Complete Mathematics Formulas Guide | Class 9 & 10 All Concepts

48 Essential Mathematics Concepts

This comprehensive mathematics formula sheet contains 48 fundamental concepts that every Class 9 and 10 student must master. These formulas cover geometry, algebra, trigonometry, mensuration, and basic arithmetic - all aligned with the Punjab Board curriculum.

Tip: Create flashcards with these formulas and practice applying them to different types of problems. Understanding when and how to use each formula is more important than mere memorization.

1. Geometry Formulas

Area Formulas

These formulas calculate the area of various geometric shapes, essential for solving geometry problems.

Equilateral Triangle

$$A = \frac{\sqrt{3}}{4}a^2$$

a = side length

Concentric Circles

$$A = \pi(R^2 - r^2)$$

R = outer radius, r = inner radius

Circle Sector

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

θ = angle in degrees

Perimeter and Circumference

Formulas for calculating boundaries of geometric shapes.

Rectangle Perimeter

$$P = 2(l + w)$$

l = length, w = width

Circle Circumference

$$C = 2\pi r$$

r = radius

Arc Length

$$L = r\theta$$

θ in radians

Volume Formulas

Formulas for calculating the volume of 3D shapes, crucial for mensuration problems.

Cuboid Volume

$$V = \text{Base Area} \times h$$

h = height

Cylinder Volume

$$V = \pi r^2 h$$

r = radius, h = height

Cone Volume

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

r = radius, h = height

Sphere Volume

$$V = \frac{4}{3} \pi r^3$$

r = radius

Hemi-Sphere Volume

$$V = \frac{2}{3} \pi r^3$$

Half of sphere volume

Special Geometry Formulas

Additional important geometry formulas for polygons and pyramids.

Polygon Interior Angle

$$\text{Angle} = \frac{n-2}{n} \times 180^\circ$$

n = number of sides

Pyramid Surface Area

$$A = 2bh + b^2$$

b = base, h = height

Square Pyramid Volume

$$V = \frac{1}{3} b^2 a$$

b = base, a = altitude

Cone Surface Area

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

l = slant height

2. Algebra Formulas

Basic Algebraic Identities

Fundamental algebraic identities used for expansion and factorization.

Sum of Squares

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

Difference Identity

$$(a+b)^2 - (a-b)^2 = 4ab$$

Trinomial Square

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

Cube Formulas

Formulas for expanding and factoring cubic expressions.

Sum Cube

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

Difference Cube

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

Sum of Cubes

$$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$

Difference of Cubes

$$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$

Quadratic Equations

Formulas related to quadratic equations and their roots.

Sum of Roots

$$\alpha + \beta = -\frac{b}{a}$$

For ax² + bx + c = 0

Product of Roots

$$\alpha \beta = \frac{c}{a}$$

For ax² + bx + c = 0

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Solution of ax² + bx + c = 0

Logarithm Formulas

Essential logarithm properties and rules.

Product Rule

$$\log_a(mn) = \log_a m + \log_a n$$

Quotient Rule

$$\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n$$

Power Rule

$$\log_a m^n = n \log_a m$$

Change of Base

$$\log_a m \cdot \log_m n = \log_a n$$

3. Number Theory & Arithmetic

LCM and HCF

Relationships between LCM (Least Common Multiple) and HCF (Highest Common Factor).

LCM × HCF

$$\text{LCM} \times \text{HCF} = A \times B$$

For two numbers A and B

Number Properties

Important number facts and conversion formulas.

Smallest 6-digit Number

$$100,000$$

cm to mm

$$\text{Multiply by } 10$$

1 cm = 10 mm

4. Trigonometry

Trigonometric Signs by Quadrant

Understanding which trigonometric functions are positive in each quadrant.

First Quadrant

$$\text{All functions positive}$$

Second Quadrant

$$\sin x \text{ and } \csc x \text{ positive}$$

Third Quadrant

$$\tan x \text{ and } \cot x \text{ positive}$$

Fourth Quadrant

$$\cos x \text{ and } \sec x \text{ positive}$$

Circle Geometry

Important circle theorems and formulas.

Angle in Semicircle

$$90^\circ$$

Angle subtended by diameter

Chord Distance

$$C = 2\sqrt{r^2 - h^2}$$

r = radius, h = perpendicular distance

5. Statistics & Measurement

Statistical Measures

Basic statistical formulas for data analysis.

Range

$$\text{Range} = X_{\text{max}} - X_{\text{min}}$$

Difference between max and min values

Area Calculation

Hero's formula for triangle area.

Hero's Formula

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

s = semi-perimeter = (a+b+c)/2

6. Coordinate Geometry

Distance Formula

Formula for calculating distance between two points in coordinate plane.

Distance Between Points

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

Pythagorean theorem application

Matrix Operations

Important matrix property for transposition.

Transpose of Product

$$(AB)^T = B^T A^T$$

Order reverses in transpose

7. Business Mathematics

Percentage Calculations

Formulas for profit, loss, and discount calculations.

Profit Percentage

$$\text{Profit %} = \frac{P}{C} \times 100$$

P = Profit, C = Cost Price

Loss Percentage

$$\text{Loss %} = \frac{L}{C} \times 100$$

L = Loss, C = Cost Price

Discount Percentage

$$\text{Discount %} = \frac{D}{M} \times 100$$

D = Discount, M = Marked Price

8. Complex Numbers

Complex Number Operations

Formulas for complex number conjugation.

Complex Conjugate

$$\text{If } Z = a + ib, \text{ then } \bar{Z} = a - ib$$

Change sign of imaginary part

9. Conversion Formulas

Temperature Conversion

Formula for converting between Celsius and Fahrenheit.

Fahrenheit to Celsius

$$F^\circ = \frac{9}{5}C^\circ + 32$$

Standard temperature conversion

Effective Study Tips for Mathematics Formulas

1. Understand Derivations

Don't just memorize formulas - understand how they are derived. This helps in remembering and applying them correctly.

2. Practice Daily

Apply formulas to solve at least 5-10 problems daily. Regular practice builds confidence and improves speed.

3. Create Formula Cards

Make flashcards with formulas on one side and examples on the other. Review them during spare time.

4. Group Study

Discuss formulas with classmates. Teaching others helps reinforce your own understanding.

Study Tools & Resources

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Practice Sheets

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Quiz Yourself

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48 Formulas Summary

Geometry Formulas

Algebra Formulas

Trigonometry Formulas

Other Formulas

Mathematics Formulas Master Sheet