17 formulas

8th Grade Math Formula Sheet

These are the formulas 8th grade math uses: slope and linear equations, exponents and scientific notation, square roots and the number system, the Pythagorean theorem, and the volume of cylinders, cones and spheres.

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet for a test.

Search every formula

Slope and linear equations

Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Calculates the slope (steepness) of a line between two points.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator
Slope-Intercept Form
$y = mx + b$
The equation of a line where m is the slope and b is the y-intercept.
$m$
Rise over run, the steepness of the line
$b$
Where the line crosses the y-axis
$x$
The independent variable
$y$
The dependent variable
Worked example and calculator
Direct Variation
$y = k x$
y varies directly with x: as x increases, y increases by a constant factor k.
$k$
The fixed ratio y/x
$x$
The independent variable
Worked example and calculator
Distance = Rate × Time
$d = r t$
Distance traveled equals speed (rate) times time, the basis of motion word problems.
$r$
How fast, e.g. miles per hour
$t$
How long, in matching time units
Worked example and calculator

Exponents and scientific notation

Product Rule for Exponents
$a^m \cdot a^n = a^{m+n}$
When multiplying with same base, add the exponents.
$a$
The common base
$m$
First exponent
$n$
Second exponent
Worked example and calculator
Quotient Rule for Exponents
$\frac{a^m}{a^n} = a^{m-n}$
When dividing with same base, subtract the exponents.
$a$
The common base
$m$
Exponent in numerator
$n$
Exponent in denominator
Worked example and calculator
Power Rule for Exponents
$(a^m)^n = a^{mn}$
When raising a power to a power, multiply the exponents.
$a$
The base
$m$
Exponent inside parentheses
$n$
Exponent outside parentheses
Worked example and calculator
Scientific Notation
$N = a \times 10^n \quad (1 \le a < 10)$
Write very large or small magnitudes as a coefficient times a power of ten.
$a$
One-digit (or 1.0–9.9…) factor with 1 ≤ a < 10
$n$
Integer power of 10
Worked example and calculator

Numbers and square roots

Real Number Hierarchy
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
Natural numbers are inside integers, integers are inside rational numbers, and rational plus irrational numbers make all real numbers.
$\mathbb{N}$
Counting numbers like 1, 2, 3...
$\mathbb{Z}$
Whole numbers and their negatives
$\mathbb{Q}$
Numbers expressible as a/b
$\mathbb{R}$
All points on the number line
Worked example and calculator
Rational Number Form
$r = \frac{a}{b},\; a,b \in \mathbb{Z},\; b \neq 0$
A rational number can be written as a fraction of two integers with a nonzero denominator.
$a$
Any integer value
$b$
Any nonzero integer value
$r$
The quotient a/b
Worked example and calculator
Square Root Bounds
$n^2 < m < (n+1)^2 \Rightarrow n < \sqrt{m} < n+1$
To estimate a nonperfect square root, find the two consecutive perfect squares around the number.
$m$
The number under the radical
$n$
Integer whose square is just below m
Worked example and calculator

Triangles and distance

Pythagorean Theorem
$a^2 + b^2 = c^2$
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is one of the most important formulas in all of mathematics.
$a$
One of the two shorter sides of the right triangle, the sides that form the right angle (units)
$b$
The other shorter side of the right triangle, together with leg a, they form the 90° corner (units)
$c$
The longest side, always opposite the right angle, this is the side you solve for most often (units)
Worked example and calculator
Distance Formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Calculates the straight-line distance between two points on a coordinate plane.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator
Triangle Angle Sum
$A + B + C = 180^\circ$
The three interior angles of any triangle always add up to 180 degrees.
$A$
A known interior angle (°)
$B$
A second known interior angle (°)
Worked example and calculator

Volume

Volume of a Cylinder
$V = \pi r^2 h$
Calculates the volume of a cylinder given its radius and height.
$r$
Radius of the circular base (units)
$h$
Height of the cylinder (units)
Worked example and calculator
Volume of a Cone
$V = \frac{1}{3}\pi r^2 h$
Calculates the volume of a cone.
$r$
Radius of the circular base (units)
$h$
Height of the cone (units)
Worked example and calculator
Volume of a Sphere
$V = \frac{4}{3}\pi r^3$
Calculates the volume of a sphere.
$r$
Radius of the sphere (units)
Worked example and calculator

Learn these formulas

Other formula sheets

8th Grade Math Formula Sheet

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet for a test.

Slope and linear equations

Slope Formula

m = \frac{y_2 - y_1}{x_2 - x_1}

Calculates the slope (steepness) of a line between two points.

$x_1$: X-coordinate of first point
$y_1$: Y-coordinate of first point
$x_2$: X-coordinate of second point
$y_2$: Y-coordinate of second point

Worked example and calculator

Slope-Intercept Form

y = mx + b

The equation of a line where m is the slope and b is the y-intercept.

$m$: Rise over run, the steepness of the line
$b$: Where the line crosses the y-axis
$x$: The independent variable
$y$: The dependent variable

Worked example and calculator

Direct Variation

y = k x

y varies directly with x: as x increases, y increases by a constant factor k.

$k$: The fixed ratio y/x
$x$: The independent variable

Worked example and calculator

Distance = Rate × Time

d = r t

Distance traveled equals speed (rate) times time, the basis of motion word problems.

$r$: How fast, e.g. miles per hour
$t$: How long, in matching time units

Worked example and calculator

Exponents and scientific notation

Product Rule for Exponents

a^m \cdot a^n = a^{m+n}

When multiplying with same base, add the exponents.

$a$: The common base
$m$: First exponent
$n$: Second exponent

Worked example and calculator

Quotient Rule for Exponents

\frac{a^m}{a^n} = a^{m-n}

When dividing with same base, subtract the exponents.

$a$: The common base
$m$: Exponent in numerator
$n$: Exponent in denominator

Worked example and calculator

Power Rule for Exponents

(a^m)^n = a^{mn}

When raising a power to a power, multiply the exponents.

$a$: The base
$m$: Exponent inside parentheses
$n$: Exponent outside parentheses

Worked example and calculator

Scientific Notation

N = a \times 10^n \quad (1 \le a < 10)

Write very large or small magnitudes as a coefficient times a power of ten.

$a$: One-digit (or 1.0–9.9…) factor with 1 ≤ a < 10
$n$: Integer power of 10

Worked example and calculator

Numbers and square roots

Real Number Hierarchy

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Natural numbers are inside integers, integers are inside rational numbers, and rational plus irrational numbers make all real numbers.

$\mathbb{N}$: Counting numbers like 1, 2, 3...
$\mathbb{Z}$: Whole numbers and their negatives
$\mathbb{Q}$: Numbers expressible as a/b
$\mathbb{R}$: All points on the number line

Worked example and calculator

Rational Number Form

r = \frac{a}{b},\; a,b \in \mathbb{Z},\; b \neq 0

A rational number can be written as a fraction of two integers with a nonzero denominator.

$a$: Any integer value
$b$: Any nonzero integer value
$r$: The quotient a/b

Worked example and calculator

Square Root Bounds

n^2 < m < (n+1)^2 \Rightarrow n < \sqrt{m} < n+1

To estimate a nonperfect square root, find the two consecutive perfect squares around the number.

$m$: The number under the radical
$n$: Integer whose square is just below m

Worked example and calculator

Triangles and distance

Pythagorean Theorem

a^2 + b^2 = c^2

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is one of the most important formulas in all of mathematics.

$a$: One of the two shorter sides of the right triangle, the sides that form the right angle (units)
$b$: The other shorter side of the right triangle, together with leg a, they form the 90° corner (units)
$c$: The longest side, always opposite the right angle, this is the side you solve for most often (units)