Mathematics Reference and Formulas Cheat Sheet

Comprehensive vocabulary flashcards covering 2D and 3D geometric formulas, coordinate geometry, algebra, exponent rules, factorials, and statistical outlier definitions from the math cheat sheet lecture notes.

Area of a Rectangle

The measure of two-dimensional space inside a rectangle, calculated as $Area = l \cdot w$ (or $b \cdot h$).

Area of a Circle

The space contained within a circle, calculated using the formula $Area = \pi r^2$.

Area of a Triangle

The two-dimensional space enclosed by a triangle, given by the formula $Area = \frac{1}{2}(bh)$ or $Area = \frac{b \cdot h}{2}$.

Area of a Trapezoid

The area calculated by averaging the parallel bases and multiplying by height: $Area = \frac{1}{2}(b_1 + b_2)h$ or $Area = \frac{b_1 + b_2}{2}h$.

Circumference of a Circle

The total distance around the outside of a circle, calculated as $C = \pi d$ or $C = 2\pi r$.

Volume of a Rectangular Prism or Cube

The three-dimensional capacity of a rectangular box or cube, calculated as $Volume = l \cdot w \cdot h$.

Volume of a Sphere

The three-dimensional space enclosed by a sphere, given by the formula $Volume = \frac{4}{3}\pi r^3$.

Volume of a Pyramid

The three-dimensional capacity of a pyramid, calculated using $Volume = \frac{1}{3} \cdot l \cdot w \cdot h$.

Volume of a Cone

The three-dimensional volume contained within a cone, given by $Volume = \frac{1}{3}\pi r^2 h$.

Volume of a Cylinder

The three-dimensional capacity of a cylinder, calculated by the formula $Volume = \pi r^2 h$.

Surface Area of a Rectangular Prism or Cube

The total exterior area of a rectangular solid, calculated as $SA = 2(lw) + 2(lh) + 2(wh)$.

Surface Area of a Sphere

The total area of the outer curved surface of a sphere, given by the formula $SA = 4\pi r^2$.

Surface Area of a Cylinder

The combined area of the circular bases and lateral curved surface: $SA = 2\pi r^2 + 2\pi rh$.

Lower Outlier Boundary (Left)

A statistical threshold for identifying extreme low values, defined by $Q_1 - (IQR \times 1.5)$.

Upper Outlier Boundary (Right)

A statistical threshold for identifying extreme high values, defined by $Q_3 + (IQR \times 1.5)$.

Speed Formula

The rate at which an object covers distance over time: $Speed = \frac{D}{t}$.

Time Formula

The duration of travel calculated by dividing distance by speed: $time = \frac{D}{s}$.

Coordinate Rotation Rules

Standard coordinate transformations about the origin, such as a $180^\circ$ rotation producing $(-x, -y)$ and a $90^\circ$ counterclockwise rotation producing $(-y, x)$.

Value of a Nickel and a Dime

Standard currency amounts where $1\text{ nickel} = 0.05$ and $1\text{ dime} = 0.10$.

Distance Formula

The formula used to compute the Euclidean distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.

Midpoint Formula

The formula that yields the coordinate halfway between two endpoints: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

Slope Formula

The steepness or rate of vertical change over horizontal change between two points: $m = \frac{y_1 - y_2}{x_1 - x_2}$.

Slope-Intercept Form

The linear equation representation $y = mx + b$, in which $m$ is the slope and $b$ is the y-intercept.

Parallel Lines (Slopes)

Lines in a plane that never intersect and possess the exact same slope.

Perpendicular Lines (Slopes)

Lines that meet at right angles, characterized by slopes that are opposite reciprocals (totally opposite).

Difference of Squares

An algebraic identity stating that the difference between two squared terms factors into $a^2 - b^2 = (a+b)(a-b)$.

Factored Form of a Quadratic Equation

A binomial product expansion formula represented by $(x+a)(x+b) = x^2 + (a+b)x + ab$.

Quotient of Powers Rule

An exponent rule stating that dividing powers with the same base subtracts the exponents: $\frac{x^a}{x^b} = x^{a-b}$.

Negative Exponent Rule

A mathematical property where a base with a negative exponent is rewritten as a reciprocal with a positive exponent, such as $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.

Zero Exponent Rule

An exponential rule establishing that any non-zero value raised to the power of zero equals one: $x^0 = 1$.

Power of a Product Rule

An algebraic rule where a product raised to an exponent equals each factor raised to that power: $(xy)^a = x^a y^a$.

Factorial ($n!$)

The product of an integer multiplied successively by all positive integers below it, illustrated by $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ and $3! = 3 \times 2 \times 1$.