Mathematics Formula Sheet — Class 10
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Mathematics Formula Sheet
68 formulas across 14 chapters — with variables explained and exam tips where needed.
Ch 1Real Numbers(2 formulas)
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes in a unique way (order apart)
Used to find HCF and LCM via prime factorisation
Irrationality proof method
Assume √p = a/b (in lowest terms) → p divides a² → p divides a → p divides b → contradiction
Applies to √2, √3, √5 and combinations like 3 + 2√5
Ch 2Polynomials(2 formulas)
Sum of zeros (quadratic)
α + β = −b/a
ax² + bx + c = 0
Product of zeros (quadratic)
αβ = c/a
2025-26 syllabus covers quadratic polynomials only
Ch 3Pair of Linear Equations in Two Variables(4 formulas)
General form
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Two equations, two unknowns
Unique solution (consistent)
a₁/a₂ ≠ b₁/b₂
Lines intersect at exactly one point
Infinitely many solutions
a₁/a₂ = b₁/b₂ = c₁/c₂
Lines coincide (dependent equations)
No solution (inconsistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Lines are parallel — no point of intersection
Ch 4Quadratic Equations(4 formulas)
Quadratic formula
x = [−b ± √(b²−4ac)] / 2a
b²−4ac is the discriminant D
D > 0: two distinct real roots; D = 0: equal roots; D < 0: no real roots
Discriminant
D = b² − 4ac
Sum of roots
x₁ + x₂ = −b/a
Product of roots
x₁ × x₂ = c/a
Ch 5Arithmetic Progressions(4 formulas)
nth term of AP
aₙ = a + (n−1)d
a = first term, d = common difference, n = term number
Sum of n terms
Sₙ = n/2 × [2a + (n−1)d]
Also written as Sₙ = n/2 × (a + l) where l is the last term
Common difference
d = aₙ₊₁ − aₙ (any consecutive pair)
Number of terms
n = [(l − a)/d] + 1
l = last term
Ch 6Triangles(3 formulas)
Basic Proportionality Theorem (BPT)
If DE ∥ BC, then AD/DB = AE/EC
Pythagoras Theorem
AC² = AB² + BC²
In right triangle, AC = hypotenuse
Converse: if AC² = AB² + BC², then angle B = 90°
Area ratio of similar triangles
Area(△ABC)/Area(△PQR) = (AB/PQ)² = (BC/QR)² = (CA/RP)²
Ch 7Coordinate Geometry(4 formulas)
Distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Section formula (internal)
P = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)]
P divides AB in ratio m:n
Midpoint formula
M = [(x₁+x₂)/2, (y₁+y₂)/2]
Area of triangle
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
If area = 0, points are collinear
Ch 8Introduction to Trigonometry(7 formulas)
Basic ratios
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Reciprocal ratios
cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Pythagorean identity 1
sin²θ + cos²θ = 1
Pythagorean identity 2
1 + tan²θ = sec²θ
Pythagorean identity 3
1 + cot²θ = cosec²θ
Standard angle values
sin 30°=½, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1
cos values: reverse order. tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90°=undefined
Complementary angles
sin(90°−θ) = cosθ, cos(90°−θ) = sinθ, tan(90°−θ) = cotθ
Ch 9Heights and Distances(5 formulas)
Angle of elevation
tan θ = height / horizontal distance
Angle measured upward from horizontal to line of sight
Angle of depression
tan θ = height / horizontal distance
Angle measured downward from horizontal to line of sight; angle of depression = angle of elevation (alternate interior angles)
tan values for standard angles
tan 30° = 1/√3 | tan 45° = 1 | tan 60° = √3
CBSE 2025-26 restricts angles to 30°, 45°, 60° only in problems
Shadow length
tan θ = height of object / length of shadow
θ = angle of elevation of sun
Longer shadow → smaller angle of elevation (sun is lower in sky)
Two-triangle problems
Set up two equations using tan for both angles; solve simultaneously
CBSE limits problems to max two right triangles
Ch 10Circles(3 formulas)
Tangent-radius angle
The tangent at any point of a circle is perpendicular to the radius through the point of contact → angle = 90°
Equal tangents from external point
PA = PB (tangents from external point P to points A and B on circle are equal)
Derived from congruent triangles OAP and OBP
Tangent length
PT = √(d² − r²)
P = external point, d = distance from P to centre, r = radius
Ch 11Areas Related to Circles(6 formulas)
Area of circle
cm²A = πr²
Circumference of circle
cmC = 2πr
Length of arc
l = (θ/360°) × 2πr
θ = central angle in degrees
Area of sector
A = (θ/360°) × πr²
Sector = 'pizza slice'
Area of minor segment
Area = Area of sector − Area of triangle
Triangle formed by the two radii and the chord
Area of major segment
Area = Area of circle − Area of minor segment
Ch 12Surface Areas and Volumes(13 formulas)
Sphere: Volume
cm³V = (4/3)πr³
Sphere: Surface area
cm²SA = 4πr²
Hemisphere: Volume
V = (2/3)πr³
Hemisphere: Curved SA
CSA = 2πr²
Cylinder: Volume
V = πr²h
Cylinder: Curved SA
CSA = 2πrh
Cone: Volume
V = (1/3)πr²h
Cone: Slant height
l = √(r² + h²)
Cone: Curved SA
CSA = πrl
Cube: Volume
V = a³
a = side length
Cube: Total SA
TSA = 6a²
Cuboid: Volume
V = l × b × h
l = length, b = breadth, h = height
Cuboid: Total SA
TSA = 2(lb + bh + lh)
Ch 13Statistics(6 formulas)
Mean (direct method)
x̄ = Σfᵢxᵢ / Σfᵢ
Mean (assumed mean)
x̄ = a + (Σfᵢdᵢ / Σfᵢ)
dᵢ = xᵢ − a, a = assumed mean
Mean (step deviation)
x̄ = a + h × (Σfᵢuᵢ / Σfᵢ)
uᵢ = (xᵢ − a)/h, h = class width, a = assumed mean
Most efficient method for equal class widths — use when numbers are large
Median
M = l + [(n/2 − cf)/f] × h
l = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class size
Mode
Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
f₁ = modal class frequency, f₀ = preceding class frequency, f₂ = succeeding class frequency
Empirical relationship
Mode = 3(Median) − 2(Mean)
Ch 14Probability(5 formulas)
Probability of event
P(E) = Number of favourable outcomes / Total outcomes
Complement rule
P(Ē) = 1 − P(E)
Impossible event
P(impossible event) = 0
Sure event
P(sure event) = 1
Range
0 ≤ P(E) ≤ 1