Comprehensive Mathematical Formulas Lecture Notes

A collection of mathematical formulas and identities covering Geometry, Trigonometry (Circle), Mensuration, Heron's Formula, Coordinate Geometry, and Polynomials.

Radius of a Circle

$$\frac{1}{2} \times \text{diameter}$$

Diameter of a Circle

$$2 \times \text{radius}$$

Circumference of a Circle

$$2\pi r$$

Formula for Pi (\pi)

$$\pi = \frac{C}{D}$$

Arc length of a sector

$$\frac{\theta}{360} \times 2\pi r$$

Area of circle

$$\pi r^2$$

Area of sector

$$\frac{\theta}{360} \times \pi r^2$$

Area of minor segment

$$\text{Area of sector} - \text{Area of triangle} = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2} \times b \times h$$

Perimeter of Square

$$4 \times \text{side} = 4a$$

Area of Square

$$\text{Side} \times \text{Side} = a^2$$

Perimeter of Rectangle

$$2(l + b)$$

Area of Rectangle

$$l \times b$$

Perimeter of Rhombus

$$4 \times \text{side}$$

Area of Rhombus

$$\frac{1}{2} \times d_1 \times d_2$$

Perimeter of 11 gram

$$2(l + b)$$

Area of 11 gram

$$(b \times h)$$

Perimeter of Triangle

$$\text{Sum of all three sides}$$

Area of Triangle

$$\frac{1}{2} \times b \times h$$

Area of St (Heron's Formula)

$\sqrt{s(s - a)(s - b)(s - c)}$ where $s = \frac{a + b + c}{2}$

Perimeter of Equilateral triangle

$$3a$$

Area of Equilateral triangle

$$\sqrt{3}a^2$$

Area of Trapezium

$$\frac{1}{2} \times (a + b) \times h$$

C.S.A of Cone

$$\pi r l$$

T.S.A of Cone

$$\pi r(l + r)$$

Volume of Cone

$$\frac{1}{3} \times \pi r^2 h$$

Slant height (l) of a Cone

$$l = \sqrt{r^2 + h^2}$$

C.S.A and T.S.A of Sphere

$$4\pi r^2$$

Volume of Sphere

$$\frac{4}{3} \times \pi r^3$$

C.S.A of Hemi-Sphere

$$2\pi r^2$$

T.S.A of Hemisphere

$$3\pi r^2$$

Volume of Hemi-sphere

$$\frac{2}{3} \times \pi r^3$$

Equation of x-axis

$$y = 0$$

Equation of y-axis

$$x = 0$$

Coordinate Quadrant I

$$(x, y) \rightarrow (+, +)$$

Coordinate Quadrant II

$$(-x, y) \rightarrow (-, +)$$

Coordinate Quadrant III

$$(-x, -y) \rightarrow (-, -)$$

Coordinate Quadrant IV

$$(x, -y) \rightarrow (+, -)$$

Point on x-axis

$$(x, 0)$$

Point on y-axis

$(0, y)$, where $x \text{ point} = 0$

Distance Formula (AB)

$$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Mid-Point Formula

$$x = \frac{x_1 + x_2}{2}, y = \frac{y_1 + y_2}{2}$$

Linear Polynomial

$$\text{Degree: 1, General Form: } ax + b$$

Quadratic Polynomial

$$\text{Degree: 2, General Form: } ax^2 + bx + c$$

Cubic Polynomial

$$\text{Degree: 3, General Form: } ax^3 + bx^2 + cx + d$$

Algebraic Identity (a+b)^2

$$a^2 + b^2 + 2ab$$

Algebraic Identity (a-b)^2

$$a^2 + b^2 - 2ab$$

Algebraic Identity a^2 - b^2

$$(a + b)(a - b)$$

Algebraic Identity (x+a)(x+b)

$$x^2 + (a + b)x + ab$$

Algebraic Identity (a+b+c)^2

$$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

Algebraic Identity (a+b)^3

$$a^3 + b^3 + 3ab(a + b)$$

Algebraic Identity (a-b)^3

$$a^3 - b^3 - 3ab(a - b)$$

Algebraic Identity a^3 + b^3

$$(a + b)(a^2 - ab + b^2)$$

Algebraic Identity a^3 - b^3

$$(a - b)(a^2 + ab + b^2)$$

Sum of Cubes Identity

$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$

Conditional Cube Identity

$$\text{if } a + b + c = 0, \text{ then } a^3 + b^3 + c^3 = 3abc$$