Math formulas
Each formula with every variable explained, when to reach for it, where it comes from, and worked examples. Not a formula sheet to copy from — a page that tells you what the letters mean.
22 formulas
Fractions
- Adding fractionsTo add fractions, give them the same bottom number and add the top numbers. The general formula uses the product of the two denominators.
- Dividing fractionsTo divide by a fraction, flip the second fraction and multiply. Only the divisor flips, and it must not be zero.
- Equivalent fractionsMultiplying or dividing the top and bottom of a fraction by the same number gives an equivalent fraction — different numbers, same amount.
- Multiplying fractionsTo multiply fractions, multiply the top numbers together and the bottom numbers together. No common denominator is needed.
- Subtracting fractionsTo subtract fractions, give them the same bottom number and subtract the top numbers. The order matters: the first numerator keeps its place on the left.
Percents
- Percent changePercent change measures how big a rise or fall was compared with where it started. The denominator is always the original value, never the new one.
- Percent of a numberTo find a percent of a number, turn the percent into a decimal and multiply. The word "of" in a percent problem always means multiply.
- Simple interestSimple interest is the principal times the rate times the time. It is always charged on the original amount, never on interest already earned.
Exponents & Roots
- Exponent rulesTo multiply powers of the same base, add the exponents. To divide them, subtract. To raise a power to another power, multiply. Every one of these is counting factors, and all of them need the bases to match.
- Pythagorean theoremIn a right triangle, the squares of the two shorter sides add to the square of the longest. The letter c is always the hypotenuse.
- Quadratic formulaThe quadratic formula gives both solutions of any equation in the form ax² + bx + c = 0, whether or not it factors.
Algebra 1
- DiscriminantThe discriminant is the part of the quadratic formula under the square root. Its sign tells you how many real solutions a quadratic has, before you solve it.
- Point-slope formPoint-slope form writes the equation of a line from one point on it and its slope, with no need to find the y-intercept first.
- Slope formulaThe slope formula gives the steepness of the line through two points: the change in y divided by the change in x.
- Slope-intercept formSlope-intercept form writes a line so its slope and its y-intercept can be read straight off the equation.
- Standard form of a lineStandard form writes a linear equation with both variables on one side. It is the only common form that can describe a vertical line.
Geometry
- Area of a circleThe area of a circle is pi times the radius squared. Square the radius first — the exponent belongs to the r alone, not to the whole product.
- Area of a rectangleThe area of a rectangle is its length times its width. A square is the special case where they are equal, giving A = s².
- Area of a triangleThe area of a triangle is half its base times its height, because any triangle is exactly half of a rectangle with the same base and height.
- Circumference of a circleThe circumference is the distance around a circle: pi times the diameter, or equivalently two pi times the radius.
- Perimeter of a rectangleThe perimeter of a rectangle is twice its length plus twice its width, because opposite sides are equal. For any other shape, add every side.
- Volume of a rectangular prismThe volume of a box is length times width times height. Equivalently it is the base area times the height, which is the form that works for any prism.