# Formulas

A collection of math formulas for lookup and reference. Each formula contains an explanation of how the formula is used, what different variables represent within the formula and examples of the formula in action.

The angle between two vectors can be calculated using the arc-cosine of their dot product divided by the product of their magnitudes.

To calculate the angle between two vectors (2D) you can use the arc-tangent 2 function.

The area between two vectors in 2D is given by the magnitude of their cross product. This formula is a generic way to find the area of any triangle given three points.

The area of a circle is given by one-half multiplied by τ (tau) multiplied by the radius of the circle squared.

The traditional formula for the area of a circle is given in terms of the geometric constant π (pi).

The area of any parallelogram is equal to the base multiplied by the height.

The Area of a rectangle is given by its width multiplied by its height.

The area of a trapezoid is given by its height multiplied by the sum of its top length and bottom length divided by two.

The area of a triangle is given by one-half multiplied by its width and height.

The circumference of a circle is given the constant τ (tau) multiplied by the radius of the circle, where τ = 2π.

The circumference of a circle is given the constant π (pi) multiplied by two times the radius of the circle.

The combination formula describes the possible combinations of r elements out of a group of n elements where order does not matter.

The compound interest formula calculates the growth of an initial value whose interest compounds over time. The frequency of when the interest is calculated and added to the initial amount can occur continuously,

The conditional probability formula shows how to calculate the probability of an event B, given that another event A has already occurred.

The cross product of two vectors can be calculated using the formal determinant.

The formula for the determinant of a two by two matrix.

The formula for the determinant of a three by three matrix.

The distance between two points, in one dimension, is given by the absolute value of the difference between the two values.

The distance between two points, in two dimensions, is given by solving Pythagorean's theorem for the length of the hypotenuse of the right triangle formed by the two points.

The dot product of two vectors is calculated by summing together the product of corresponding elements.

The dot product can be geometrically interpreted as the magnitude of the two vectors multiplied by the cosine of the angle between them.

Euler's Formula returns the point on the unit circle in the complex plane when given an angle.

The expected value, describes the most likely value of a probability distribution. It also describes where a probability distribution is centered.

To calculate the expected value of a discrete distribution, multiply the events of the distribution by the probability of the event occuring.

The magnitude of the cross product can be given as the magnitude of the two vectors multiplied by the sine of the angles between them.

The magnitude of a vector is given by the square root of the sum of its components squared.

The x-coordinate of the mid-point is calculated by averaging the x-coordinates of the two end points. The y-coordinate is calculated by averaging the y-coordinates of the two points.

The permutation formula describes the possible permutations of r elements out of a group of n elements where order does matter.

The number of permutations of n distinct items is given by n factorial. A permutation is a unique ordering or arrangement of the set of items.

To convert a point from polar coordinates to cartesian coordinates, the trigonometric functions cosine and sine can be used.

The population growth formula models the exponential growth of a function. Note, that this formula models unbounded population growth. For bounded growth, see logistic growth.

The population mean formula gives the average value of the whole population.

The quadratic formula solves for the x-intercepts of a two-degree polynomial (quadratic) equation.

The sample mean, also called the arithmetic mean, is the average of a sample space. To calculate the sample mean, sum all the data points in a sample space and then divide by the number of elements.

The sample standard deviation formula calculates the standard deviation of a sample population. The sample deviation is denoted with the Latin letter s, where the population standard deviation is denoted by the greek lower-case sigma symbol.

The slope of a line is calculated by finding the change in y over the change in x using two points on the line.

The standard deviation formula calculates the standard deviation of population data. The standard deviation value is denoted by the symbol σ (sigma) and measures how far the data is distributed around the population's mean.

The volume of a cone is given by one-third multiplied by PI, the radius of its base squared, and its height.

The volume of a cube is given by the length of the cube length raised to the third power.

The volume of a cylinder is given by 1/2 τ (tau) multiplied by the radius squared and height.

The volume of a cylinder is equal to PI multiplied by its radius squared and its height.

To calculate the volume of a rectangular prism multiply its height, width, and length together.

The volume of a sphere is given by two-thirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.