As students gear up for their Class 10 board exams, having a strong grasp of key mathematical formulas is essential. This essential formula sheet consolidates the crucial mathematical formulas from the Class 10 NCERT textbook, spanning algebra, geometry, trigonometry, and more. It serves as a quick reference to help students review and ensure they have the tools needed for solving a variety of mathematical problems efficiently in their exams

**Algebra**

**Linear Equations in One and Two Variables:**- Slope formula: m = (y2 – y1) / (x2 – x1)
- Equation of a line in slope-intercept form: y = mx + c
- Equation of a line in two-point form: y – y1 = m(x – x1)

**Polynomials:**- Expansion of basic identities: (a + b)², (a – b)², (a + b)(a – b)
- Factorization techniques: splitting the middle term, grouping, using identities

**Quadratic Equations:**- Quadratic formula: x = (-b ± √(b² – 4ac)) / 2a
- Discriminant: b² – 4ac (determines the nature of roots – real and equal, real and distinct, or complex)

**Arithmetic Progressions (AP):**- nth term: a + (n – 1)d
- Sum of n terms: Sn = n/2 (2a + (n – 1)d)

**Geometry**

**Triangles:**- Pythagorean Theorem: a² + b² = c² (right-angled triangles)
- Sum of angles in a triangle: 180°
- Exterior angle of a triangle: equal to the sum of the remote interior angles
- Area of a triangle: ½ * base * height
- Heron’s Formula (for any triangle): Area = √s(s – a)(s – b)(s – c), where s = semi-perimeter (perimeter / 2)

**Circles:**- Circumference: 2πr
- Area: πr²
- Arc length: (θ/360°) * 2πr (θ in degrees)
- Sector area: (θ/360°) * πr² (θ in degrees)

**Quadrilaterals:**- Properties of parallelograms (opposite sides equal and parallel, opposite angles equal, diagonals bisect each other)
- Properties of rectangles (all angles 90°, opposite sides equal and parallel, diagonals equal)
- Properties of squares (all sides equal, all angles 90°, diagonals bisect each other at right angles)
- Area of a parallelogram: base * height
- Area of a rectangle: length * breadth
- Area of a square: side²

**Trigonometry (introduced in Class 10)**

**Ratios in Right-Angled Triangles (SOH CAH TOA):**- sin θ = opposite/hypotenuse
- cos θ = adjacent/hypotenuse
- tan θ = opposite/adjacent
- Basic trigonometric identities: sin² θ + cos² θ = 1, tan θ = sin θ / cos θ

**Angles and their Measures:**- Conversion between degrees and radians: π radians = 180°

**Mensuration**

**Areas and Volumes:**- Area of a rectangle: length * breadth
- Area of a square: side²
- Area of a triangle: ½ * base * height
- Area of a parallelogram: base * height
- Area of a circle: πr²
- Volume of a cube: side³
- Volume of a cuboid: length * breadth * height
- Surface area of a cube: 6a² (a = side)
- Surface area of a cuboid: 2(lb + bh + hl) (l = length, b = breadth, h = height)

**Statistics (introduced in Class 10)**

**Measures of Central Tendency:**- Mean: Average = Sum of observations / Number of observations
- Median: Middle value when data is arranged in ascending or descending order
- Mode: Most frequent value in the data set

**Remember:** This list is extensive, but it might not be entirely exhaustive. Refer to your NCERT textbook for any additional formulas or specific variations.

**Focus on understanding the concepts behind the formulas and how to apply them effectively in problem-solving.**