The Math Formulas Students Look Up Most

I have a folder on my laptop called forgot-again. It is a screenshot graveyard of the same dozen formulas I keep re-Googling: the quadratic formula, the distance between two points, sphere volume, the law of cosines. After watching myself search "area of a circle formula" for what felt like the hundredth time, I built the Math Formula Reference so the lookup takes three seconds instead of a tab full of ad-stuffed homework sites.

This post is the short version of what lives in that tool: the formulas students actually look up, grouped by topic, with one worked example so you are not staring at bare symbols. Skim it, bookmark it, and when you blank on a denominator at 11pm, you will know where it is.

Geometry: areas and volumes you keep forgetting

These are the ones I forget first, because the shapes feel obvious until you need an exact number.

Circle area: A = πr²
Circle circumference: C = 2πr
Triangle area: A = ½ × base × height
Rectangle area: A = length × width
Trapezoid area: A = ½(b₁ + b₂) × h
Sphere volume: V = (4/3)πr³
Sphere surface area: A = 4πr²
Cylinder volume: V = πr²h
Cone volume: V = (1/3)πr²h
Cube volume: V = s³

The trap here is mixing up volume and surface area. Sphere volume is (4/3)πr³, but surface area is 4πr². For r = 3 both happen to equal 36π, which is a coincidence that has fooled plenty of people on a test. The fix is to read the units off the answer: cm³ means volume, cm² means area. If your formula and your units disagree, the formula is wrong.

Algebra: the quadratic formula and friends

The quadratic formula is the single most-searched formula on the planet, and it is worth memorizing once and for all. For any equation in the form ax² + bx + c = 0:

x = (−b ± √(b²−4ac)) / 2a

The piece under the square root, b²−4ac, is the discriminant. If it is positive you get two real roots, if it is zero you get one, and if it is negative you get none in the real numbers. That one expression tells you the shape of the answer before you finish the arithmetic.

Two more algebra workhorses worth keeping close:

Difference of squares: a² − b² = (a + b)(a − b)
Perfect square: (a + b)² = a² + 2ab + b²

Worked example: solving 2x² + 5x − 3 = 0

Let me plug real numbers into the quadratic formula so you can see it move. Here a = 2, b = 5, and c = −3.

Find the discriminant: b²−4ac = 5² − 4(2)(−3) = 25 + 24 = 49.
Square root it: √49 = 7.
Assemble the formula: x = (−5 ± 7) / (2 × 2) = (−5 ± 7) / 4.
Split the ±: x = (−5 + 7) / 4 = 2/4 = 0.5, and x = (−5 − 7) / 4 = −12/4 = −3.

So the roots are x = 0.5 and x = −3. The most common mistake is dropping the 2a denominator and dividing by a alone — that would turn 0.5 into 1 and quietly wreck the whole answer. The entire numerator divides by 2a, never just a.

The Pythagorean theorem and coordinate geometry

For a right triangle with legs a and b and hypotenuse c:

a² + b² = c²

This one formula spawns half of coordinate geometry. The distance between two points (x₁, y₁) and (x₂, y₂) is just the Pythagorean theorem wearing coordinates:

d = √((x₂−x₁)² + (y₂−y₁)²)

For the points (1, 2) and (4, 6), that gives d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5. A clean 3-4-5 triangle hiding in the plane.

The midpoint of those same two points is the average of each coordinate:

M = ((x₁+x₂)/2, (y₁+y₂)/2)

And slope, the rate a line rises or falls, is rise over run:

m = (y₂−y₁) / (x₂−x₁)

Keep your subtraction order consistent. If y₂ goes first on top, x₂ goes first on the bottom. Flip one without the other and you get the slope's sign backwards, which sends your line the wrong way.

Basic trigonometry: ratios and the laws

Trig starts with three ratios in a right triangle, summarized by SOH-CAH-TOA:

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

The special-angle values are worth memorizing because they show up everywhere: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, and cos runs the same values in reverse.

Once triangles stop being right-angled, two laws take over. The law of sines ties each side to the sine of its opposite angle:

a / sin A = b / sin B = c / sin C

The law of cosines generalizes the Pythagorean theorem to any triangle:

c² = a² + b² − 2ab·cos C

The thing to nail here is pairing: the angle C sits opposite side c, so the formula uses cos C, never cos A. Match each angle to the side across from it before you substitute anything, and the law of cosines becomes hard to get wrong.

A double-angle identity rounds out the set: sin(2θ) = 2 sin θ cos θ. Test it at θ = 30°: 2 × (1/2) × (√3/2) = √3/2, which is exactly sin 60°. It checks out, because 2 × 30° is 60°.

How to actually use a reference like this

A formula sheet is only useful if it loads fast and the formula is one search away. When I am mid-homework I do not want a 300 KB page that animates LaTeX for two seconds before I can read "a² + b² = c²". So the Math Formula Reference renders everything in plain Unicode symbols, weighs under 25 KB, works offline, and lets you search English and Chinese names at the same time. Filter to one topic, print a single category as a study sheet, done.

For the moment you have a formula and need a number out of it, hand the arithmetic to the Scientific Calculator so a stray keystroke does not cost you the answer. Reference to look it up, calculator to run it — that is the whole loop.

The goal is not to memorize a hundred formulas. It is to recognize which one you need, recall its shape, and check it in seconds. Get that loop fast enough and the formulas stop being something you fear forgetting.

Made by Toolora · Updated 2026-06-13