Smart Math Formula Calculator
Smart Math Formula Calculator
Algebra Formulas
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
Arithmetic Sequence: aₙ = a₁ + (n - 1)d
Geometric Sequence: aₙ = a₁ * r^(n - 1)
Sum of Arithmetic Series: Sₙ = n/2 (a₁ + aₙ)
Sum of Geometric Series: Sₙ = a₁ (1 - rⁿ) / (1 - r)
Logarithm: logₐ(b) = c ⇒ a^c = b
Change of Base: logₐ(b) = logₖ(b) / logₖ(a)
Exponential Growth: A = P(1 + r)^t
Exponential Decay: A = P(1 - r)^t
Slope of a Line: m = (y₂ - y₁) / (x₂ - x₁)
Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint Formula: M = [(x₁ + x₂)/2, (y₁ + y₂)/2]
Binomial Theorem: (a + b)^n = Σ [n! / (k!(n - k)!)] a^(n - k) b^k
Determinant of 2x2 Matrix: |A| = ad - bc
Determinant of 3x3 Matrix: |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
Inverse of 2x2 Matrix: A⁻¹ = (1/|A|) * [d, -b; -c, a]
Euler's Formula: e^(ix) = cos(x) + i sin(x)
De Moivre's Theorem: (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)
Trigonometry Formulas
Pythagorean Theorem: a² + b² = c²
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ - sin²θ
tan(2θ) = 2 tanθ / (1 - tan²θ)
sin(θ/2) = ±√[(1 - cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)]
Law of Sines: a / sinA = b / sinB = c / sinC
Law of Cosines: c² = a² + b² - 2ab cosC
Law of Tangents: (a - b)/(a + b) = tan[(A - B)/2] / tan[(A + B)/2]
Area of Triangle: (1/2)ab sinC
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
Arc Length: s = rθ
Sector Area: A = (1/2) r²θ
Radian to Degree: degrees = radians * (180/π)
Degree to Radian: radians = degrees * (π/180)
Unit Circle: x² + y² = 1
Geometry Formulas
Area of Rectangle: A = l * w
Perimeter of Rectangle: P = 2(l + w)
Area of Triangle: A = (1/2) * b * h
Perimeter of Triangle: P = a + b + c
Area of Circle: A = πr²
Circumference of Circle: C = 2πr
Area of Trapezoid: A = (1/2) * (b₁ + b₂) * h
Volume of Cube: V = s³
Surface Area of Cube: SA = 6s²
Volume of Rectangular Prism: V = l * w * h
Surface Area of Rectangular Prism: SA = 2(lw + lh + wh)
Volume of Cylinder: V = πr²h
Surface Area of Cylinder: SA = 2πr(r + h)
Volume of Sphere: V = (4/3)πr³
Surface Area of Sphere: SA = 4πr²
Volume of Cone: V = (1/3)πr²h
Surface Area of Cone: SA = πr(r + l)
Pythagorean Theorem: a² + b² = c²
Distance Between Two Points: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint Formula: M = [(x₁ + x₂)/2, (y₁ + y₂)/2]
Slope of a Line: m = (y₂ - y₁) / (x₂ - x₁)
Equation of a Line: y = mx + b
Area of Parallelogram: A = b * h
Perimeter of Parallelogram: P = 2(a + b)
Area of Rhombus: A = (1/2) * d₁ * d₂
Perimeter of Rhombus: P = 4a
Circle Formulas
Area of Circle: A = πr²
Circumference of Circle: C = 2πr
Arc Length: L = rθ
Sector Area: A = (1/2) r²θ
Equation of Circle: (x - h)² + (y - k)² = r²
Tangent to Circle: y = mx ± r√(1 + m²)
Chord Length: L = 2r sin(θ/2)
Segment Area: A = (1/2) r² (θ - sinθ)
Central Angle: θ = L / r
Inscribed Angle: θ = (1/2) * Arc Length