9713. Common Math Formulas
Math Formulas

Common math formulas.

1. Geometry

Shape	Formula	Explanation
Square	$P = 4 a$	where `a` = any edge
Rectangle	$P = 2 l + 2 w$	where `l` = length and `w` = width
Triangle	$P = a + b + c$	where `a` = side, `b` = base, and `c` = side
Circle	$P = π d$ or $P = 2 π r$	where $π$ = 3.14, `d` = diameter and `r` = radius

Shape	Formula	Explanation
Square	$A = a^{2}$	where `a` = any side of the square
Rectangle	$A = l w$	where `l` = length and `w` = width
Parallelogram	$A = b h$	where `b` = base and `h` = height
Triangle	$A = \frac{1}{2} b h$	where `b` = base and `h` = height
Triangle	$A = \| \frac{(A_{x} (B_{y} - C_{y}) + B_{x} (C_{y} - A_{y}) + C_{x} (A_{y} - B_{y})}{2} \|$	where ( $A_{x}$ , $A_{y}$ ) are the `x` and `y` coordinates of the point `A`, etc.
Circle	$A = π r^{2}$	where $π$ = 3.14 and `r` = radius
Trapezoid	$A = \frac{a + b}{2} h$	where `a` = top base, `b` = bottom base, and `h` = height
Sphere	$S = 4 π r^{2}$	where `S` = surface area, $π$ = 3.14 and `r` = radius
Cube	$S = 6 a^{2}$	where `a` = any edge
Cylinder	$S = 2 π r h$	where $π$ = 3.14, `r` = radius, and `h` = height

Shape	Formula	Explanation
Cube	$V = a^{3}$	where `a` = any edge
Rectangular Container	$V = l w h$	where `l` = length, `w` = width, and `h` = height
Square Pyramid	$V = \frac{1}{3} b^{2} h$	where `b` = base length, `h` = height
Cylinder	$V = π r^{2} h$	where $π$ = 3.14, `r` = radius, and `h` = height
Cone	$V = \frac{1}{3} π r^{2} h$	where $π$ = 3.14, `r` = radius, and `h` = height
Sphere	$V = \frac{4}{3} π r^{3}$	where $π$ = 3.14, `r` = radius
Right Circular Cylinder	$V = π r^{2} h$	where $π$ = 3.14, `r` = radius, and `h` = height

Function	Formula
Sine	$\sin θ = \frac{o p p o s i t e}{h y p o t e n u s e}$
Cosine	$\cos θ = \frac{a d j a c e n t}{h y p o t e n u s e}$
Tangent	$\tan θ = \frac{o p p o s i t e}{a d j a c e n t}$ , or $\tan θ = \frac{\sin θ}{\cos θ}$
Cosecant	$\csc θ = \frac{1}{\sin θ}$
Secant	$\sec θ = \frac{1}{\cos θ}$
Cotangent	$\cot θ = \frac{1}{\tan θ}$ , or $\cot θ = \frac{\cos θ}{\sin θ}$
Equation	$\sin^{2} θ + \cos^{2} θ = 1$

Title	Formula	Explanation
Distance between two points	$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$	where ( $x_{1}$ , $y_{1}$ ) and ( $x_{2}$ , $y_{2}$ ) are two points on a coordinate plane
Slope of a line	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$	where ( $x_{1}$ , $y_{1}$ ) and ( $x_{2}$ , $y_{2}$ ) are two points on a coordinate plane
Equation of a line	$y = m x + b$	where `m` is the slope and `b` is the y-intercept
Quadratic Equation	$a x^{2} + b x + c = 0$	where `a` and `b` are coefficients and `c` is constant
Quadratic formula	$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$	where `a` and `b` are coefficients and `c` is constant
Equation of a circle	$(x - h)^{2} + (y - k)^{2} = r^{2}$	where `r` is the radius and `(h, k)` is the center
Logarithm Equation	$\log_{b} x = y$ , $b^{y} = x$
Logarithm Equation	$l o g_{b} x y = l o g_{b} x + l o g_{b} y$
Logarithm Equation	$l o g_{b} \frac{x}{y} = l o g_{b} x - l o g_{b} y$

Title	Formula	Explanation
Product Rule	$a^{n} \times a^{m} = a^{n + m}$	where `a` is the base, `n` and `m` are the exponents
Power Rule	$(a^{n})^{m} = a^{n m}$	where `a` is the base, `n` and `m` are the exponents
Quotient Rule	$\frac{a^{n}}{a^{m}} = a^{n - m}$	where `a` is the base, `n` and `m` are the exponents
Negative Exponent	$a^{- n} = \frac{1}{a^{n}}$	where `a` is the base, `n` is the exponent

S_{n} = 1 + 2 + 3 + . . . + n

then reverse the series and rewrite that as

S_{n} = n + (n - 1) + (n - 2) + . . . + 1

Adding the two together

2 S_{n} = n (n + 1)

S_{n} = \frac{n (n + 1)}{2}

S_{n} = 2^{0} + 2^{1} + 2^{2} + . . . + 2^{n} = 2^{n + 1} - 1

Proofs: Look at these values in binary way.

Example 1:

S_{3} = 2^{0} + 2^{1} + 2^{2} + 2^{3} = 1111 (B i n a r y) = 2^{3 + 1} - 1

Example 2:

S_{5} = 2^{0} + 2^{1} + 2^{2} + . . . + 2^{5} = 111111 (B i n a r y) = 2^{6 + 1} - 1

Permutation:

P (n, r) = \frac{n!}{(n - r)!}

Example: Choose 2 numbers from array [1,2,3,4], return the total number of all possible permutations.

P (4, 2) = \frac{4!}{(4 - 2)!} = \frac{4!}{(2)!} = \frac{24}{2} = 12

Combination:

C (n, r) = \frac{n!}{r! (n - r)!}

Example: Choose 2 numbers from array [1,2,3,4], return the total number of all possible combinations.

C (4, 2) = \frac{4!}{2! (4 - 2)!} = \frac{4!}{2! (2!)} = \frac{24}{2 * 2} = 6