Secondary math formulas (grades 5–11)

Quick reference for middle and high school. Browse by topic or search by name.

Identities & factoring

Square of a sum

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
Identities & factoring

Square of a difference

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
Identities & factoring

Difference of squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
Identities & factoring

Cube of a sum

(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Linear equations

Linear equation in one variable

ax+b=0  ⇒  x=−baax + b = 0 \;\Rightarrow\; x = -\frac{b}{a}
a≠0a \neq 0
Linear equations

Proportion

ab=cd  ⇔  ad=bc\frac{a}{b} = \frac{c}{d} \;\Leftrightarrow\; ad = bc
b and d must be nonzero.
Quadratic equations

Standard form

ax2+bx+c=0ax^2 + bx + c = 0
a≠0a \neq 0
Quadratic equations

Discriminant

D=b2−4acD = b^2 - 4ac
D > 0: two roots; D = 0: one; D < 0: no real roots.
Quadratic equations

Quadratic formula

x=−b±D2ax = \frac{-b \pm \sqrt{D}}{2a}
Quadratic equations

Vieta’s formulas

x1+x2=−bax_1 + x_2 = -\frac{b}{a}
x1⋅x2=cax_1 \cdot x_2 = \frac{c}{a}
For x2+px+q=0x^2 + px + q = 0: x1+x2=−p,  x1x2=qx_1 + x_2 = -p,\; x_1 x_2 = q.
Powers & roots

Product of powers (same base)

am⋅an=am+na^m \cdot a^n = a^{m+n}
Powers & roots

Quotient of powers (same base)

aman=am−n\frac{a^m}{a^n} = a^{m-n}
Powers & roots

Power of a power

(am)n=amn(a^m)^n = a^{mn}
Powers & roots

Square root of a product

ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
For a≥0a \geq 0 and b≥0b \geq 0.
Functions

Linear function

y=kx+by = kx + b
k is the slope, b is the y-intercept.
Functions

Quadratic function

y=ax2+bx+cy = ax^2 + bx + c
a≠0a \neq 0
Functions

Parabola vertex

xv=−b2ax_v = -\frac{b}{2a}
yv=f(xv)y_v = f(x_v)
Plane geometry

Pythagorean theorem

c2=a2+b2c^2 = a^2 + b^2
In a right triangle, c is the hypotenuse.
Plane geometry

Circumference

C=2πRC = 2\pi R
Plane geometry

Area of a circle

S=πR2S = \pi R^2
Plane geometry

Trapezoid area

S=a+b2⋅hS = \frac{a + b}{2} \cdot h
a and b are the bases, h is the height.
Solid geometry

Volume of a rectangular prism

V=abcV = abc
Solid geometry

Volume of a cylinder

V=πR2hV = \pi R^2 h
Solid geometry

Volume of a cone

V=13πR2hV = \frac{1}{3} \pi R^2 h
Solid geometry

Volume of a sphere

V=43πR3V = \frac{4}{3} \pi R^3
Trigonometry

Sine of an acute angle

sin⁡α=ac\sin \alpha = \frac{a}{c}
a is the opposite leg, c is the hypotenuse.
Trigonometry

Cosine of an acute angle

cos⁡α=bc\cos \alpha = \frac{b}{c}
Trigonometry

Tangent of an acute angle

tg⁡α=ab=sin⁡αcos⁡α\tg \alpha = \frac{a}{b} = \frac{\sin \alpha}{\cos \alpha}
Trigonometry

Pythagorean trigonometric identity

sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1
Sequences & progressions

nth term of an arithmetic progression

an=a1+d(n−1)a_n = a_1 + d(n - 1)
d is the common difference.
Sequences & progressions

Sum of n terms (arithmetic)

Sn=(a1+an)⋅n2S_n = \frac{(a_1 + a_n) \cdot n}{2}
Sequences & progressions

nth term of a geometric progression

bn=b1⋅qn−1b_n = b_1 \cdot q^{n-1}
q is the common ratio, q≠0q \neq 0.
Sequences & progressions

Sum of n terms (geometric)

Sn=b1⋅qn−1q−1S_n = b_1 \cdot \frac{q^n - 1}{q - 1}
q≠1q \neq 1
Coordinate plane

Distance between two points

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

Midpoint of a segment

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