, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles add to $180^\circ$. These hold only when the lines are known to be parallel. In any triangle the longest side lies opposite the largest angle, and each side is shorter than the sum of the other two.

Lessons Angles, parallel lines and triangles

Similar and congruent triangles

Similar triangles have equal corresponding angles and proportional corresponding sides. Congruent triangles are similar with scale factor $1$. List the vertices in matching order so that corresponding parts line up.

Equal angles alone never prove congruence, and SSA is not a general test. A line parallel to one side of a triangle cuts off a smaller triangle similar to the whole. For example, if $\triangle ABC \sim \triangle DEF$ with $AB = 6$, $DE = 9$ and $BC = 8$, then

Lessons Similar and congruent triangles

Right triangles

In a right triangle with legs $a$ and $b$ and hypotenuse $c$, the side opposite the right angle,

Conversely, a triangle whose sides satisfy this equation has a right angle opposite $c$. Common whole-number triples are $3, 4, 5$ and $5, 12, 13$ and $8, 15, 17$, and any positive multiple of a triple is again a right triangle. The two special right triangles have fixed side ratios.

The shortest side of a $30^\circ$-$60^\circ$-$90^\circ$ triangle lies opposite the $30^\circ$ angle and is half the hypotenuse. For example a square with side $5$ has diagonal $5\sqrt{2}$, and an equilateral triangle with side $10$ has height $5\sqrt{3}$.

Lessons Pythagorean theorem and special right triangles

Sine, cosine and tangent

For an acute angle $\theta$ in a right triangle, SOH CAH TOA gives

The ratios depend only on the angle, because all right triangles with that angle are similar. The two acute angles of a right triangle are complementary, so