SAT Math formulas: reference sheet and what to learn

01Use the reference sheet as a reference

College Board's Bluebook reference-sheet example includes rectangle, triangle, and circle area; circle circumference; rectangular-solid, cylinder, sphere, cone, and rectangular-pyramid volume; the Pythagorean theorem; and the two special right triangles. It also gives basic circle and triangle angle facts.

You can look up a supplied formula, but you still have to identify its variables. In $A=\pi r^2$, r is radius. In $A=\frac{1}{2}bh$, h is perpendicular height. A formula list can't make that choice for you.

Practice with the reference open at first. Once you know where things are, try a few questions without looking. Memorization is helpful when it removes a lookup; using the right dimensions is what makes the result correct.

02Learn linear relationships and their units

Slope is $m=\frac{y_2-y_1}{x_2-x_1}$ when $x_2\ne x_1$. A line can be written as $y=mx+b$ or $y-y_1=m(x-x_1)$. Parallel nonvertical lines share a slope; perpendicular nonvertical lines have slopes whose product is $-1$.

For points $(2,5)$ and $(6,13)$, $m=(13-5)/(6-2)=2$. Substitution gives $5=2(2)+b$, so $b=1$ and the line is $y=2x+1$.

Distance is $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, and midpoint is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. For $(1,2)$ and $(7,10)$, the distance is 10 and midpoint is $(4,6)$. The distance formula is the Pythagorean theorem applied to coordinate differences.

03Recognize each quadratic form

Standard form is $y=a{x}^2+bx+c$. Vertex form is $y=a(x-h{)}^2+k$, with vertex $(h,k)$. Factored form is $y=a(x-r_1)(x-r_2)$, making the roots visible.

The quadratic formula is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ for $a\ne 0$. The discriminant $b^2-4ac$ is positive for two distinct real roots, zero for one repeated real root, and negative for no real roots.

For $x^2-6x+9=0$, the discriminant is zero and the root is 3. You could also factor $(x-3{)}^2=0$. Choose the form that answers the question rather than forcing the quadratic formula onto every quadratic.

04Turn percent and rate statements into models

Percent change is $\frac{\text{new}-\text{old}}{\text{old}}\cdot 100\%$. Growth uses $V=V_0(1+r{)}^t$ and decay uses $V=V_0(1-r{)}^t$, when r is a decimal rate per unit of t.

A value falling 15% each year has multiplier 0.85. Starting at 200, after two years it is $200(0.85{)}^2=144.5$. The exponent counts applications of the multiplier.

For constant speed, $d=rt$. For a data set, $\text{mean}=\text{sum}/\text{count}$. For equally likely outcomes, $P=\text{favorable}/\text{eligible}$. Label the units and the eligible group; formulas with vague variable definitions invite the wrong denominator.

05Separate a constant change from a repeated percentage

An account gaining 20 dollars each month can be modeled by $A=500+20t$. An account gaining 4% each month uses $A=500(1.04{)}^t$. After two months, the first has 540 dollars and the second has 540.80 dollars.

The difference comes from the second percentage applying to the updated amount. A stated percentage rate needs its time unit, and a fixed addition needs its units too. Don't choose an exponential model just because money appears in the question.

06Add the circle and trig relationships

A coordinate circle uses $(x-h{)}^2+(y-k{)}^2=r^2$. Arc length is $\frac{\theta }{360^{\circ }}\cdot 2\pi r$ in degrees, or $r\theta$ in radians. Sector area is $\frac{\theta }{360^{\circ }}\cdot \pi r^2$ in degrees, or $\frac{1}{2}{r}^2\theta$ in radians.

In a right triangle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. The acute angles are complementary, so $\sin \theta =\cos (90^{\circ }-\theta )$.

For radius 6 and central angle $\pi /3$, the arc length is $2\pi$ and sector area is $6\pi$. That example needs both the correct formula and the radian interpretation of the angle.

07Make a formula useful by attaching a question

For each formula you review, solve one problem that uses it and one where it would be the wrong choice. Compare a right triangle with a non-right triangle, or percent of with percent greater than. Those pairs test recognition.

In the Math bank, record whether you forgot the formula, mislabeled a quantity, or made an arithmetic error. Use the Desmos guide for graphing checks, but keep exact algebra when a question asks about a parameter or an exact expression. Knowing when a formula applies is part of learning it.