107 formulas · 4 official domains

Digital SAT Math Formula Sheet

Every rule you need, grouped the way the exam is built — with a plain-language note on when to use it and a Desmos shortcut where the calculator is faster than algebra.

107 shown

Algebra

Lines, slopes, systems and inequalities — the highest-frequency domain.

Lines & slope

Slope from two points
m = $\frac{y2 − y1}{x2 − x1}$
Rise over run; negative slope falls left to right.
Worked example
Points (1, 2) and (4, 11): m = $\frac{11 − 2}{(4}$ − 1) = $\frac{9}{3}$ = 3.
Slope-intercept form
y = mx + b
b is the y-intercept, the value when x = 0.
Desmos: Type y=mx+b and add sliders for m and b to see the line move.
Worked example
y = −2x + 5 has slope −2 and crosses the y − axis at (0, 5).
Point-slope form
y − y1 = m(x − x1)
Fastest when you know one point and the slope.
Standard form
Ax + By = C
Slope = -A/B; x-intercept = C/A; y-intercept = C/B.
Parallel / perpendicular
m1 = m2 ; m1 · m2 = −1
Perpendicular slopes are opposite reciprocals.
Midpoint
M = ( $\frac{x1 + x2}{2}$ , $\frac{y1 + y2}{2)}$
Worked example
(2, 3) and (8, −1): midpoint = ( $\frac{2 + 8}{2}$ , (3 + (−1))/2) = (5, 1).
Distance
d = $\sqrt{(x2 − x1)^2 + (y2 − y1)^2}$
Worked example
(1, 1) to (4, 5): d = $\sqrt{3^2 + 4^2}$ = $\sqrt{25}$ = 5.

Systems of equations

One solution
a1/a2 ≠ b1/b2
Different slopes — the lines cross exactly once.
No solution
a1/a2 = b1/b2 ≠ c1/c2
Same slope, different intercept — parallel lines.
Infinitely many
a1/a2 = b1/b2 = c1/c2
The two equations are multiples of each other.
Solve by graphing
y1(x) = y2(x)
Desmos: Graph both equations and click the intersection point to read it.

Inequalities & absolute value

Flip rule
if c < 0 then a < b = > ac > bc
Multiplying or dividing by a negative reverses the sign.
Absolute value equation
|ax + b| = c = > ax + b = c or ax + b = −c
Absolute value inequality
|x − a| < b = > a − b < x < a + b
Distance from a is less than b.

Expressions & word models

Linear growth model
y = (start) + (rate)x
Constant amount added per unit — always linear.
Direct proportion
y = kx
Inverse proportion
y = $\frac{k}{x}$

Advanced Math

Quadratics, polynomials, exponentials, radicals and rational functions.

Quadratics

Standard form
y = ax² + bx + c
c is the y-intercept; a > 0 opens up, a < 0 opens down.
Vertex form
y = a(x − h)² + k
Vertex is (h, k) — best form for max/min questions.
Worked example
y = 2(x − 3)² + 1 has vertex (3, 1) and opens upward.
Factored form
y = a(x − r1)(x − r2)
r1 and r2 are the x-intercepts (zeros).
Axis of symmetry / vertex x
x = $\frac{−b}{2a}$
Also the average of the two roots.
Worked example
y = x² − 6x + 2: x = $\frac{−(−6)}{(2}$ · 1) = 3.
Quadratic formula
x = (−b +- $\sqrt{b^2 − 4ac}$ )/(2a)
Worked example
x² − 5x + 6 = 0: x = (5 ± $\sqrt{25 − 24}$ )/2, so x = 3 or x = 2.
Discriminant
D = b² − 4ac
D > 0 two real roots; D = 0 one; D < 0 none.
Worked example
x² + 4x + 7: b² − 4ac = 16 − 28 = −12 < 0, so no real solutions.
Sum & product of roots
r1 + r2 = $\frac{−b}{a}$ ; r1 · r2 = $\frac{c}{a}$
Desmos: Graph the parabola and click the vertex and zeros to read exact values.

Exponents & radicals

Product rule
x^a · x^b = x^(a + b)
Quotient rule
x^ $\frac{a}{x}$ ^b = x^(a − b)
Power of a power
(x^a)^b = x^(ab)
Negative exponent
x^(−a) = $\frac{1}{x}$ ^a
Zero exponent
x⁰ = 1
Rational exponent
x^( $\frac{a}{b}$ ) = root(b, x^a)
Radical product
$\sqrt{a}$ · $\sqrt{b}$ = $\sqrt{ab}$

Exponential growth & decay

Growth / decay model
y = a(1 + r)^t
r > 0 growth, r < 0 decay; a is the initial amount.
Worked example
An account of 500 dollars growing 4% per year for 3 years: 500(1.04)³ ≈ 562.43.
Compounded n times per year
A = P(1 + $\frac{r}{n}$ )^(nt)
Half-life form
y = a( $\frac{1}{2)}$ ^( $\frac{t}{h}$ )
h is the half-life in the same units as t.
Percent change per period
b = 1 + r
Base b > 1 grows, 0 < b < 1 decays.

Polynomials & rational functions

Factor theorem
P(r) = 0 ≤ > (x − r) is a factor
Remainder theorem
P(x)/(x − r) leaves remainder P(r)
Difference of squares
a² − b² = (a + b)(a − b)
Perfect square
a² +- 2ab + b² = (a +- b)²
Sum/difference of cubes
a³ +- b³ = (a +- b)(a² -+ ab + b²)
Vertical asymptote
denominator = 0
Provided that factor does not cancel — a canceled factor is a hole.
Horizontal asymptote
y = $\frac{\text{leading coeff top}}{\text{leading coeff bottom}}$
Only when the degrees are equal; if top degree is smaller, y = 0.

Function behaviour

Shift up / down
f(x) + k
Shift left / right
f(x − h)
Positive h shifts right.
Vertical stretch
a · f(x)
Reflection over x-axis
−f(x)
Composition
(f o g)(x) = f(g(x))
Average rate of change
(f(b) − f(a))/(b − a)

Problem-Solving and Data Analysis

Ratios, percents, units, statistics and study design.

Percents

Percent of a number
part = ( $\frac{p}{100)}$ · whole
Percent change
(new - old)/old · 100
Increase is positive, decrease is negative.
Worked example
Price rises from 80 to 92: $\frac{92 − 80}{80}$ · 100 = 15% increase.
Successive changes
final = start · (1 + r1)(1 + r2)
Never just add the percents.
Percent-of-percent
a% of b% = $\frac{ab}{10000}$

Ratios, rates & units

Proportion
$\frac{a}{b}$ = $\frac{c}{d}$ = > ad = bc
Unit rate
rate = quantity/time
Distance
d = rt
Worked example
(1, 1) to (4, 5): d = $\sqrt{3^2 + 4^2}$ = $\sqrt{25}$ = 5.
Work / density style
total = rate · amount
Unit conversion
value · $\frac{\text{target unit}}{\text{given unit}}$
Cancel units diagonally so only the target unit remains.
Scale factor for area / volume
k² ; k³

Statistics

Mean
mean = $\frac{\text{sum of values}}{n}$
Worked example
Scores 70, 80, 90,100: mean = $\frac{340}{4}$ = 85.
Total from mean
sum = mean · n
Median
middle value of an ordered list
Average the two middle values when n is even.
Range
max - min
Standard deviation
spread about the mean
Tighter clustering means smaller standard deviation; no computation is required.
Weighted mean
$\frac{n1 \cdot m1 + n2 \cdot m2}{n1 + n2}$

Probability & two-way tables

Simple probability
P = $\frac{\text{favorable}}{\text{total}}$
Complement
P(not A) = 1 − P(A)
Conditional probability
P(A given B) = $\frac{A \text{and }B}{\text{total }B}$
Read the row or column total the condition names.
Worked example
12 of 40 students take art; 3 of those 12 take music: P(music | art) = $\frac{3}{12}$ = $\frac{1}{4}$ .
Independent events
P(A and B) = P(A) · P(B)

Study design & inference

Margin of error interval
estimate +- margin
Larger random samples shrink the margin of error.
Generalising results
random sample = > population it was drawn from
Causation
randomised controlled experiment only
Observational data supports association, never cause.
Line of best fit
y = mx + b (predicted)
Desmos: Paste the table and use y1 ~ mx1 + b to regress it.

Geometry and Trigonometry

Angles, triangles, circles, solids and right-triangle trig.

Angles & lines

Straight line
sum of angles = 180 deg
Around a point
sum of angles = 360 deg
Parallel lines cut by a transversal
corresponding = equal ; co-interior = 180 deg
Triangle angle sum
A + B + C = 180 deg
Worked example
Angles 50 deg and 65 deg: third angle = 180 − 115 = 65 deg.
Exterior angle
exterior = sum of remote interior
Polygon interior angle sum
(n − 2) · 180 deg
Worked example
Hexagon: (6 − 2) · 180 = 720 deg.

Triangles

Area
A = $\frac{(1}{2}$ )bh
Worked example
Base 10, height 4: A = $\frac{(1}{2)}$ (10)(4) = 20.
Pythagorean theorem
a² + b² = c²
Worked example
Legs 6 and 8: c = $\sqrt{36 + 64}$ = 10.
Common triples
3 − 4 − 5 ; 5 − 12 − 13 ; 8 − 15 − 17 ; 7 − 24 − 25
Multiples of these appear constantly — memorise them.
45-45-90
s , s , s · $\sqrt{2}$
30-60-90
x , x · $\sqrt{3}$ , 2x
Similar triangles
corresponding sides in proportion
Equal angles guarantee similarity; set up a ratio.

Circles

Area
A = pi · r²
Worked example
Base 10, height 4: A = $\frac{(1}{2)}$ (10)(4) = 20.
Circumference
C = 2 · pi · r
Equation of a circle
(x − h)² + (y − k)² = r²
Center (h, k), radius r; complete the square to get here.
Arc length
s = (theta/360) · 2 · pi · r
Sector area
A = (theta/360) · pi · r²
Radians
s = r · theta
Degrees to radians
rad = deg · pi/180
Central vs inscribed angle
central = 2 · inscribed
Both must open onto the same arc.

Area, perimeter & solids

Rectangle
A = l · w
Parallelogram
A = bh
Trapezoid
A = $\frac{(1}{2)}$ (b1 + b2)h
Rectangular solid
V = l · w · h
Cylinder
V = pi · r² · h
Cone
V = $\frac{(1}{3)}$ · pi · r² · h
Sphere
V = $\frac{(4}{3)}$ · pi · r³
Pyramid
V = $\frac{(1}{3)}$ · l · w · h
Cylinder surface area
SA = 2 · pi · r² + 2 · pi · r · h

Right-triangle trigonometry

Sine
sin(theta) = opposite/hypotenuse
Cosine
cos(theta) = adjacent/hypotenuse
Tangent
tan(theta) = opposite/adjacent
Complementary identity
sin(theta) = cos(90 deg - theta)
Pythagorean identity
sin²(theta) + cos²(theta) = 1
Tangent as a ratio
tan(theta) = sin(theta)/cos(theta)

Algebra

📈Lines & slope

🔀Systems of equations

⚖️Inequalities & absolute value

🧮Expressions & word models

Advanced Math

🅿️Quadratics

⚡Exponents & radicals

🌱Exponential growth & decay

🧩Polynomials & rational functions

🔧Function behaviour

Problem-Solving and Data Analysis

％Percents

🔁Ratios, rates & units

📊Statistics

🎲Probability & two-way tables

🔬Study design & inference