Math Formula Sheet

A comprehensive collection of mathematical formulas and equations for quick reference.

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solves quadratic equations in the form ax² + bx + c = 0

Algebra

Pythagorean Theorem

a2+b2=c2a^2 + b^2 = c^2

Relates the lengths of the sides of a right triangle

Geometry

Area of a Circle

A=πr2A = \pi r^2

Calculates the area of a circle with radius r

Geometry

Volume of a Sphere

V=43πr3V = \frac{4}{3}\pi r^3

Calculates the volume of a sphere with radius r

Geometry

Derivative Power Rule

ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}

Basic rule for finding the derivative of a power function

Calculus

Integration Power Rule

∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C

Basic rule for integrating a power function

Calculus

Exponential Function

f(x)=exf(x) = e^x

The natural exponential function

Algebra

Natural Logarithm

ln⁡(ex)=x\ln(e^x) = x

The natural logarithm function

Algebra

Sine and Cosine

sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1

Fundamental trigonometric identity

Algebra

Volume of a Cylinder

V=πr2hV = \pi r^2h

Calculates the volume of a cylinder with radius r and height h

Geometry

Chain Rule

ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)

Rule for finding the derivative of a composite function

Calculus

Integration by Parts

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du

Formula for integrating products of functions

Calculus

Arithmetic Sequence

an=a1+(n−1)da_n = a_1 + (n-1)d

nth term of an arithmetic sequence with first term a₁ and common difference d

Algebra

Geometric Sequence

an=a1rn−1a_n = a_1r^{n-1}

nth term of a geometric sequence with first term a₁ and common ratio r

Algebra

Arithmetic Series Sum

Sn=n2(a1+an)=n2[2a1+(n−1)d]S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]

Sum of n terms of an arithmetic sequence

Algebra

Geometric Series Sum

Sn=a11−rn1−rS_n = a_1\frac{1-r^n}{1-r}

Sum of n terms of a geometric sequence where r ≠ 1

Algebra

Surface Area of Sphere

A=4πr2A = 4\pi r^2

Calculates the surface area of a sphere with radius r

Geometry

Volume of Cone

V=13πr2hV = \frac{1}{3}\pi r^2h

Calculates the volume of a cone with radius r and height h

Geometry

Area of Triangle

A=12bh=12absin⁡(C)A = \frac{1}{2}bh = \frac{1}{2}ab\sin(C)

Area using base and height, or two sides and included angle

Geometry

Product Rule

ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Rule for finding the derivative of a product of functions

Calculus

Quotient Rule

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Rule for finding the derivative of a quotient of functions

Calculus

Trigonometric Derivatives

ddxsin⁡(x)=cos⁡(x),ddxcos⁡(x)=−sin⁡(x)\frac{d}{dx}\sin(x) = \cos(x), \frac{d}{dx}\cos(x) = -\sin(x)

Basic derivatives of sine and cosine functions

Calculus

Distance Formula

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Distance between two points in a plane

Geometry

Midpoint Formula

(x1+x22,y1+y22)(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})

Coordinates of the midpoint between two points

Geometry

Slope Formula

m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}

Slope of a line through two points

Algebra

Compound Interest

A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}

Amount after t years with principal P, rate r, compounded n times per year

Algebra