📐 Math Grades 6–12
Formula Sheets by Course
The formulas you'll use most in each course, on one page each. Print the one for your course and keep it in your binder.
Learn How it works
A formula is a tool, not a magic answer. For each one, know what each letter means and when to use it. A good habit: write the formula, substitute the numbers, then solve, showing each step.
Examples See it done
Pythagorean theorem. A ladder reaches 12 ft up a wall and its base is 5 ft from the wall. How long is it?
a² + b² = c² → 5² + 12² = c² → 25 + 144 = 169 → c = √169 = 13 ft.
Quadratic formula. Solve x² − 5x + 6 = 0. a = 1, b = −5, c = 6.
x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2 → x = 3 or x = 2.
Try it Formula sheets
Pre-Algebra (grades 6–8)
| Topic | Formula |
|---|---|
| Order of operations | PEMDAS: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right) |
| Percent | part = percent × whole · percent change = (new − old) / old × 100% |
| Simple interest | I = Prt |
| Rate | distance = rate × time (d = rt) |
| Area: rectangle / triangle | A = lw · A = ½bh |
| Area: circle · circumference | A = πr² · C = 2πr = πd |
| Volume: rectangular prism | V = lwh |
| Mean | sum of values ÷ number of values |
| Integers | (−)(−) = + · (−)(+) = − |
Algebra 1
| Topic | Formula |
|---|---|
| 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 |
| Exponent rules | aᵐ·aⁿ = aᵐ⁺ⁿ · aᵐ/aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ · a⁰ = 1 · a⁻ⁿ = 1/aⁿ |
| Special products | (a + b)² = a² + 2ab + b² · (a − b)(a + b) = a² − b² |
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a |
| Discriminant | b² − 4ac: >0 two real roots, =0 one, <0 none (real) |
| Vertex of y = ax² + bx + c | x = −b / 2a |
| Exponential growth / decay | y = a(1 + r)ˣ · y = a(1 − r)ˣ |
Geometry
| Topic | Formula |
|---|---|
| Pythagorean theorem | a² + b² = c² |
| Distance | d = √((x₂ − x₁)² + (y₂ − y₁)²) |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) |
| Angle sums | Triangle: 180° · polygon interior sum: (n − 2)·180° |
| Special right triangles | 45-45-90: x, x, x√2 · 30-60-90: x, x√3, 2x |
| Trig ratios (SOH-CAH-TOA) | sin = opp/hyp · cos = adj/hyp · tan = opp/adj |
| Area: parallelogram / trapezoid | A = bh · A = ½(b₁ + b₂)h |
| Circle: arc length / sector | arc = (θ/360)·2πr · sector = (θ/360)·πr² |
| Volume | cylinder πr²h · cone ⅓πr²h · sphere (4/3)πr³ · pyramid ⅓Bh |
| Surface area | sphere 4πr² · cylinder 2πr² + 2πrh |
| Circle equation | (x − h)² + (y − k)² = r² |
Algebra 2
| Topic | Formula |
|---|---|
| Logarithms | log_b(x) = y ⇔ bʸ = x · log(xy) = log x + log y · log(x/y) = log x − log y · log(xⁿ) = n log x |
| Change of base | log_b(x) = log x / log b |
| Compound interest | A = P(1 + r/n)ⁿᵗ · continuous: A = Peʳᵗ |
| Complex numbers | i = √−1 · i² = −1 |
| Arithmetic sequence | aₙ = a₁ + (n − 1)d · Sₙ = n(a₁ + aₙ)/2 |
| Geometric sequence | aₙ = a₁rⁿ⁻¹ · Sₙ = a₁(1 − rⁿ)/(1 − r) · S∞ = a₁/(1 − r), |r| < 1 |
| Vertex form | y = a(x − h)² + k, vertex (h, k) |
| Inverse functions | swap x and y, then solve for y |
| Binomial theorem | (a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ |
| Permutations / combinations | P(n,r) = n!/(n − r)! · C(n,r) = n!/(r!(n − r)!) |
Trigonometry & Precalculus
| Topic | Formula |
|---|---|
| Unit circle | (cos θ, sin θ); radians = degrees × π/180 |
| Pythagorean identity | sin²θ + cos²θ = 1 · 1 + tan²θ = sec²θ |
| Reciprocal | csc = 1/sin · sec = 1/cos · cot = 1/tan |
| Law of sines | a/sin A = b/sin B = c/sin C |
| Law of cosines | c² = a² + b² − 2ab cos C |
| Triangle area (SAS) | A = ½ab sin C |
| Double angle | sin 2θ = 2 sin θ cos θ · cos 2θ = cos²θ − sin²θ |
| Sum formulas | sin(A ± B) = sin A cos B ± cos A sin B · cos(A ± B) = cos A cos B ∓ sin A sin B |
| Sinusoid | y = A sin(B(x − C)) + D: amplitude |A|, period 2π/B |
| Vectors | |v| = √(a² + b²) · u·v = |u||v|cos θ |
| Limits (intro) | lim(x→0) sin x / x = 1 |
Practice Check yourself
Answer, then tap Check. You get an explanation either way.
(8 − 2)/(3 − 1) = 6/2 = 3.
(6 − 2) × 180 = 720.