Top 25 Algebraic Formulas

• 1. \((a + b)^2 = a^2 + 2ab + b^2\)
• 2. \((a - b)^2 = a^2 - 2ab + b^2\)
• 3. \(a^2 - b^2 = (a - b)(a + b)\)
• 4. \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
• 5. \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)
• 6. \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• 7. \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• 8. \(a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \dots + b^{n-1})\)
• 9. \(a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + \dots + b^{n-1})\) (for odd \(n\))
• 10. \(a^m \cdot a^n = a^{m+n}\)
• 11. \(\frac{a^m}{a^n} = a^{m-n}\)
• 12. \((a^m)^n = a^{mn}\)
• 13. \(a^0 = 1\)
• 14. \(a^{-n} = \frac{1}{a^n}\)
• 15. \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)
• 16. \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
• 17. \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
• 18. \(a^2 + b^2 = (a + b)^2 - 2ab\)
• 19. \(a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)\)
• 20. \(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)\)
• 21. \(x^2 + (a + b)x + ab = (x + a)(x + b)\)
• 22. \(a^4 - b^4 = (a^2 + b^2)(a + b)(a - b)\)
• 23. \(a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)\)
• 24. \(a^5 + b^5 = (a + b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4)\)
• 25. \(a^6 - b^6 = (a^3 + b^3)(a^3 - b^3)\)

Top 25 Trigonometry Formulas

• 1. \(\sin^2 \theta + \cos^2 \theta = 1\)
• 2. \(1 + \tan^2 \theta = \sec^2 \theta\)
• 3. \(1 + \cot^2 \theta = \csc^2 \theta\)
• 4. \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
• 5. \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
• 6. \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)
• 7. \(\sin 2\theta = 2 \sin \theta \cos \theta\)
• 8. \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)
• 9. \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)
• 10. \(\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}\)
• 11. \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\)
• 12. \(\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}\)
• 13. \(\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\)
• 14. \(\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)\)
• 15. \(\cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\)
• 16. \(\cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)\)
• 17. \(\sin \theta \cdot \cos \phi = \frac{1}{2} [\sin(\theta + \phi) + \sin(\theta - \phi)]\)
• 18. \(\cos \theta \cdot \cos \phi = \frac{1}{2} [\cos(\theta + \phi) + \cos(\theta - \phi)]\)
• 19. \(\sin \theta \cdot \sin \phi = \frac{1}{2} [\cos(\theta - \phi) - \cos(\theta + \phi)]\)
• 20. \(\sin^3 \theta = \frac{3 \sin \theta - \sin 3\theta}{4}\)
• 21. \(\cos^3 \theta = \frac{3 \cos \theta + \cos 3\theta}{4}\)
• 22. \(\tan^3 \theta = \frac{3 \tan \theta - \tan 3\theta}{1 - 3 \tan^2 \theta}\)
• 23. \(\sinh x = \frac{e^x - e^{-x}}{2}\)
• 24. \(\cosh x = \frac{e^x + e^{-x}}{2}\)
• 25. \(\tanh x = \frac{\sinh x}{\cosh x}\)

Top 25 Geometry Formulas

• 1. Area of a rectangle: \(A = l \times w\)
• 2. Perimeter of a rectangle: \(P = 2(l + w)\)
• 3. Area of a square: \(A = s^2\)
• 4. Perimeter of a square: \(P = 4s\)
• 5. Area of a triangle: \(A = \frac{1}{2} \times b \times h\)
• 6. Perimeter of a triangle: \(P = a + b + c\)
• 7. Area of a parallelogram: \(A = b \times h\)
• 8. Perimeter of a parallelogram: \(P = 2(a + b)\)
• 9. Area of a trapezoid: \(A = \frac{1}{2} \times (b_1 + b_2) \times h\)
• 10. Circumference of a circle: \(C = 2\pi r\)
• 11. Area of a circle: \(A = \pi r^2\)
• 12. Volume of a cube: \(V = s^3\)
• 13. Surface area of a cube: \(SA = 6s^2\)
• 14. Volume of a rectangular prism: \(V = l \times w \times h\)
• 15. Surface area of a rectangular prism: \(SA = 2(lw + lh + wh)\)
• 16. Volume of a cylinder: \(V = \pi r^2 h\)
• 17. Surface area of a cylinder: \(SA = 2\pi r(h + r)\)
• 18. Volume of a sphere: \(V = \frac{4}{3}\pi r^3\)
• 19. Surface area of a sphere: \(SA = 4\pi r^2\)
• 20. Volume of a cone: \(V = \frac{1}{3}\pi r^2 h\)
• 21. Surface area of a cone: \(SA = \pi r(r + l)\)
• 22. Pythagorean theorem: \(a^2 + b^2 = c^2\)
• 23. Distance between two points: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
• 24. Slope of a line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• 25. Midpoint formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

Arithmetic Formulas

• 1. Perimeter of a rectangle: \(P = 2(l + w)\)
• 2. Area of a rectangle: \(A = l \times w\)
• 3. Side length of a square: \(s = \sqrt{A}\)
• 4. Perimeter of a square: \(P = 4s\)
• 5. Area of a square: \(A = s^2\)
• 6. Diagonal of a rectangle: \(d = \sqrt{l^2 + w^2}\)
• 7. Area of a rhombus: \(A = \frac{1}{2} \times d_1 \times d_2\)
• 8. Perimeter of a rhombus: \(P = 4s\)
• 9. Area of a trapezoid: \(A = \frac{1}{2} \times (b_1 + b_2) \times h\)
• 10. Cube root: \(\sqrt[3]{x}\)
• 11. Square root: \(\sqrt{x}\)
• 12. Simple interest: \(I = P \times r \times t\)
• 13. Compound interest: \(A = P \left(1 + \frac{r}{n}\right)^{nt}\)
• 14. Profit: \(Profit = SP - CP\)
• 15. Loss: \(Loss = CP - SP\)
• 16. Percentage: \(\% = \left( \frac{Part}{Whole} \right) \times 100\)
• 17. Ratio: \(a : b\)
• 18. Proportion: \(\frac{a}{b} = \frac{c}{d}\)
• 19. Average: \(Avg = \frac{Sum}{Number of terms}\)
• 20. Time and distance: \(Distance = Speed \times Time\)
• 21. Time and work: \(Work = Rate \times Time\)
• 22. Prime number: A number greater than 1 with no positive divisors other than 1 and itself.
• 23. Sum of numbers: \(Sum = \frac{n(n + 1)}{2}\)
• 24. Divisibility rules: Check divisibility by 2, 3, 5, etc.
• 25. Multiplication rules: \(a \times b = b \times a\)

All Circle Formulas

• 1. Circumference: \(C = 2\pi r\)
• 2. Area: \(A = \pi r^2\)
• 3. Diameter: \(D = 2r\)
• 4. Arc length: \(L = r \theta\) (where \(\theta\) is in radians)
• 5. Sector area: \(A = \frac{1}{2} r^2 \theta\)
• 6. Chord length: \(C = 2r \sin \left( \frac{\theta}{2} \right)\)
• 7. Tangent: A line touching the circle at one point.
• 8. Secant: A line intersecting the circle at two points.
• 9. Central angle: \(\theta = \frac{L}{r}\)
• 10. Inscribed angle: \(\theta = \frac{1}{2} \times \text{intercepted arc}\)