SAT Math Formula Reference (All Domains)

SAT Math Formula Reference

A complete reference of the formulas worth knowing for the Digital SAT Math section. The left column is the formula in LaTeX; the right column is a quick way to understand or remember it.

Note on the reference sheet: The Digital SAT provides a small set of geometry formulas (areas, circumference, volumes, special right triangles, the fact that a circle has $360^{\circ }=2\pi$ radians, and a triangle’s angles sum to $180^{\circ }$). Everything else below must be memorized. Formulas the test gives you are marked [given].

Status: verification: pending — every formula in this file should be independently checked by a separate math AI agent before publishing.

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1. Algebra (Linear Equations, Lines, Systems)

Formula	How to Understand / Remember
$m=\frac{y_2-y_1}{x_2-x_1}$	Slope = rise over run: change in $y$ divided by change in $x$. Keep the points in the same order top and bottom.
$y=mx+b$	Slope-intercept form. $m$ tilts the line, $b$ is where it bumps into the $y$-axis.
$y-y_1=m(x-x_1)$	Point-slope form. Use it the instant you know one point and a slope — no need to solve for $b$.
$Ax+By=C$	Standard form. Handy for intercepts: set $x=0$ for the $y$-intercept, $y=0$ for the $x$-intercept.
$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$	Distance = Pythagorean theorem on the plane: the segment is the hypotenuse of a right triangle with legs $\Delta x$ and $\Delta y$.
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$	Midpoint = just average the two $x$‘s and the two $y$‘s.
$m_1=m_2$	Parallel lines have equal slopes (same steepness, never meet).
$m_1\cdot m_2=-1$	Perpendicular slopes are negative reciprocals: flip and change the sign.
$y=kx$	Direct variation: $y$ and $x$ rise together; $k$ is the constant ratio $y/x$.
$y=\frac{k}{x}$	Inverse variation: one goes up, the other goes down; the product $xy=k$ stays constant.
$|x|=a\Rightarrow x=a\text{ or }x=-a$	Absolute value = distance from 0, so it opens into two cases, $+a$ and $-a$.

Linear system — number of solutions (lines $y=m_{1}x+b_1$, $y=m_{2}x+b_2$):

Condition	Solutions	Memory
$m_1\ne m_2$	Exactly one	Different slopes → lines cross once.
$m_1=m_2,b_1\ne b_2$	None	Same slope, different intercept → parallel, never meet.
$m_1=m_2,b_1=b_2$	Infinitely many	Same line written twice.

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2. Advanced Math (Quadratics, Polynomials, Exponents, Functions)

Formula	How to Understand / Remember
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$	The quadratic formula for $a{x}^2+bx+c=0$. Sing it: ”$x$ equals minus $b$, plus or minus root $b$ squared minus $4ac$, all over $2a$.”
$\Delta =b^2-4ac$	The discriminant. $\Delta >0$: two real roots; $\Delta =0$: one (repeated); $\Delta <0$: no real roots. It is the part under the root sign.
$y=a(x-h{)}^2+k$	Vertex form. The vertex is $(h,k)$ — note the sign flip on $h$. $a$ controls width and direction (up if $a>0$).
$x=-\frac{b}{2a}$	Axis of symmetry / vertex $x$-coordinate of $y=a{x}^2+bx+c$. It is the quadratic formula without the root part (the center between the two roots).
$x_1+x_2=-\frac{b}{a},x_1{x}_2=\frac{c}{a}$	Vieta’s: sum of roots $=-b/a$, product $=c/a$. Lets you find/check roots without solving.
$a^2-b^2=(a+b)(a-b)$	Difference of squares. Spot two perfect squares with a minus between them.
$a^2\pm 2ab+b^2=(a\pm b{)}^2$	Perfect-square trinomial. The middle term is twice the product of the two square roots.
$a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$	Sum/difference of cubes. Mnemonic SOAP: Same sign, Opposite sign, Always Positive.
$x^a\cdot x^b=x^{a+b}$	Multiply same base → add exponents.
$\frac{x^a}{x^b}=x^{a-b}$	Divide same base → subtract exponents.
$(x^a{)}^b=x^{ab}$	Power of a power → multiply exponents.
$x^{-a}=\frac{1}{x^a},x^0=1$	Negative exponent = reciprocal; anything (nonzero) to the 0 power is 1.
$x^{a/b}=\sqrt[b]{x^a}={\left(\sqrt[b]{x}\right)}^a$	Fractional exponent: denominator is the root, numerator is the power.
$\sqrt{ab}=\sqrt{a}\sqrt{b},\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$	Roots split over multiplication and division (but not over $+$ or $-$).
$y=a(1+r{)}^t\text{(growth)},y=a(1-r{)}^t\text{(decay)}$	Exponential change: start $a$, rate $r$ per period, $t$ periods. Plus grows, minus decays.
$A=P{\left(1+\frac{r}{n}\right)}^{nt}$	Compound interest: rate split into $n$ periods/year, compounded $nt$ times. More compounding $n$ → slightly more money.
${\log }_b(x)=y⟺b^y=x$	A log is an exponent: “to what power must $b$ go to give $x$?”

Function transformations of $y=f(x)$:

Transformation	Effect	Memory
$f(x)+k$	Shift up $k$	Outside the function → vertical, and intuitive direction.
$f(x)-k$	Shift down $k$
$f(x+h)$	Shift left $h$	Inside the function → horizontal and opposite of the sign.
$f(x-h)$	Shift right $h$
$-f(x)$	Reflect over the $x$-axis	Negative outside flips it upside down.
$f(-x)$	Reflect over the $y$-axis	Negative inside flips it left-right.
$af(x),|a|>1$	Vertical stretch	Big multiplier → taller/skinnier.
$af(x),0<|a|<1$	Vertical shrink	Small multiplier → shorter/wider.

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3. Problem-Solving and Data Analysis

Formula	How to Understand / Remember
$\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\%$	Always divide the change by the original (old) amount. Positive = increase, negative = decrease.
$\text{part}=\text{percent}\times \text{whole}$	”Percent of” means multiply. “What percent” → solve $\text{percent}=\text{part}/\text{whole}$.
$\text{new}=\text{old}\times (1+r)$	Increase by $r$ → multiply by $1+r$. A 20% raise is $\times 1.20$.
$\text{new}=\text{old}\times (1-r)$	Decrease by $r$ → multiply by $1-r$. A 20% discount is $\times 0.80$.
$\text{mean}=\frac{\sum x}{n}$	Average = total of values divided by how many there are.
median	Middle value when sorted (average the two middles if $n$ is even). Resistant to outliers.
mode	The value that appears most often.
range $=\max -\min$	Spread from smallest to largest.
$P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}$	Probability is a fraction of the whole, always between 0 and 1.
$P(\text{not }A)=1-P(A)$	Complement: everything else adds up to 1.
$\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc$	Proportions → cross-multiply. The backbone of ratio, scale, and unit-rate problems.
$\text{average speed}=\frac{\text{total distance}}{\text{total time}}$	Use totals, not the average of two speeds (a classic trap).
$\text{quantity}\times \frac{\text{new unit}}{\text{old unit}}$	Unit conversion: multiply by a fraction equal to 1 so the old units cancel.

Concept-only (no plug-in formula needed on the SAT): A larger standard deviation means data is more spread out from the mean. A larger or more random sample generally lowers the margin of error. Over equal steps, a linear model adds a constant amount; an exponential model multiplies by a constant factor.

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4. Geometry and Trigonometry

4a. Area and Perimeter

Formula	How to Understand / Remember
$A=\ell w,P=2(\ell +w)$	Rectangle: area is length times width; perimeter walks all four sides.
$A=\frac{1}{2}bh$ [given]	Triangle: half of the base-times-height “box” it fits inside. $h$ is perpendicular to the base.
$A=bh$	Parallelogram: a “leaned” rectangle, still base times perpendicular height.
$A=\frac{1}{2}(b_1+b_2)h$	Trapezoid: average the two parallel sides, then times the height.
$A=\pi r^2$ [given]	Circle area. “Area is $\pi r$-squared.”
$C=2\pi r=\pi d$ [given]	Circumference. Diameter $d=2r$, so both forms match.

4b. Circles, Arcs, and Sectors

Formula	How to Understand / Remember
$(x-h{)}^2+(y-k{)}^2=r^2$	Equation of a circle: center $(h,k)$ (sign flip), radius $r$ (note $r^2$ on the right).
$\text{arc length}=\frac{\theta }{360^{\circ }}\cdot 2\pi r$	An arc is a fraction of the full circumference — the fraction is the angle out of $360^{\circ }$.
$\text{sector area}=\frac{\theta }{360^{\circ }}\cdot \pi r^2$	A sector is the same fraction of the full area. (Same fraction idea as arc length.)
$s=r\theta ,A_{\text{sector}}=\frac{1}{2}{r}^2\theta$	When $\theta$ is in radians: arc length $=r\theta$; sector area $=\frac{1}{2}{r}^2\theta$.

4c. Right Triangles and Trigonometry

Formula	How to Understand / Remember
$a^2+b^2=c^2$	Pythagorean theorem: the two legs squared add to the hypotenuse squared ($c$ is always opposite the right angle).
$45^{\circ }\text{-}45^{\circ }\text{-}90^{\circ }:x,x,x\sqrt{2}$ [given]	Isosceles right triangle: two equal legs $x$, hypotenuse is leg $\times \sqrt{2}$.
$30^{\circ }\text{-}60^{\circ }\text{-}90^{\circ }:x,x\sqrt{3},2x$ [given]	Short leg $x$ (opposite $30^{\circ }$), long leg $x\sqrt{3}$ (opposite $60^{\circ }$), hypotenuse $2x$ (opposite $90^{\circ }$).
$\sin \theta =\frac{\text{opp}}{\text{hyp}},\cos \theta =\frac{\text{adj}}{\text{hyp}},\tan \theta =\frac{\text{opp}}{\text{adj}}$	SOH-CAH-TOA. Each ratio is named opposite / adjacent / hypotenuse relative to the angle $\theta$.
$\tan \theta =\frac{\sin \theta }{\cos \theta }$	Tangent is sine over cosine (opp/hyp ÷ adj/hyp = opp/adj).
${\sin }^2\theta +{\cos }^2\theta =1$	Pythagorean identity — it is $a^2+b^2=c^2$ on the unit circle (hypotenuse 1).
$\sin \theta =\cos (90^{\circ }-\theta )$	Cofunction rule: sine of an angle = cosine of its complement (that’s what “co” means).
$\text{radians}=\text{degrees}\times \frac{\pi }{180^{\circ }}$	Convert by multiplying by $\frac{\pi }{180}$; go back with $\frac{180}{\pi }$. Half-circle $180^{\circ }=\pi$.

4d. Volume and Surface Area

Formula	How to Understand / Remember
$V=\ell wh$ [given]	Rectangular box: area of the base $(\ell w)$ stacked $h$ high.
$V=\pi r^{2}h$ [given]	Cylinder: circle area $(\pi r^2)$ stacked $h$ high.
$V=\frac{4}{3}\pi r^3$ [given]	Sphere. “Four-thirds pi r-cubed.”
$V=\frac{1}{3}\pi r^{2}h$ [given]	Cone = one-third of the cylinder with the same base and height.
$V=\frac{1}{3}Bh$ [given]	Pyramid = one-third of the prism with the same base area $B$ and height.
$SA=4\pi r^2$	Sphere surface area — exactly 4 times the area of its great circle ($\pi r^2$).
$SA=2\pi r^2+2\pi rh$	Cylinder surface = two circle caps $+$ the side “label” (a rectangle of width $2\pi r$, height $h$).

4e. Angles and Similarity

Formula	How to Understand / Remember
angles of a triangle sum to $180^{\circ }$	Always — split any triangle’s corners and they form a straight line.
interior angles of an $n$-gon $=(n-2)\cdot 180^{\circ }$	A polygon splits into $(n-2)$ triangles, each contributing $180^{\circ }$.
complementary $=90^{\circ }$, supplementary $=180^{\circ }$	“Complementary” comes before “Supplementary” in the alphabet, just as $90<180$.
similar triangles: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$	Same shape, scaled: all corresponding sides share one ratio $k$.
area ratio $=k^2$, volume ratio $=k^3$	If lengths scale by $k$, areas scale by $k^2$ and volumes by $k^3$.

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Related Entries

• Linear_Equations · Quadratic_Equations · Quadratic_Functions_Parabolas · Exponential_Functions_Logarithms
• Right_Triangles_Pythagorean · Trigonometric_Functions · Properties_Circles · Volume_Surface_Area · Coordinate_Geometry
• Percentages_Percent_Change · Statistical_Measures · Probability

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All formulas above are pending independent verification by a separate math AI agent (verification: pending).