Every formula, worked example, and free calculator for percentages, fractions, algebra, geometry, trigonometry, volume, number conversions, and 600+ more math calculations — all in one place. From basic arithmetic to advanced calculus concepts, organized by topic so you find what you need instantly.
Calculate percentages, percent change, percent difference, increase, decrease, and ratio-to-percent conversions instantly.
Percentages express a value as a fraction of 100. Every percentage calculation traces back to one of four fundamental formulas. Knowing which one to apply is the key skill — the arithmetic itself is always simple.
Percentage of a number: Result = (P / 100) x Number
What % is X of Y: P = (X / Y) x 100
Percent Change: ((New - Old) / |Old|) x 100
Percent Difference: (|A - B| / ((A + B) / 2)) x 100
Reverse % (find total): Total = Part / (P / 100)
These two are commonly confused. Percent change compares a new value to an original value and has direction — it can be positive (increase) or negative (decrease). Percent difference compares two values without a defined reference point, using the average of both as the denominator. Use percent difference when neither value is the "original."
A percentage increase multiplies the original by (1 + rate). A percentage decrease multiplies by (1 - rate). To reverse an increase — finding the original before a 20% markup — divide by 1.2, not subtract 20% from the result. Subtracting 20% from a marked-up price removes less than was added, because the percentage is now applied to a larger base.
| Scenario | Formula | Example |
|---|---|---|
| % of a number | (P/100) x N | 15% of 80 = 12 |
| X is what % of Y | (X/Y) x 100 | 12 is 15% of 80 |
| % increase | ((New-Old)/Old) x 100 | 80 to 92 = +15% |
| % decrease | ((Old-New)/Old) x 100 | 80 to 68 = -15% |
| Find the original | New / (1 + P/100) | 92 / 1.15 = 80 |
| Basis point | 1 bps = 0.01% | 25 bps = 0.25% |
Add, subtract, multiply, and divide fractions, simplify complex fractions, and convert between fractions, decimals, and percentages.
Fractions follow specific rules for each arithmetic operation. The most important rule: addition and subtraction require a common denominator; multiplication and division do not.
Add/Subtract: Find LCD, convert, then add/subtract numerators: a/b + c/d = (ad+bc)/bd
Multiply: Numerator x Numerator / Denominator x Denominator: a/b x c/d = ac/bd
Divide: Multiply by the reciprocal: a/b ÷ c/d = a/b x d/c = ad/bc
Simplify: Divide both numerator and denominator by their GCD
The Least Common Denominator (LCD) is the smallest number both denominators divide into evenly. The Greatest Common Divisor (GCD) is the largest number that divides both numerator and denominator — used to simplify fractions to lowest terms. The Euclidean algorithm finds the GCD efficiently: repeatedly replace the larger number with the remainder of dividing the two, until the remainder is zero.
A mixed number (2 3/4) converts to an improper fraction by multiplying the whole by the denominator and adding the numerator: 2 x 4 + 3 = 11, so 2 3/4 = 11/4. To reverse, divide: 11 ÷ 4 = 2 remainder 3 = 2 3/4.
| From | To | Method | Example |
|---|---|---|---|
| Fraction | Decimal | Divide numerator by denominator | 3/8 = 0.375 |
| Fraction | Percent | (Numerator / Denominator) x 100 | 3/8 = 37.5% |
| Decimal | Fraction | Numerator = decimal x 10^digits, simplify | 0.375 = 375/1000 = 3/8 |
| Decimal | Percent | Multiply by 100 | 0.375 = 37.5% |
| Percent | Decimal | Divide by 100 | 37.5% = 0.375 |
| Percent | Fraction | P/100, then simplify | 37.5% = 375/1000 = 3/8 |
Quadratic equations, polynomial operations, factoring, completing the square, FOIL method, and algebraic expressions.
The quadratic formula solves any equation in the form ax² + bx + c = 0. The key is the discriminant (b² - 4ac): it tells you how many real solutions exist before you do any other calculation.
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Discriminant: D = b^2 - 4ac
D > 0: two distinct real solutions
D = 0: one repeated solution (x = -b/2a)
D < 0: no real solutions (complex roots only)
Completing the square rewrites ax² + bx + c in vertex form a(x + h)² + k. This is the method behind the quadratic formula. Steps: move the constant to the right, add (b/2a)² to both sides, factor the perfect square trinomial on the left. The vertex of the parabola is (-h, k).
FOIL (First, Outer, Inner, Last) expands two binomials: (a + b)(c + d) = ac + ad + bc + bd. For trinomials and higher, distribute every term in the first polynomial across every term in the second. Collect like terms (same variable and exponent) last.
Difference of squares: a^2 - b^2 = (a+b)(a-b)
Perfect square: (a+b)^2 = a^2 + 2ab + b^2
Sum of cubes: a^3 + b^3 = (a+b)(a^2-ab+b^2)
Difference of cubes: a^3 - b^3 = (a-b)(a^2+ab+b^2)
|x| = x when x ≥ 0 and |x| = -x when x < 0. Absolute value equations |x| = a have two solutions: x = a and x = -a (when a > 0). Absolute value inequalities |x| < a yield -a < x < a; |x| > a yields x < -a OR x > a.
Area, perimeter, and properties of all 2D shapes — triangles, rectangles, circles, polygons, and special shapes.
Rectangle: A = length x width
Square: A = side^2
Triangle: A = (1/2) x base x height
Circle: A = pi x r^2
Trapezoid: A = (1/2) x (a + b) x height (a, b = parallel sides)
Parallelogram: A = base x height
Rhombus: A = (d1 x d2) / 2 (d1, d2 = diagonals)
Regular Polygon: A = (1/2) x perimeter x apothem
Sector (circle): A = (theta/360) x pi x r^2
Ellipse: A = pi x a x b (a, b = semi-axes)
For a triangle where all three sides are known but no height is given, use Heron's formula. Calculate the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s-a)(s-b)(s-c)). The Pythagorean theorem applies only to right triangles: the square of the hypotenuse equals the sum of squares of both legs.
A circle is fully defined by one measurement. From radius r: diameter d = 2r, circumference C = 2πr, area A = πr². From circumference: r = C / (2π). From area: r = √(A/π). Sector area = (θ/360) x πr² where θ is the central angle in degrees. Arc length = (θ/360) x 2πr.
| Shape | Area Formula | Perimeter Formula |
|---|---|---|
| Square (side s) | s² | 4s |
| Rectangle (l x w) | l x w | 2(l + w) |
| Triangle (b, h) | (1/2) bh | a + b + c |
| Circle (r) | πr² | 2πr |
| Parallelogram | bh | 2(a + b) |
| Trapezoid (a,b,h) | (1/2)(a+b)h | a+b+c+d |
| Regular Hexagon (s) | (3√3/2)s² | 6s |
| Regular Octagon (s) | 2(1+√2)s² | 8s |
Volume and surface area calculations for cubes, spheres, cylinders, cones, prisms, and pyramids.
Cube: V = s^3
Rectangular box: V = l x w x h
Cylinder: V = pi x r^2 x h
Sphere: V = (4/3) x pi x r^3
Cone: V = (1/3) x pi x r^2 x h
Triangular prism: V = (1/2) x b x h x l (b, h = triangle base/height, l = length)
Pyramid (rect. base):V = (1/3) x l x w x h
Hemisphere: V = (2/3) x pi x r^3
Surface area is the total 2D area covering the outside of a 3D shape — it determines how much material you need to wrap or coat the shape. Volume is the 3D space inside. Both are important: surface area for painting, wrapping, and heat transfer problems; volume for filling, shipping, and buoyancy problems.
| Shape | Volume | Surface Area |
|---|---|---|
| Cube (s) | s³ | 6s² |
| Box (l,w,h) | lwh | 2(lw+lh+wh) |
| Sphere (r) | (4/3)πr³ | 4πr² |
| Cylinder (r,h) | πr²h | 2πr(r+h) |
| Cone (r,h) | (1/3)πr²h | πr(r+l) where l=slant |
| Triangular prism | (1/2)bhl | bh + l(a+b+c) |
Sine, cosine, tangent, inverse trig functions, law of sines, law of cosines, half-angle, double-angle, and triangle solvers.
sin(A) = Opposite / Hypotenuse
cos(A) = Adjacent / Hypotenuse
tan(A) = Opposite / Adjacent (= sin/cos)
csc(A) = 1/sin(A) | sec(A) = 1/cos(A) | cot(A) = 1/tan(A)
For non-right triangles, use the Law of Sines when you know two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS).
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C)
Also: cos(C) = (a^2 + b^2 - c^2) / (2ab)
These identities derive exact trig values for angles that are multiples or fractions of known angles, without a calculator.
Double angle: sin(2A) = 2sin(A)cos(A)
Double angle: cos(2A) = cos^2(A) - sin^2(A) = 1 - 2sin^2(A)
Half angle: sin(A/2) = +/- sqrt((1-cos(A))/2)
Half angle: cos(A/2) = +/- sqrt((1+cos(A))/2)
Pythagorean: sin^2(A) + cos^2(A) = 1
Simplify ratios, solve proportions, convert between ratios, fractions and percentages, find unit rates, and apply the golden ratio.
A ratio compares two or more quantities. It can be expressed as 3:4, as the fraction 3/4, or in words as "3 to 4." Ratios are simplified by dividing all parts by their GCD. Equivalent ratios scale up or down by the same factor.
If a/b = c/d, then a x d = b x c (cross multiplication)
To find x: a/b = x/d => x = (a x d) / b
Unit rate: Rate per single unit = total quantity / number of units
Golden ratio: phi = (1 + sqrt(5)) / 2 = 1.61803398...
When two figures are similar, all corresponding sides are in the same ratio (the scale factor). Areas scale by the square of the scale factor; volumes scale by the cube. If scale factor = k, then area ratio = k² and volume ratio = k³.
Convert between binary, decimal, octal, hexadecimal, and any base — plus scientific notation, standard form, and number representations.
Computers work in binary (base 2). Humans work in decimal (base 10). Hex (base 16) and octal (base 8) are shorthand for binary — each hex digit represents exactly 4 binary bits; each octal digit represents exactly 3 bits. Converting between bases involves expressing the number as powers of the target base.
Decimal to Binary: Repeatedly divide by 2; remainders (LSB to MSB) form the binary number
Binary to Decimal: Sum of each bit x 2^position (rightmost = position 0)
Decimal to Hex: Repeatedly divide by 16; digits 10-15 = A-F
Hex to Binary: Each hex digit = 4-bit group (e.g., F = 1111, A = 1010)
Scientific notation writes numbers as a × 10^n, where 1 ≤ |a| < 10. To convert: count how many places you move the decimal point. Moving left = positive exponent; moving right = negative. Example: 0.000456 = 4.56 × 10³. Adding/subtracting requires matching exponents first; multiplying/dividing handles the coefficients and adds/subtracts exponents.
| Decimal | Binary | Octal | Hex |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 8 | 1000 | 10 | 8 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 255 | 11111111 | 377 | FF |
Arithmetic and geometric sequences, Fibonacci, factorials, prime numbers, GCD, LCM, and number patterns.
Arithmetic nth term: a_n = a_1 + (n-1)d (d = common difference)
Arithmetic sum: S_n = n/2 x (a_1 + a_n) = n/2 x (2a_1 + (n-1)d)
Geometric nth term: a_n = a_1 x r^(n-1) (r = common ratio)
Geometric sum (r!=1): S_n = a_1 x (1 - r^n) / (1 - r)
Fibonacci: F(n) = F(n-1) + F(n-2), starting F(1)=1, F(2)=1
Factorial: n! = n x (n-1) x ... x 2 x 1, 0! = 1
The Greatest Common Divisor (GCD, also GCF) is the largest number dividing two integers exactly. The Least Common Multiple (LCM) is the smallest number both integers divide into. They are related: GCD(a,b) x LCM(a,b) = a x b. The Euclidean algorithm finds GCD efficiently without factoring.
A prime number has exactly two factors: 1 and itself. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic). To check if n is prime, test divisibility by all primes up to √n. Prime factorization is the foundation for finding GCD and LCM of large numbers.
Vector addition, dot product, cross product, magnitude, unit vectors, vector projections, and matrix operations.
Magnitude: |v| = sqrt(vx^2 + vy^2 + vz^2)
Unit vector: u = v / |v|
Dot product: a · b = ax*bx + ay*by + az*bz = |a||b|cos(theta)
Cross product: a x b = (ay*bz-az*by, az*bx-ax*bz, ax*by-ay*bx)
Angle between: cos(theta) = (a · b) / (|a| x |b|)
Projection of a onto b: proj = (a · b / |b|^2) x b
The dot product produces a scalar — use it to find the angle between two vectors or to check orthogonality (perpendicular vectors have a dot product of zero). The cross product produces a vector perpendicular to both inputs — use it for finding surface normals, torque, and areas of parallelograms. The cross product is only defined in 3D.
Length, area, volume, weight, temperature, time, speed, and all common unit conversions between metric and imperial systems.
The metric system uses powers of 10. Every unit has a base (meter, gram, liter, second) with prefixes: kilo- (10³), hecto- (10²), deca- (10), deci- (10¹), centi- (10²), milli- (10³), micro- (10&sup6;), nano- (10&sup9;). Converting between metric units is always a matter of multiplying or dividing by a power of 10.
Length: 1 inch = 2.54 cm | 1 foot = 30.48 cm | 1 mile = 1.609 km
Weight: 1 pound = 453.592 g | 1 kg = 2.20462 lbs | 1 oz = 28.35 g
Temperature: F = C x 9/5 + 32 | C = (F - 32) x 5/9 | K = C + 273.15
Volume: 1 gallon = 3.78541 L | 1 cup = 240 mL | 1 fl oz = 29.57 mL
Speed: 1 mph = 1.60934 km/h | 1 m/s = 3.6 km/h = 2.237 mph
Area conversions square the linear conversion factor. If 1 foot = 30.48 cm, then 1 square foot = 30.48² = 929 cm². Volume conversions cube it: 1 cubic foot = 30.48³ = 28,317 cm³. This means converting acres to square feet or cubic meters to liters requires applying these squared or cubed factors.
Mean, median, mode, weighted average, geometric mean, harmonic mean, and measures of central tendency.
Arithmetic mean: (x1 + x2 + ... + xn) / n
Weighted mean: (w1*x1 + w2*x2 + ... + wn*xn) / (w1+w2+...+wn)
Geometric mean: (x1 x x2 x ... x xn)^(1/n) = n-th root of the product
Harmonic mean: n / (1/x1 + 1/x2 + ... + 1/xn)
Median: Middle value when sorted (average two middle if even n)
Mode: Most frequently occurring value(s)
Use the arithmetic mean for symmetric data without extreme outliers — scores, temperatures. Use the median for skewed data or when outliers distort the mean — household income, house prices. Use the mode for categorical data — most popular product. Use the geometric mean for rates of change, growth rates, and ratios. Use the harmonic mean for rates where the denominator represents time or distance — average speed over a fixed distance.
| Average Type | Formula | Best For |
|---|---|---|
| Arithmetic Mean | Σx / n | Symmetric distributions, exam scores |
| Weighted Mean | Σ(wx) / Σw | GPA, investment returns |
| Geometric Mean | n-th root of product | Growth rates, ratios, investments |
| Harmonic Mean | n / Σ(1/x) | Speed, rates with fixed denominator |
| Median | Middle value | Skewed data, income, prices |
| Mode | Most frequent | Categorical data, voting |
Square roots, cube roots, nth roots, exponent rules, logarithms, natural log, exponential growth and decay.
Product rule: a^m x a^n = a^(m+n)
Quotient rule: a^m / a^n = a^(m-n)
Power rule: (a^m)^n = a^(mn)
Negative exponent: a^(-n) = 1/a^n
Zero exponent: a^0 = 1 (for a != 0)
Log definition: log_b(x) = y means b^y = x
Change of base: log_b(x) = log(x) / log(b) = ln(x) / ln(b)
Log rules: log(xy) = log(x)+log(y) | log(x/y) = log(x)-log(y)
Power log rule: log(x^n) = n*log(x)
Exponential growth: A = A&sub0; × e^(rt) where r > 0. Exponential decay: A = A&sub0; × e^(-rt) where r > 0. Doubling time = ln(2)/r. Half-life = ln(2)/r. Both growth and decay are characterized by the same formula structure — the sign of r determines the direction.
Distance, midpoint, slope, line equations, endpoint, and coordinate geometry calculations.
Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2)
Midpoint formula: M = ((x1+x2)/2, (y1+y2)/2)
Slope: m = (y2-y1) / (x2-x1)
Slope-intercept form: y = mx + b
Point-slope form: y - y1 = m(x - x1)
Standard form: Ax + By = C
Endpoint (given mid): x2 = 2*Mx - x1, y2 = 2*My - y1
Parallel lines: Same slope (m1 = m2)
Perpendicular lines: Slopes multiply to -1 (m1 x m2 = -1)
Slope is rise over run — the change in y per unit change in x. A slope of 2 means the line rises 2 units for every 1 unit moved right. Positive slope rises left to right; negative falls; zero is horizontal; undefined (infinite) is vertical. The y-intercept is where the line crosses the y-axis (set x = 0). The x-intercept is where it crosses the x-axis (set y = 0).
All formulas and reference data in this guide are drawn from established mathematics standards and authoritative references: