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formula section

SAT Math formulas, organized by pattern

Search 67 high-value formulas, learn when each one applies, and see which basics are already on the Bluebook reference sheet. The real advantage is not memorizing a wall of symbols. It is recognizing the question pattern that calls for each formula.

9 skill groups67 formulasPattern Master links included

1. Recognize

Search the quantity or question type, not only the formula name.

2. Match

Choose the formula whose inputs and units match what the question gives.

3. Verify

Check units, scale, and the exact quantity the stem asks you to report.

Showing 67 of 67 formulas

linear equations and functions

Linear equations and functions

Rates, lines, and systems form the highest-return foundation of SAT Math.

Linear equation

Derive quickly
ax+b=c⟹x=c−baax+b=c\quad\Longrightarrow\quad x=\frac{c-b}{a}

Isolate one variable when a is not zero.

Learn the question pattern

Slope

Know cold
m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Find vertical change per horizontal unit between two points.

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Point-slope form

Know cold
y−y1=m(x−x1)y-y_1=m(x-x_1)

Write a line from one point and a slope.

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Average rate of change

Know cold
f(b)−f(a)b−a\frac{f(b)-f(a)}{b-a}

Compare output change across an interval.

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Parallel and perpendicular slopes

Know cold
m∥=m,m⊥=−1mm_{\parallel}=m,\qquad m_{\perp}=-\frac{1}{m}

Build a related line through a new point.

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System solution count

Derive quickly
m1=m2, b1≠b2⇒0 solutions;same line⇒∞ solutionsm_1=m_2,\ b_1\ne b_2\Rightarrow 0\text{ solutions};\qquad \text{same line}\Rightarrow\infty\text{ solutions}

Classify a system without solving it.

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quadratics and nonlinear equations

Quadratics and nonlinear equations

Use the form that exposes the feature the question asks for.

Quadratic formula

Know cold
x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Solve ax^2 + bx + c = 0 when factoring is not efficient.

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Discriminant

Know cold
D=b2−4acD=b^2-4ac

D < 0 gives 0 real roots, D = 0 gives 1, and D > 0 gives 2.

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Axis of symmetry

Know cold
x=−b2ax=-\frac{b}{2a}

Find the horizontal location of a quadratic's vertex.

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Factored form

Derive quickly
y=a(x−r1)(x−r2)y=a(x-r_1)(x-r_2)

Read zeros r1 and r2 directly.

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Root sum and product

Know cold
r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1r_2=\frac{c}{a}

Answer root relationship questions without solving.

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Absolute value split

Derive quickly
∣u∣=c⟹u=c or u=−c|u|=c\quad\Longrightarrow\quad u=c\ \text{or}\ u=-c

Solve absolute-value equations when c is nonnegative.

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exponents and factoring

Exponents and factoring

These identities reveal the hidden familiar expression in hard questions.

Product of powers

Know cold
am⋅an=am+na^m\cdot a^n=a^{m+n}

Combine like bases under multiplication.

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Quotient of powers

Know cold
aman=am−n\frac{a^m}{a^n}=a^{m-n}

Combine like bases under division.

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Negative and fractional exponents

Know cold
a−n=1an,a1/n=ana^{-n}=\frac{1}{a^n},\qquad a^{1/n}=\sqrt[n]{a}

Rewrite reciprocals and radicals.

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Difference of squares

Know cold
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Factor a subtraction of two perfect squares.

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Perfect-square trinomial

Know cold
a2±2ab+b2=(a±b)2a^2\pm2ab+b^2=(a\pm b)^2

Recognize a repeated linear factor.

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Difference of cubes

Know cold
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factor two perfect cubes being subtracted.

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Sum of cubes

Know cold
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factor two perfect cubes being added.

Learn the question pattern

exponential change and interest

Exponential change and interest

Repeated percent change uses multipliers, not repeated addition.

Exponential growth

Know cold
A(t)=A0(1+r)tA(t)=A_0(1+r)^t

Model repeated growth by rate r per period.

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Exponential decay

Know cold
A(t)=A0(1−r)tA(t)=A_0(1-r)^t

Model repeated decay by rate r per period.

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Compound interest

Know cold
A=P(1+rn)ntA=P\left(1+\frac{r}{n}\right)^{nt}

Model interest compounded n times per year.

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Simple interest

Know cold
A=P(1+rt)A = P(1 + rt)

Model interest applied only to the original principal.

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Doubling time

Know cold
A(t)=A0⋅2t/dA(t)=A_0\cdot2^{t/d}

Model a quantity that doubles every d time units.

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Half-life

Know cold
A(t)=A0(12)t/hA(t)=A_0\left(\frac{1}{2}\right)^{t/h}

Model a quantity that halves every h time units.

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ratios, rates, percentages, and units

Ratios, rates, percentages, and units

Set the reference quantity and make units cancel before calculating.

Proportion

Derive quickly
ab=cd⟹ad=bc\frac{a}{b}=\frac{c}{d}\quad\Longrightarrow\quad ad=bc

Solve equivalent ratios.

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Density

Derive quickly
density=amountspace\text{density}=\frac{\text{amount}}{\text{space}}

Relate mass to volume, population to area, or another per-unit measure.

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Percent of a whole

Derive quickly
part=percent100⋅whole\text{part}=\frac{\text{percent}}{100}\cdot\text{whole}

Find a portion or reverse to find the whole.

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Percent change

Know cold
new−oldold⋅100%\frac{\text{new}-\text{old}}{\text{old}}\cdot100\%

Measure change relative to the original value.

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Growth and decay multipliers

Know cold
growth: 1+r;decay: 1−r\text{growth}:\ 1+r;\qquad \text{decay}:\ 1-r

Apply or reverse a percent increase or decrease.

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data and statistics

Data and statistics

Know what each statistic measures and how transformations affect it.

Mean

Derive quickly
xˉ=∑i=1nxin\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}

Find an arithmetic average or reconstruct a total.

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Weighted mean

Know cold
xˉw=∑i=1nwixi∑i=1nwi\bar{x}_w=\frac{\sum_{i=1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i}

Average values with unequal importance or frequency.

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Range

Derive quickly
range=xmax⁡−xmin⁡\text{range}=x_{\max}-x_{\min}

Measure the full spread of a data set.

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Interquartile range

Know cold
IQR=Q3−Q1\mathrm{IQR}=Q_3-Q_1

Measure the spread of the middle half.

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Outlier fences

Know cold
Q1−1.5(IQR),Q3+1.5(IQR)Q_1-1.5(\mathrm{IQR}),\qquad Q_3+1.5(\mathrm{IQR})

Flag unusually low or high values.

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Residual

Know cold
residual=y−y^=observed−predicted\text{residual}=y-\hat{y}=\text{observed}-\text{predicted}

Measure a model's signed prediction error.

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Confidence interval

Derive quickly
estimate±margin of error\text{estimate}\pm\text{margin of error}

Find a plausible range for a population value.

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probability

Probability

Define the sample space first, especially when the question adds a condition.

Basic probability

Derive quickly
P(A)=favorable outcomestotal outcomesP(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}

Find the chance of one event in an equally likely sample space.

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Complement

Know cold
P(Ac)=1−P(A)P(A^c)=1-P(A)

Solve not or at-least-one questions through the opposite event.

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Addition rule

Know cold
P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)

Combine events without double-counting their overlap.

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Conditional probability

Know cold
P(A∣B)=P(A∩B)P(B)P(A\mid B)=\frac{P(A\cap B)}{P(B)}

Restrict the denominator to cases where B occurred.

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Independent events

Know cold
P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)

Multiply when one event does not change the other.

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coordinate geometry and trigonometry

Coordinate geometry and trigonometry

Label the points or triangle before selecting a relationship.

Distance

Know cold
d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Find the straight-line distance between two points.

Learn the question pattern

Midpoint

Know cold
M=(x1+x22,y1+y22)M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Find the point halfway between two endpoints.

Learn the question pattern

Pythagorean theorem

Provided in Bluebook
a2+b2=c2a^2 + b^2 = c^2

Relate the legs and hypotenuse of a right triangle.

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45-45-90 triangle

Provided in Bluebook
x:x:x2x:x:x\sqrt{2}

Relate sides in an isosceles right triangle.

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30-60-90 triangle

Provided in Bluebook
x:x3:2xx:x\sqrt{3}:2x

Relate short leg, long leg, and hypotenuse.

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Trig ratios

Know cold
sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta=\frac{\mathrm{opp}}{\mathrm{hyp}},\quad \cos\theta=\frac{\mathrm{adj}}{\mathrm{hyp}},\quad \tan\theta=\frac{\mathrm{opp}}{\mathrm{adj}}

Connect an acute angle with two right-triangle sides.

Learn the question pattern

Complementary trig

Know cold
sin⁡θ=cos⁡(90∘−θ)\sin\theta=\cos(90^{\circ}-\theta)

Switch between complementary acute angles.

Learn the question pattern

geometry, area, volume, and circles

Geometry, area, volume, and circles

The Bluebook reference sheet provides many basics, but recognition and scale rules still matter.

Circle area and circumference

Provided in Bluebook
A=πr2,C=2πrA=\pi r^2,\qquad C=2\pi r

Use radius, not diameter, in both formulas.

Learn the question pattern

Prism and cylinder volume

Provided in Bluebook
V=Bh,Vcylinder=πr2hV=Bh,\qquad V_{\mathrm{cylinder}}=\pi r^2h

Multiply constant base area by perpendicular height.

Learn the question pattern

Pyramid and cone volume

Provided in Bluebook
V=13Bh,Vcone=13πr2hV=\frac{1}{3}Bh,\qquad V_{\mathrm{cone}}=\frac{1}{3}\pi r^2h

Use one third of the matching prism or cylinder.

Learn the question pattern

Scale factors

Know cold
length:k,area:k2,volume:k3\text{length}:k,\qquad \text{area}:k^2,\qquad \text{volume}:k^3

Scale similar figures in the correct dimension.

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Polygon interior sum

Know cold
S=(n−2)⋅180∘S=(n-2)\cdot180^{\circ}

Find the total interior-angle measure of an n-gon.

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Circle equation

Know cold
(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Read center (h, k) and radius r.

Learn the question pattern

Radians and arc length

Know cold
θrad=θdeg⋅π180,s=rθ\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\cdot\frac{\pi}{180},\qquad s=r\theta

Convert angle units and find arc length.

Learn the question pattern

Inscribed angle

Know cold
m∠=12m(intercepted arc^)m\angle=\frac{1}{2}m(\widehat{\text{intercepted arc}})

Connect an angle on a circle to its intercepted arc.

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Turn formulas into recognition

Pattern Master teaches the clue, the move, the hard-version disguise, and the traps behind every Math family.

Open Pattern Master

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