Class 9 — Maths Formula Sheet

√a × √b = √(ab)
√a / √b = √(a/b)
(√a)² = a

Basic operations with surds (irrational square roots).

1/(a + √b) = (a − √b) / (a² − b)

Multiply by conjugate to remove surd from denominator.

If p(a) = 0, then (x − a) is a factor of p(x)

A polynomial p(x) has (x−a) as a factor iff p(a) = 0.

Remainder when p(x) ÷ (x − a) = p(a)

Substitute x = a to find the remainder directly.

(a+b+c)² = a²+b²+c²+2ab+2bc+2ca

Square of a trinomial.

💡 (1+2+3)² = 1+4+9+4+12+6 = 36

a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)

Special factorisation for sum of cubes minus 3abc.

If a+b+c = 0, then a³+b³+c³ = 3abc

Simplification when sum of three terms is zero.

d = √[(x₂−x₁)² + (y₂−y₁)²]

Distance between two points in the plane.

💡 A(1,2), B(4,6): d=√(9+16)=5

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint of a line segment.

💡 A(2,4), B(6,8): M=(4,6)

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Point dividing AB in ratio m:n.

📌 m:n = ratio, A(x₁,y₁), B(x₂,y₂)

c² = a² + b²

In a right triangle, square of hypotenuse = sum of squares of other two sides.

📌 c = hypotenuse, a, b = other sides

💡 3²+4²=5² ✓

If c² = a² + b², then ∠C = 90°

If the square of the longest side equals sum of squares of others, angle is 90°.

A = √[s(s−a)(s−b)(s−c)]
where s = (a+b+c)/2

Area of a triangle when all three sides are known.

📌 a, b, c = sides, s = semi-perimeter

💡 a=3,b=4,c=5: s=6, A=√(6×3×2×1)=6

SA = 4πr²

Total surface area of a sphere.

📌 r = radius

V = (4/3)πr³

Volume enclosed by a sphere.

📌 r = radius

CSA = 2πr²

Curved surface area of a hemisphere.

TSA = 3πr²

Total surface area including flat circular face.

CSA = πrl

Curved surface area of a cone.

📌 r = radius, l = slant height

TSA = πr(r + l)

Total surface area of a cone including base.

📌 l = slant height = √(r²+h²)

V = (1/3)πr²h

Volume of a cone.

📌 r = base radius, h = height

l = √(r² + h²)

Slant height from apex to base edge.

📌 r = radius, h = height