This math formula and equation reference sheet brings together essential algebra, geometry, and trigonometry formulas, with guidance on when to use each one, how to rearrange it, and which units or assumptions to check. Use it for homework help, exam preparation, or as a starting point before working through a detailed solution.
Overview
A formula is a relationship between quantities. A formula sheet is useful when you know what information a problem gives you but need to identify the relationship that connects those values. The goal is not to memorize every equation without context. Instead, learn to recognize the type of problem, select a suitable formula, substitute carefully, and check whether the result makes sense.
Before using any equation, identify the unknown. Mark the values you know, write down their units, and describe what the question is asking. This small step prevents a common mistake: choosing a familiar formula that does not match the information provided.
For a full method-selection guide, see Equation Solver Guide: Which Method Should You Use for Each Algebra Problem? If you need a broader walkthrough of solving equations, continue with How to Solve Equations Step by Step.
How to estimate
In a formula-based problem, estimating means setting up a repeatable calculation from the given inputs. Use this five-step process:
- Classify the problem. Decide whether it involves an equation, a shape, a right triangle, a percentage, or a conversion.
- Choose the relationship. Select the formula whose variables match the known values and the unknown.
- Rearrange before substituting when needed. Isolate the unknown first. This makes the calculation easier to inspect and reuse.
- Substitute with consistent units. Do not mix centimeters and meters, degrees and radians, or squared and unsquared units without converting.
- Check the result. Substitute the answer back into the original relationship or compare it with a rough estimate.
Core algebra formulas
The distributive property is a(b + c) = ab + ac. It is useful when removing parentheses. The reverse process, factoring, rewrites a sum as a product; for example, ab + ac = a(b + c).
For a linear equation in the form ax + b = c, isolate the variable by subtracting b and then dividing by a:
x = (c - b) / a, provided a is not zero.
The slope between two points, (x1, y1) and (x2, y2), is:
m = (y2 - y1) / (x2 - x1).
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. The point-slope form is y - y1 = m(x - x1).
For a quadratic equation ax2 + bx + c = 0, the quadratic formula is:
x = (-b ± √(b2 - 4ac)) / 2a.
The expression b2 - 4ac is the discriminant. It helps describe the number and type of solutions, but calculate it only after identifying a, b, and c correctly.
Geometry formulas
For a rectangle, the perimeter and area are:
- P = 2l + 2w
- A = lw
For a triangle, the area is A = 1/2bh, where b is the base and h is the perpendicular height. The height must meet the base at a right angle; a slanted side is not automatically the height.
For a circle, use C = 2πr for circumference and A = πr2 for area. If the diameter d is given, remember that d = 2r.
The Pythagorean theorem applies to right triangles: a2 + b2 = c2, where c is the hypotenuse, the side opposite the right angle. It does not apply to every triangle.
Common volume formulas include V = lwh for a rectangular prism, V = πr2h for a cylinder, and V = 1/3πr2h for a cone. Area uses square units; volume uses cubic units.
Trigonometry formulas
For a right triangle, the basic ratios are:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
A useful memory aid is SOH-CAH-TOA. Use the ratio that contains the side or angle you know and the quantity you need. The inverse functions, sin-1, cos-1, and tan-1, can find an angle from a ratio. Check whether your calculator is set to degrees or radians before calculating.
Inputs and assumptions
Correct formulas still produce incorrect answers when the inputs or assumptions are wrong. Record each value with its label before entering it into a calculator. For example, distinguish a radius from a diameter, a side length from an area, and a height from a slanted edge.
Keep units consistent. If a rectangle has a length of 2 meters and a width of 75 centimeters, convert one measurement before finding the area. The result should be expressed in square meters or square centimeters, not a mixture of both. For trigonometry, confirm the angle mode. For algebra, note restrictions created by denominators, square roots, or logarithms.
Rounding is another assumption to make explicit. Keep several digits during intermediate steps and round at the end unless instructions say otherwise. In science and measurement problems, the required precision may depend on significant figures. See the significant figures reference when the problem uses measured quantities.
A formula sheet supports learning, but it does not replace showing the setup. Write the formula, substitute the values, simplify, and state the answer with units. This creates an explanation that can be checked by a teacher, tutor, or future you.
Worked examples
Example 1: Rearranging a linear equation
Suppose 3x + 7 = 22. Subtract 7 from both sides: 3x = 15. Divide by 3: x = 5. Check the answer in the original equation: 3(5) + 7 = 22, so the result is consistent.
Example 2: Finding the area of a circle
If a circle has a radius of 4 centimeters, use A = πr2. Substitute the radius: A = π(4)2 = 16π. The exact area is 16π cm2. A decimal approximation can be calculated if the question requests one, but retain the exact form when appropriate.
Example 3: Using a trigonometric ratio
Suppose a right triangle has an angle of 30 degrees and a hypotenuse of 10 units. If the opposite side is unknown, use sin(30°) = opposite / 10. Rearranging gives opposite = 10sin(30°). With the calculator in degree mode, the opposite side is 5 units. The side is shorter than the hypotenuse, which provides a quick reasonableness check.
For graphing, tables, and more complicated expressions, a calculator can help verify a manual solution. The guide on using a graphing calculator for algebra and precalculus can help you choose an appropriate checking method.
When to recalculate
Revisit a formula whenever an input changes, the question supplies a different unit, or you discover that an assumption was incorrect. Recalculate after changing a radius to a diameter, replacing an estimated measurement with an exact one, or switching between degree and radian mode. In a multi-step problem, update every dependent quantity rather than changing only the final line.
Use this reference again before a test to review formulas by category, then practice choosing formulas without looking. A useful routine is to cover the formula, identify the unknown from a sample question, write the relationship from memory, and compare your setup with the sheet. For a longer review routine, use the seven-day math test study plan.
Finally, keep a short mistake checklist: wrong formula, incorrect substitution, inconsistent units, calculator mode, algebra error, and premature rounding. Checking these items turns a formula sheet into a practical study tool rather than a list to memorize. If a problem involves chemical equations or scientific measurements, pair this page with the balancing chemical equations guide and the significant-figures reference before submitting your work.
