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Math Formula Reference — 100+ Formulas, Search & Filter

Math formula reference — algebra, geometry, trigonometry, calculus, statistics, all in one place.

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Showing 114 / 114 formulas

Algebra (22)

Quadratic Formula
一元二次求根公式
x = (−b ± √(b² − 4ac)) / 2a
Example: 2x² − 4x − 6 = 0 → x = (4 ± √(16 + 48))/4 = 3 or −1

Roots of ax² + bx + c = 0 when a ≠ 0.

Discriminant
判别式
Δ = b² − 4ac
Example: For x² + 2x + 5 = 0: Δ = 4 − 20 = −16 (no real roots)

Δ > 0: two real roots; Δ = 0: one repeated root; Δ < 0: two complex roots.

Vieta's Formulas
韦达定理
x₁ + x₂ = −b/a, x₁ · x₂ = c/a

Sum and product of roots of ax² + bx + c = 0.

Difference of Squares
平方差公式
a² − b² = (a + b)(a − b)
Example: x² − 9 = (x + 3)(x − 3)

Factor the difference of two perfect squares.

Perfect Square (sum)
完全平方和
(a + b)² = a² + 2ab + b²
Example: (x + 3)² = x² + 6x + 9

Expansion of the square of a sum.

Perfect Square (difference)
完全平方差
(a − b)² = a² − 2ab + b²
Example: (x − 5)² = x² − 10x + 25

Expansion of the square of a difference.

Cube (sum)
完全立方和
(a + b)³ = a³ + 3a²b + 3ab² + b³

Expansion of the cube of a sum.

Cube (difference)
完全立方差
(a − b)³ = a³ − 3a²b + 3ab² − b³

Expansion of the cube of a difference.

Sum of Cubes
立方和公式
a³ + b³ = (a + b)(a² − ab + b²)

Factor the sum of two cubes.

Difference of Cubes
立方差公式
a³ − b³ = (a − b)(a² + ab + b²)

Factor the difference of two cubes.

Binomial Theorem
二项式定理
(a + b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ, k = 0..n
Example: (a + b)³ = a³ + 3a²b + 3ab² + b³

Expansion of (a + b)ⁿ using binomial coefficients C(n,k) = n! / (k!(n−k)!).

Arithmetic Sequence (nth term)
等差数列通项
aₙ = a₁ + (n − 1)d
Example: 1, 4, 7, 10, … → a₁₀ = 1 + 9·3 = 28

nth term of an arithmetic sequence with first term a₁ and common difference d.

Arithmetic Series Sum
等差数列求和
Sₙ = n(a₁ + aₙ) / 2 = n·a₁ + n(n−1)d/2
Example: 1 + 2 + … + 100 = 100·101/2 = 5050

Sum of the first n terms of an arithmetic sequence.

Geometric Sequence (nth term)
等比数列通项
aₙ = a₁ · qⁿ⁻¹
Example: 3, 6, 12, 24, … → a₅ = 3·2⁴ = 48

nth term with first term a₁ and common ratio q.

Geometric Series Sum
等比数列求和
Sₙ = a₁(1 − qⁿ) / (1 − q), q ≠ 1

Sum of the first n terms when the common ratio q ≠ 1.

Infinite Geometric Series
无穷等比级数
S = a₁ / (1 − q), |q| < 1

Sum to infinity exists only when |q| < 1.

Logarithm Product Rule
对数乘法
logₐ(xy) = logₐ x + logₐ y

Log of a product is the sum of logs.

Logarithm Quotient Rule
对数除法
logₐ(x/y) = logₐ x − logₐ y

Log of a quotient is the difference of logs.

Logarithm Power Rule
对数幂运算
logₐ(xⁿ) = n · logₐ x

Log of a power moves the exponent out front.

Change of Base
换底公式
logₐ b = logc b / logc a
Example: log₂ 10 = ln 10 / ln 2 ≈ 3.32

Convert between log bases — c is any new base.

Exponent Sum Rule
同底数幂相乘
aᵐ · aⁿ = aᵐ⁺ⁿ

When multiplying same bases, add exponents.

Triangle Inequality (abs)
绝对值三角不等式
|a + b| ≤ |a| + |b|

Absolute value of a sum is at most the sum of absolute values.

Geometry (28)

Triangle Area
三角形面积
S = (1/2) · b · h
Example: b = 6, h = 4 → S = 12

Half base times height. Works for any triangle.

Heron's Formula
海伦公式
S = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
Example: a=3,b=4,c=5 → s=6, S=√(6·3·2·1)=6

Area of a triangle from its three side lengths.

Pythagorean Theorem
勾股定理
a² + b² = c²
Example: 3² + 4² = 5²

In a right triangle, the square of the hypotenuse equals the sum of squares of legs.

Equilateral Triangle Area
等边三角形面积
S = (√3 / 4) · a²

Area of an equilateral triangle with side a.

Rectangle Area
矩形面积
S = length × width

Length times width.

Rectangle Perimeter
矩形周长
P = 2(length + width)

Twice the sum of length and width.

Square Area
正方形面积
S = a²

Side squared.

Parallelogram Area
平行四边形面积
S = b · h

Base times perpendicular height.

Trapezoid Area
梯形面积
S = (a + b) · h / 2
Example: a=3, b=5, h=4 → S = 16

Average of the two parallel sides times the height.

Rhombus Area
菱形面积
S = (d₁ · d₂) / 2

Half the product of the two diagonals.

Circle Area
圆面积
S = π · r²
Example: r = 5 → S = 25π ≈ 78.54

π times radius squared.

Circle Circumference
圆周长
C = 2π · r = π · d

Diameter times π, or 2π times radius.

Circular Sector Area
扇形面积
S = (1/2) · r² · θ (θ in radians)

Half radius squared times central angle in radians.

Arc Length
弧长
L = r · θ (θ in radians)

Radius times central angle in radians.

Ellipse Area
椭圆面积
S = π · a · b

Semi-major axis a times semi-minor axis b times π.

Cube Volume
立方体体积
V = a³

Edge length cubed.

Cube Surface Area
立方体表面积
S = 6 · a²

Six times the area of one face.

Cuboid Volume
长方体体积
V = length × width × height

Product of the three edge lengths.

Cuboid Surface Area
长方体表面积
S = 2(lw + lh + wh)

Twice the sum of the three pairwise face areas.

Cylinder Volume
圆柱体积
V = π · r² · h
Example: r=3, h=10 → V = 90π ≈ 282.74

Base area times height.

Cylinder Surface Area
圆柱表面积
S = 2π · r² + 2π · r · h

Two circular bases plus lateral surface.

Cone Volume
圆锥体积
V = (1/3) · π · r² · h

One-third base area times height.

Cone Surface Area
圆锥表面积
S = π · r² + π · r · ℓ, ℓ = √(r² + h²)

Base plus lateral surface; ℓ is the slant height.

Sphere Volume
球体积
V = (4/3) · π · r³
Example: r=3 → V = 36π ≈ 113.10

Four-thirds π times radius cubed.

Sphere Surface Area
球表面积
S = 4π · r²

Four π times radius squared.

Distance Between Two Points (2D)
两点距离公式(平面)
d = √((x₂ − x₁)² + (y₂ − y₁)²)

Euclidean distance between (x₁, y₁) and (x₂, y₂).

Midpoint Formula
中点坐标公式
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Midpoint of the segment between two points.

Slope of a Line
直线斜率
k = (y₂ − y₁) / (x₂ − x₁)

Rise over run between two points.

Trigonometry (24)

sin/cos/tan of 30°
30° 三角函数值
sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3

Standard values at 30° (π/6).

sin/cos/tan of 45°
45° 三角函数值
sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1

Standard values at 45° (π/4).

sin/cos/tan of 60°
60° 三角函数值
sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Standard values at 60° (π/3).

sin/cos/tan of 90°
90° 三角函数值
sin 90° = 1, cos 90° = 0, tan 90° = undefined

Standard values at 90° (π/2). tan is undefined.

Pythagorean Identity
平方关系
sin²θ + cos²θ = 1

The fundamental trig identity, true for any θ.

1 + tan² = sec²
正切与正割关系
1 + tan²θ = sec²θ

Derived from sin²+cos²=1 by dividing by cos².

1 + cot² = csc²
余切与余割关系
1 + cot²θ = csc²θ

Derived from sin²+cos²=1 by dividing by sin².

tan as sin/cos
正切定义
tan θ = sin θ / cos θ

tan equals sin divided by cos.

Law of Sines
正弦定理
a / sin A = b / sin B = c / sin C = 2R

In any triangle, sides are proportional to sines of opposite angles. R is the circumradius.

Law of Cosines
余弦定理
c² = a² + b² − 2ab · cos C
Example: a=3, b=4, C=60° → c² = 9 + 16 − 12 = 13

Generalisation of the Pythagorean theorem to any triangle.

Triangle Area (SAS)
三角形面积(两边夹角)
S = (1/2) · a · b · sin C

Area from two sides and the included angle.

sin(α + β)
正弦和角公式
sin(α + β) = sin α · cos β + cos α · sin β

Sine of a sum identity.

sin(α − β)
正弦差角公式
sin(α − β) = sin α · cos β − cos α · sin β

Sine of a difference identity.

cos(α + β)
余弦和角公式
cos(α + β) = cos α · cos β − sin α · sin β

Cosine of a sum identity.

cos(α − β)
余弦差角公式
cos(α − β) = cos α · cos β + sin α · sin β

Cosine of a difference identity.

tan(α + β)
正切和角公式
tan(α + β) = (tan α + tan β) / (1 − tan α · tan β)

Tangent of a sum identity.

Double Angle (sin)
二倍角(正弦)
sin 2θ = 2 · sin θ · cos θ

Double-angle identity for sine.

Double Angle (cos)
二倍角(余弦)
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Three equivalent forms of the cosine double-angle identity.

Double Angle (tan)
二倍角(正切)
tan 2θ = 2 · tan θ / (1 − tan²θ)

Double-angle identity for tangent.

Half Angle (sin)
半角(正弦)
sin(θ/2) = ±√((1 − cos θ) / 2)

Sign depends on the quadrant of θ/2.

Half Angle (cos)
半角(余弦)
cos(θ/2) = ±√((1 + cos θ) / 2)

Sign depends on the quadrant of θ/2.

Sum to Product (sin + sin)
和差化积(正弦和)
sin α + sin β = 2 · sin((α+β)/2) · cos((α−β)/2)

Convert sum of sines to product.

Product to Sum (sin·cos)
积化和差(正弦×余弦)
sin α · cos β = (1/2)[sin(α+β) + sin(α−β)]

Convert product of sin and cos to sum.

Degrees ↔ Radians
角度与弧度互换
rad = deg · π / 180, deg = rad · 180 / π
Example: 90° = π/2 ≈ 1.5708 rad

Convert between degrees and radians.

Calculus (22)

Derivative Definition
导数定义
f'(x) = lim (h→0) [f(x + h) − f(x)] / h

Limit definition of the derivative.

Power Rule (derivative)
幂函数求导
(xⁿ)' = n · xⁿ⁻¹
Example: (x³)' = 3x²

Derivative of x to the n.

Derivative of a Constant
常数求导
(c)' = 0

A constant has derivative zero.

Derivative of sin x
sin x 求导
(sin x)' = cos x

Derivative of sine is cosine.

Derivative of cos x
cos x 求导
(cos x)' = −sin x

Derivative of cosine is negative sine.

Derivative of tan x
tan x 求导
(tan x)' = sec²x = 1 / cos²x

Derivative of tangent.

Derivative of eˣ
eˣ 求导
(eˣ)' = eˣ

eˣ is its own derivative — that's what makes e special.

Derivative of aˣ
aˣ 求导
(aˣ)' = aˣ · ln a

General exponential function.

Derivative of ln x
ln x 求导
(ln x)' = 1 / x, x > 0

Natural log differentiates to reciprocal.

Derivative of logₐ x
logₐ x 求导
(logₐ x)' = 1 / (x · ln a)

General log derivative.

Sum Rule
加法法则
(f + g)' = f' + g'

Derivative of a sum is the sum of derivatives.

Product Rule
乘法法则
(f · g)' = f' · g + f · g'

Differentiation of a product.

Quotient Rule
除法法则
(f / g)' = (f' · g − f · g') / g²

Differentiation of a quotient.

Chain Rule
链式法则
(f(g(x)))' = f'(g(x)) · g'(x)
Example: (sin(x²))' = cos(x²) · 2x

Differentiation of a composite function.

Integral of xⁿ
xⁿ 积分
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1
Example: ∫ x² dx = x³/3 + C

Reverse power rule for integration. Add the integration constant C.

Integral of 1/x
1/x 积分
∫ (1/x) dx = ln|x| + C

The exception to the power rule when n = −1.

Integral of eˣ
eˣ 积分
∫ eˣ dx = eˣ + C

eˣ is its own antiderivative too.

Integral of sin x
sin x 积分
∫ sin x dx = −cos x + C

Antiderivative of sine.

Integral of cos x
cos x 积分
∫ cos x dx = sin x + C

Antiderivative of cosine.

Integration by Parts
分部积分
∫ u dv = u · v − ∫ v du
Example: ∫ x · eˣ dx, u = x, dv = eˣ dx → x·eˣ − eˣ + C

Reverse of the product rule. Pick u and dv so the new integral is simpler.

u-Substitution
换元积分
∫ f(g(x)) · g'(x) dx = ∫ f(u) du, u = g(x)

Substitute an inner function to simplify the integral.

Fundamental Theorem of Calculus
微积分基本定理
∫ₐᵇ f(x) dx = F(b) − F(a), F' = f

Connects differentiation and integration.

Statistics (18)

Arithmetic Mean
算术平均值
x̄ = (Σ xᵢ) / n
Example: {2, 4, 6, 8} → x̄ = 20 / 4 = 5

Sum of all values divided by the count.

Median
中位数
middle value of sorted data (avg of two middles if even count)
Example: {1, 3, 5, 7} → median = (3 + 5) / 2 = 4

The middle value when sorted; average the two centre values if n is even.

Mode
众数
value(s) that occur most frequently

The most frequent value; a dataset may have several modes or none.

Population Variance
总体方差
σ² = (1/N) · Σ (xᵢ − μ)²

Average squared deviation from the population mean μ.

Sample Variance
样本方差
s² = (1/(n − 1)) · Σ (xᵢ − x̄)²

Use n − 1 (Bessel correction) when estimating from a sample.

Standard Deviation
标准差
σ = √(σ²), s = √(s²)

Square root of the variance — same units as the data.

Range
极差
R = max − min

Difference between the largest and smallest values.

Covariance
协方差
Cov(X, Y) = (1/n) · Σ (xᵢ − x̄)(yᵢ − ȳ)

Measures how two variables change together. Sign tells direction; magnitude is unit-dependent.

Pearson Correlation
皮尔逊相关系数
r = Cov(X, Y) / (σx · σy)

Unitless correlation in [−1, 1]. r = 1 perfect positive, r = −1 perfect negative.

Z-Score
标准分(Z)
z = (x − μ) / σ
Example: μ=70, σ=10, x=85 → z = 1.5

How many standard deviations a value is from the mean.

Normal Distribution PDF
正态分布概率密度
f(x) = (1 / (σ · √(2π))) · e^(−(x − μ)² / (2σ²))

The classic bell curve, fully characterized by μ and σ.

68-95-99.7 Rule
68-95-99.7 法则
P(|x − μ| ≤ k·σ) ≈ 68% (k=1), 95% (k=2), 99.7% (k=3)

Approximate areas under the normal curve at 1/2/3 standard deviations.

Binomial Probability
二项分布概率
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
Example: Flip 10 fair coins, exactly 5 heads → C(10,5)·0.5¹⁰ ≈ 0.246

Probability of exactly k successes in n independent trials with success probability p.

Binomial Mean & Variance
二项分布均值与方差
E(X) = n · p, Var(X) = n · p · (1 − p)

Expected value and variance of a binomial random variable.

Permutation
排列数
P(n, k) = n! / (n − k)!
Example: P(5, 2) = 5! / 3! = 20

Number of ordered arrangements of k items chosen from n.

Combination
组合数
C(n, k) = n! / (k! · (n − k)!)
Example: C(5, 2) = 10

Number of unordered selections of k items from n.

Expected Value (discrete)
离散期望
E(X) = Σ xᵢ · P(xᵢ)

Weighted average of all possible outcomes by their probabilities.

Bayes' Theorem
贝叶斯定理
P(A | B) = P(B | A) · P(A) / P(B)

Update probabilities given new evidence.

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Free interactive math formula reference for students and teachers. Over 100 essential formulas organized into five categories: Algebra (quadratic formula, difference of squares, perfect square, sum/difference of cubes, arithmetic and geometric series, binomial theorem), Geometry (triangle area, circle circumference and area, sphere volume, cylinder, cone, cube, cuboid, trapezoid, ellipse), Trigonometry (sin/cos/tan tables, law of sines, law of cosines, sum-to-product, product-to-sum, double-angle identities), Calculus (derivative rules, chain rule, integral basics, integration by parts, substitution), and Statistics (mean, variance, standard deviation, normal distribution, binomial distribution). Every formula displays in clean Unicode math symbols (∑ ∫ √ π ≤ ≥ ≠) with no KaTeX or MathJax dependency — the entire reference loads in under 25 KB and works offline. Search across English and Chinese names simultaneously, or filter by category to focus on exactly what you need. Each entry includes the formula itself, a plain English and Chinese explanation, and a concrete worked example so you're not just looking at symbols. 100% client-side; no patterns or queries leave the tab. Pair with our Percentage Calculator and Unit Converter for quick numeric work.

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