Formula and notation reference

Circle

A = πr², C = 2πr

Rectangle

A = lw

Triangle

A = $\frac{\text{1}}{\text{2}}$ bh; h is perpendicular to the chosen base.

Right triangle

c² = a² + b²; c is the hypotenuse.

45°–45°–90°

Side lengths s, s, s$\sqrt{\text{2}}$; the last is the hypotenuse.

30°–60°–90°

Side lengths x, x$\sqrt{\text{3}}$, 2x, opposite the respective angles.

Rectangular prism

V = lwh

Right circular cylinder

V = πr² h

Sphere

V = $\frac{\text{4}}{\text{3}}$πr³

Right circular cone

V = $\frac{\text{1}}{\text{3}}$πr² h

Right rectangular pyramid

V = $\frac{\text{1}}{\text{3}}$ lwh

Full circle

360° = 2π radians

Triangle angle sum

180°

m = $\frac{{\text{y}}_{\text{2}}\text{−}{\text{y}}_{\text{1}}}{{\text{x}}_{\text{2}}\text{−}{\text{x}}_{\text{1}}}$

Slope; requires x₂ ≠ x₁. Use consistent subtraction order.

y = mx + b

m is slope; b is the output at x = 0.

y − y₁ = m(x − x₁)

Point-slope form for a nonvertical line.

Ax + By = C

Standard form; not both A, B zero. If B ≠ 0, slope is −A/B.

Parallel / perpendicular lines

Distinct parallel nonvertical lines have equal slopes. Perpendicular finite slopes have product −1; horizontal and vertical lines form the separate case.

x = h; y = k

Vertical line; horizontal line. Vertical slope is undefined; horizontal slope is zero.

ax + b = c

If a ≠ 0, x = (c − b)/a. If a = 0, compare b and c to distinguish no solutions from every real input.

Multiply or divide an inequality

Reverse its direction when multiplying or dividing by a negative number. Do not divide by an unknown-sign expression without analyzing cases.

Linear system solution counts

One intersection, no intersection, or the same line. Check constants as well as variable coefficients for proportionality.

$\frac{\text{a}}{\text{b}}$ + $\frac{\text{c}}{\text{d}}$ = $\frac{\text{ad + bc}}{\text{bd}}$

Requires b, d ≠ 0. A common denominator combines entire numerators.

$\frac{\text{a}}{\text{b}}$ ÷ $\frac{\text{c}}{\text{d}}$ = $\frac{\text{ad}}{\text{bc}}$

Requires b, c, d ≠ 0. Division by zero is undefined.

a(b + c) = ab + ac

Distribute to every term, including a negative sign outside the parentheses.

Scientific notation

a × 10ⁿ with 1 ≤ |a| < 10 for a nonzero number; n is an integer.

(a ± b)² = a² ± 2ab + b²

Do not omit the middle term.

a² − b² = (a − b)(a + b)

Difference of squares.

x^m xⁿ = x^m+n; (x^m)ⁿ = x^mn

For rational-exponent use without extra cases, take x > 0. Integer-exponent cases allow broader domains.

x^m/xⁿ = x^m−n; x⁻ⁿ = 1/xⁿ

Require a nonzero base and defined real expressions.

x^p/q = $\sqrt[\text{q}]{{\text{x}}^{\text{p}}}$

For a clean real-valued rule, take x > 0, integer p, and positive integer q. Odd-root negative-base cases require care.

$\sqrt{{\text{x}}^{\text{2}}}$ = |x|

The principal square root is nonnegative.

ax² + bx + c = 0

x = $\frac{\text{−b ±}$ \sqrt{{\text{b}}^{\text{2}}\text{− 4ac}}$}{\text{2a}}$

Requires a ≠ 0 to use quadratic rules.

The quadratic formula. A real root requires a nonnegative radicand.

D = b² − 4ac

D > 0: two distinct real roots; D = 0: one; D < 0: none. Restrictions may reject algebraic roots.

h = −b/(2a); k = f (h)

Vertex (h, k); minimum when a > 0, maximum when a < 0.

f(x) = a(x − h)² + k

Vertex form. Factored form a(x − r)(x − s) displays zeros when defined.

r + s = −b/a; rs = c/a

Root sum and product, including a repeated root counted twice.

|u| = d

If d > 0, solve u = d or u = −d. If d = 0, solve u = 0. If d < 0, no real solution.

Q(t) = Q₀b^t/T

b > 0 is the multiplier per interval of length T > 0. Growth p% gives b = 1 + p/100; decay gives b = 1 − p/100 > 0.

g(x) = a f (b(x − h)) + k

For a, b ≠ 0, a point (u, v) maps to (h + u/b, av + k). Check the new domain as well as the shape.

Vertical / supplementary angles

Vertical angles are equal. A linear pair sums to 180°. Parallel-line angle rules require parallel lines.

Triangle relationships

Exterior angle equals the sum of the remote interior angles. Equal sides face equal angles. Similarity needs justified matching angles or side ratios.

sin θ = $\frac{\text{opposite}}{\text{hypotenuse}}$

Right triangle, acute angle θ; label sides relative to that angle.

cos θ = $\frac{\text{adjacent}}{\text{hypotenuse}}$

tan θ = $\frac{\text{opposite}}{\text{adjacent}}$.

sin θ = cos(90° − θ)

Complementary acute-angle relationship.

rθ; $\frac{\text{1}}{\text{2}}$ r²θ

Arc length; sector area. These versions require θ in radians.

$\frac{\text{θ}}{\text{360°}}$(2πr); $\frac{\text{θ}}{\text{360°}}$(πr²)

Degree versions of arc length and sector area.

Inscribed angle / tangent

An inscribed angle is half its intercepted arc measure. A tangent is perpendicular to the radius at the tangency point.

(x − h)² + (y − k)² = r²

Circle center (h, k) and radius r > 0. Complete both squares if needed.

d = $\sqrt{\text{(}{\text{x}}_{\text{2}}\text{−}{\text{x}}_{\text{1}}{\text{)}}^{\text{2}}\text{+ (}{\text{y}}_{\text{2}}\text{−}{\text{y}}_{\text{1}}{\text{)}}^{\text{2}}}$

Coordinate distance; derived from a right triangle.

Midpoint ($\frac{{\text{x}}_{\text{1}}\text{+}{\text{x}}_{\text{2}}}{\text{2}}$, $\frac{{\text{y}}_{\text{1}}\text{+}{\text{y}}_{\text{2}}}{\text{2}}$)

Average coordinates separately.

Similar figures: length factor s > 0

Perimeter factor s; area factor s²; volume factor s³. Only use the rule after similarity is established.

Parallelogram A = bh

Use perpendicular height, not a slanted side.

Trapezoid A = $\frac{\text{1}}{\text{2}}$(b₁ + b₂)h

b₁, b₂ are the parallel bases.

Prism V = Bh; pyramid V = Bh/3

B is base area, not a base edge. The general forms extend the supplied rectangular-base cases.

Surface area

Add exposed faces only; remove shared interior faces. For a sphere, S = 4πr².

rate = quantity/time

Average speed is total distance divided by total time, not generally the average of speeds.

Ratio a : b

Represent matching quantities as ak, bk for a common scale factor k.

Unit conversion

Multiply by conversion fractions equal to 1. Square or cube the factor for area or volume units.

percent change = $\frac{\text{new − old}}{\text{old}}$ 100%

Uses the original nonzero base. A positive change is an increase; a negative one is a decrease.

New = old ×(1 ± p/100)

Multiply successive factors; divide by the factor to reverse a change.

Percentage points

The arithmetic difference of percentages. Not the same as relative percent change.

$\overline{x}$ = $\frac{\text{∑}{\text{x}}_{\text{i}}}{\text{n}}$

Mean; total = n$\overline{x}$. Count frequencies when data are in a table.

Combined mean = $\frac{\sum n_j\overline{x_j}}{\sum n_j}$

Pool group totals and counts. Equal weighting of group means requires equal group sizes.

Median / range / IQR

Median is the middle of sorted data; range is maximum minus minimum; IQR is Q₃ − Q₁.

y_i = ax_i + b

Mean becomes a$\overline{x}$ + b; standard deviation becomes |a| times the old one. Adding b alone does not change spread.

Residual = observed − predicted

For a fitted model, positive residual means the observation is above its prediction.

P(A | B) = $\frac{\text{P(A ∩ B)}}{\text{P(B)}}$

Requires P(B) > 0. In a count table, restrict the denominator to B.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

“At least one”; subtract the overlap counted twice.

P(A^c) = 1 − P(A)

Complement: outcomes not in A.

Independent events

P(A ∩ B) = P(A)P(B). Sampling without replacement usually changes the second conditional probability.

Sample to population

Estimate a population count with population size × sample proportion when the sampling method supports that inference. An estimate is not an exact census.

Estimate ± margin

Reported interval endpoints are estimate minus margin and estimate plus margin. Match fraction units or percentage-point units.

Larger random samples

With method, confidence level, and other relevant factors comparable, larger samples generally yield smaller margins of error. They do not automatically remove selection bias.

Random sampling

Helps justify generalizing to the population sampled, subject to study quality and uncertainty.

Random assignment

Helps justify a causal treatment comparison in a well-conducted experiment. It does not automatically make a volunteer sample representative.

Association

Two variables vary together. This alone does not identify a cause; confounding or reverse direction may be possible.

f(x)

Output of function f at input x, not the product f · x. f (a + h) changes the input; f (a) + h changes the output.

(x, y)

Ordered pair: input/horizontal coordinate first, output/vertical coordinate second.

= versus ≈

Exact equality versus approximation. Preserve exact expressions until final rounding is required.

≠, <, ≤

Not equal; strictly less; less than or equal. Equality is included only in the last relation.

[a, b], (a, b)

As intervals, square brackets include endpoints and parentheses exclude them. Context distinguishes an open interval from an ordered pair.

∩ and ∪

Intersection (both) and union (at least one).

√, |x|, π

Principal nonnegative square root; distance of x from zero; the circle constant.

$\hat{y}$, $\overline{x}$

A model’s predicted value; the mean of the data values.