Lets have a look at some of the important formulas NCERT Class 9 Mathematics book for all the chapters.
CBSE Maths Formulas For Class 9 from NCERT Book
Chapter 1: Number Systems
1. \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \)
2. \( \sqrt{a} \div \sqrt{b} = \sqrt{a \div b} \)
3. \( (\sqrt{a})^2 = a \)
4. \( \sqrt{m^2} = |m| \)
5. Rationalizing Factor: For \( \frac{1}{\sqrt{a}}, \text{ multiply by } \sqrt{a} \).
6. \( a^{m} \cdot a^{n} = a^{m+n} \)
7. \( a^{m} \div a^{n} = a^{m-n} \)
8. \( (a^{m})^{n} = a^{mn} \)
9. \( a^{0} = 1 \), where \( a \neq 0 \).
10. \( a^{-m} = \frac{1}{a^m} \).
Chapter 2: Polynomials
1. For a polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \):
Degree = Highest power of \( x \).
2. Remainder Theorem: If \( p(x) \) is divided by \( x – a \), the remainder = \( p(a) \).
3. Factor Theorem: \( x – a \) is a factor of \( p(x) \) if \( p(a) = 0 \).
4. Algebraic Identities:
– \( (a + b)^2 = a^2 + 2ab + b^2 \)
– \( (a – b)^2 = a^2 – 2ab + b^2 \)
– \( a^2 – b^2 = (a – b)(a + b) \)
– \( (x + a)(x + b) = x^2 + (a + b)x + ab \).
5. Cubic Identities:
– \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \)
– \( (a – b)^3 = a^3 – b^3 – 3ab(a – b) \)
– \( a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)
– \( a^3 – b^3 = (a – b)(a^2 + ab + b^2) \).
Chapter 3: Coordinate Geometry
1. Distance Formula:
\( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points.
2. Midpoint Formula:
\( \text{Midpoint } = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
3. Section Formula (Internal Division):
\( \text{Point dividing in ratio } m:n = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \).
4. Area of a Triangle:
\( \text{Area} = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right| \), where \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) are the vertices.
Chapter 4: Linear Equations in Two Variables
1. General form of a linear equation in two variables:
\( ax + by + c = 0 \)
where \( a, b, \text{ and } c \) are constants, and \( a \neq 0 \text{ or } b \neq 0 \).
2. Solution of a linear equation:
A pair \( (x, y) \) that satisfies the equation \( ax + by + c = 0 \).
3. Equation of a straight line:
\( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
Chapter 5: Introduction to Euclid’s Geometry
(No specific formulas, but important postulates include:)
1. A straight line can be drawn from any point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any center and radius.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side.
Chapter 6: Lines and Angles
1. Angle Sum Property of a Triangle:
\( \text{Sum of interior angles} = 180^\circ \).
2. Vertically Opposite Angles:
\( \text{If two lines intersect, } \angle 1 = \angle 3 \text{ and } \angle 2 = \angle 4 \).
3. Linear Pair Axiom:
\( \angle 1 + \angle 2 = 180^\circ \text{ (on a straight line)} \).
4. Parallel Lines:
– Corresponding Angles: \( \angle 1 = \angle 2 \).
– Alternate Interior Angles: \( \angle 3 = \angle 4 \).
– Co-Interior Angles: \( \angle 5 + \angle 6 = 180^\circ \).
Chapter 7: Triangles
1. Pythagoras Theorem (for a right triangle):
\( \text{In } \triangle ABC, \text{ if } \angle C = 90^\circ, \text{ then } AB^2 = BC^2 + AC^2 \).
2. Converse of Pythagoras Theorem:
\( \text{If } AB^2 = BC^2 + AC^2, \text{ then } \angle C = 90^\circ \).
3. Area of an Equilateral Triangle:
\( \text{Area} = \frac{\sqrt{3}}{4} a^2 \), where \( a \) is the side of the triangle.
4. Heron’s Formula for Area of a Triangle:
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \), where:
\( s = \frac{a+b+c}{2} \), and \( a, b, c \) are the sides of the triangle.
5. Similar Triangles (Basic Proportionality Theorem):
If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides the two sides in the same ratio:
\( \frac{DE}{BC} = \frac{AE}{AC} = \frac{BD}{AB} \).
Chapter 8: Quadrilaterals
1. Sum of Interior Angles of a Quadrilateral:
\( \text{Sum of all interior angles} = 360^\circ \).
2. Parallelogram Properties:
– Opposite sides are equal: \( AB = CD, \; BC = AD \).
– Opposite angles are equal: \( \angle A = \angle C, \; \angle B = \angle D \).
– Diagonals bisect each other: \( AO = CO \text{ and } BO = DO \) (where \( O \) is the point of intersection).
3. Area of a Parallelogram:
\( \text{Area} = \text{Base} \times \text{Height} \).
4. Area of a Rhombus:
\( \text{Area} = \frac{1}{2} \times (\text{Diagonal}_1 \times \text{Diagonal}_2) \).
Chapter 9: Circles
1. Circumference of a Circle:
\( \text{Circumference} = 2\pi r \), where \( r \) is the radius.
2. Area of a Circle:
\( \text{Area} = \pi r^2 \).
3. Length of an Arc:
\( \text{Length of arc} = \frac{\text{Angle subtended by arc at center}}{360^\circ} \times 2\pi r \).
4. Area of a Sector:
\( \text{Area of sector} = \frac{\text{Angle subtended by sector at center}}{360^\circ} \times \pi r^2 \).
Chapter 10: Heron’s Formula
1. Area of a Triangle (Heron’s Formula):
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \), where:
\( s = \frac{a+b+c}{2} \), and \( a, b, c \) are the sides of the triangle.
Chapter 11: Surface Areas and Volumes
1. Surface Area of a Sphere:
\( \text{Surface Area} = 4\pi r^2 \).
2. Volume of a Sphere:
\( \text{Volume} = \frac{4}{3} \pi r^3 \).
3. Surface Area of a Hemisphere:
\( \text{Curved Surface Area} = 2\pi r^2 \).
\( \text{Total Surface Area} = 3\pi r^2 \).
4. Volume of a Hemisphere:
\( \text{Volume} = \frac{2}{3} \pi r^3 \).
5. Surface Area of a Cylinder:
\( \text{Curved Surface Area} = 2\pi r h \).
\( \text{Total Surface Area} = 2\pi r(h + r) \).
6. Volume of a Cylinder:
\( \text{Volume} = \pi r^2 h \).
7. Surface Area of a Cone:
\( \text{Curved Surface Area} = \pi r l \), where \( l = \sqrt{r^2 + h^2} \).
\( \text{Total Surface Area} = \pi r(l + r) \).
8. Volume of a Cone:
\( \text{Volume} = \frac{1}{3} \pi r^2 h \).
