Math Formulas — Complete Reference Sheet
Every essential math formula in one place, grouped by topic. Click any formula card to use an interactive calculator, or bookmark this page as a quick reference.
Algebra
Quadratic Formula
Quadratic Solver →x = (−b ± √(b²−4ac)) / 2aFinds the roots of any quadratic equation ax² + bx + c = 0.
Slope of a Line
m = (y₂ − y₁) / (x₂ − x₁)Measures the steepness and direction of a line through two points.
Distance Between Two Points
d = √((x₂−x₁)² + (y₂−y₁)²)Straight-line distance between any two coordinate points.
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)Coordinates of the exact center point between two endpoints.
Point-Slope Form
y − y₁ = m(x − x₁)Equation of a line given its slope and one point on the line.
Difference of Squares
a² − b² = (a + b)(a − b)Factoring identity useful for simplifying expressions and solving equations.
Geometry
Circumference of a Circle
C = 2πrTotal perimeter (boundary length) of a circle.
Area of a Triangle
Triangle Calculator →A = ½bhArea of a triangle given its base b and perpendicular height h.
Pythagorean Theorem
Pythagorean Theorem Calculator →a² + b² = c²Relates the three sides of a right triangle; c is the hypotenuse.
Area of a Rectangle
A = lwArea of a rectangle with length l and width w.
Perimeter of a Rectangle
P = 2(l + w)Total boundary length of a rectangle.
Volume of a Sphere
V = (4/3)πr³Volume enclosed by a sphere of radius r.
Volume of a Cylinder
V = πr²hVolume of a right circular cylinder with radius r and height h.
Trigonometry
Pythagorean Identity
sin²θ + cos²θ = 1The fundamental identity relating sine and cosine for any angle θ.
Tangent Identity
tan θ = sin θ / cos θDefines tangent as the ratio of sine to cosine.
Law of Cosines
Triangle Calculator →c² = a² + b² − 2ab·cos(C)Generalizes the Pythagorean theorem to any triangle; relates all three sides and the angle C opposite side c.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)Ratio of each side of a triangle to the sine of its opposite angle is constant.
Double Angle — Sine
sin(2θ) = 2·sin θ·cos θExpresses the sine of double an angle in terms of sine and cosine.
Double Angle — Cosine
cos(2θ) = cos²θ − sin²θExpresses the cosine of double an angle; equivalent forms exist using the Pythagorean identity.
Statistics
Population Standard Deviation
Standard Deviation Calculator →σ = √(Σ(x − x̄)² / n)Measures how spread out values are around the mean for an entire population.
Population Variance
σ² = Σ(x − x̄)² / nThe square of the standard deviation; average squared deviation from the mean.
Sample Standard Deviation
s = √(Σ(x − x̄)² / (n−1))Standard deviation for a sample; uses n−1 (Bessel's correction) for an unbiased estimate.
Probability
P(A) = favorable outcomes / total outcomesBasic probability of event A occurring in a sample space of equally likely outcomes.
Calculus
Power Rule (Differentiation)
d/dx(xⁿ) = n·xⁿ⁻¹Differentiates a power function; multiply by the exponent, then reduce the exponent by 1.
Chain Rule
d/dx[f(g(x))] = f′(g(x))·g′(x)Differentiates a composite function; multiply the outer derivative by the inner derivative.
Product Rule
d/dx[u·v] = u′v + uv′Differentiates the product of two functions.
Quotient Rule
d/dx[u/v] = (u′v − uv′) / v²Differentiates the quotient of two functions.
Power Rule (Integration)
∫xⁿ dx = xⁿ⁺¹/(n+1) + CAntiderivative of a power function; raise the exponent by 1 then divide by the new exponent.
Fundamental Theorem of Calculus
∫ₐᵇ f(x) dx = F(b) − F(a)Connects differentiation and integration; evaluates a definite integral using the antiderivative F.
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