Class 8 Maths Formulas Handbook (Chapter-wise)

Chapter-wise formulas and key results for Class 8 Mathematics (Ganita Prakash), updated as each chapter is added.

Class 8 Maths Formulas Handbook (Chapter-wise)

Chapter 1: A Square and A Cube

• Square of n: n² = n×n. Perfect square condition (prime factorisation): every prime’s exponent is even.
• Cube of n: n³ = n×n×n. Perfect cube condition (prime factorisation): every prime’s exponent is a multiple of 3.
• Numbers between n² and (n+1)²: 2n.
• Sum of first n consecutive odd numbers: 1+3+5+…+(2n−1) = n².
• nth run of n consecutive odd numbers sums to n³ (e.g. 7+9+11 = 3³).
• Trailing zeros: squares always have an even number of trailing zeros; cubes always have a multiple-of-3 number of trailing zeros.
• Last-digit rule for cube roots: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9 (last digit of n maps to last digit of n³, and vice versa for the cube root).
• (a+1)² shortcut: (a+1)² = a² + 2a + 1.

Chapter 2: Power Play

• Product rule: a^m×aⁿ = a^m+n
• Quotient rule: a^m÷aⁿ = a^m−n (a≠0)
• Power of a power: (a^m)ⁿ = a^mn
• Power of a product/quotient: a^m×b^m = (ab)^m; a^m÷b^m = (a÷b)^m
• Zero exponent: a⁰ = 1 (a≠0)
• Negative exponent: a⁻ⁿ = 1/aⁿ
• Standard form: x×10^y, where 1≤x<10
• See full Chapter 2 Solutions

Chapter 3: A Story of Numbers

• Landmark numbers of a base-n system: n⁰=1, n, n², n³…
• Place-value system: a symbol’s position determines its value (e.g. Mesopotamian base-60, Hindu base-10)
• Zero: acts as a placeholder AND a number, resolving the ambiguity of additive/landmark systems
• Any number can be converted to any base by repeated division and taking remainders
• See full Chapter 3 Solutions

Chapter 4: Quadrilaterals

• Angle sum of a quadrilateral = 360 degrees
• Rectangle: diagonals equal and bisect each other
• Square: diagonals equal, bisect each other, perpendicular, bisect vertex angles
• Rhombus: diagonals perpendicular bisectors of each other (not necessarily equal)
• Kite: one diagonal is perpendicular bisector of the other (adjacent sides equal, not opposite)
• Trapezium: co-interior angles on the parallel sides sum to 180 degrees
• See full Chapter 4 Solutions

Chapter 5: Number Play

• Divisibility by 9: digit sum divisible by 9
• Divisibility by 11: alternating digit-sum difference is 0 or a multiple of 11
• Digital root: repeatedly sum digits until one digit remains; equals remainder mod 9 (digital root 9 = remainder 0)
• Product of n consecutive integers is always divisible by n! (2 consec: div by 2, 3 consec: div by 6, 4 consec: div by 24, 5 consec: div by 120)
• Numbers leaving remainder r on division by d: general form dn+r
• See full Chapter 5 Solutions

Chapter 6: We Distribute, Yet Things Multiply

• Distributive property: a(b+c) = ab+ac — expands any product of multi-term expressions term by term
• Identity 1A: (a+b)² = a²+2ab+b²
• Identity 1B: (a−b)² = a²−2ab+b²; also (a−b)² = (b−a)² always
• Identity 1C: (a+b)(a−b) = a²−b² — used for fast mental multiplication of numbers equidistant from a round number
• General pattern: (a−b)(aⁿ+aⁿ⁻¹b+…+bⁿ) = aⁿ⁺¹−bⁿ⁺¹
• 2(a²+b²) = (a+b)²+(a−b)² — holds for integers, negatives and fractions alike
• See full Chapter 6 Solutions

Chapter 7: Proportional Reasoning-1

• Ratio: a:b = a/b; must express both terms in the same unit before comparing
• Proportion: a:b::c:d is true when a×d = b×c (cross products equal)
• Simplest form: divide both terms of a ratio by their HCF
• Rule of Three (unitary method): find the value for one unit first, then scale to the required amount
• Direct proportion: ratio between the two quantities stays constant as both increase together
• Inverse proportion: product of the two quantities stays constant (one increases as the other decreases) — e.g. speed × time = fixed distance
• Sharing in ratio a:b: divide total into (a+b) equal parts, then give a parts and b parts respectively
• See full Chapter 7 Solutions

Chapter 8: Fractions in Disguise

• Percentage: x% = x/100 of a quantity.
• x% of y = y% of x = xy/100.
• Profit%/Loss% = (Profit or Loss / CP) x 100.
• Discount: SP = MP x (1 – discount%/100).
• Successive % changes multiply as factors, not add.
• Simple Interest: SI = P x r x t / 100.
• Compound Interest: Amount = P(1 + r/100)^t.
• GST/tax: Final price = Price x (1 + GST%/100).
• Reverse percentage: Original = Final / (1 +/- r/100).

See full Chapter 8 Solutions.

Chapter 9: The Baudhayana-Pythagoras Theorem

• Baudhayana-Pythagoras theorem: (hypotenuse)^2 = (leg 1)^2 + (leg 2)^2.
• Isosceles right triangle: hypotenuse = side x root(2).
• Baudhayana (Pythagorean) triple: (a,b,c) with a^2+b^2=c^2.
• Primitive triple: HCF(a,b,c) = 1, e.g. (3,4,5), (5,12,13), (8,15,17).
• Odd-square method: for odd n, (n, (n^2-1)/2, (n^2+1)/2) is a primitive triple.
• Hypotenuse is always the longest side of a right triangle.
• Lattice-grid square rule: area x is possible exactly when x = p^2+q^2 for integers p,q.
• Area of equilateral triangle (side a) via altitude: (root(3)/4) x a^2.

See full Chapter 9 Solutions.

Chapter 10: Proportional Reasoning 2

• Dividing a whole in a ratio: each share = (part’s ratio number / sum of ratio numbers) x total.
• Pie chart angle for a category = (category count / total count) x 360 degrees.
• Direct proportion: x/y = constant. Inverse proportion: x times y = constant.
• Test for inverse proportion: check x1y1 = x2y2 = x3y3 … for every pair.
• Common inverse cases: workers vs days, taps vs fill-time, speed vs time (fixed distance), machines vs days.
• Common direct cases: quantity bought vs price (fixed rate), pages vs reading time (fixed speed).
• Combined work rate = sum of individual rates (1/time each); total time = 1 / combined rate.

See full Chapter 10 Solutions.

Chapter 11: Exploring Some Geometric Themes

• Fractal: a shape built by repeating the same rule at smaller scales (Sierpinski Triangle, Sierpinski Carpet, Koch Snowflake).
• Sierpinski Triangle holes at step n = (3^n – 1)/2. Area remaining = (3/4)^n x original.
• Koch Snowflake sides at step n = 3 x 4^n. Perimeter = 3 x (4/3)^n x original side (grows without bound).
• Cube has exactly 11 distinct nets.
• Prism (n-sided base): Faces = n+2, Edges = 3n, Vertices = 2n.
• Pyramid (n-sided base): Faces = n+1, Edges = 2n, Vertices = n+1.
• Euler’s formula: Vertices – Edges + Faces = 2 for any convex polyhedron.
• Cylinder net = rectangle (2 pi r x h) + 2 circles of radius r. Cone net = sector (arc = 2 pi r) + 1 circle of radius r.

See full Chapter 11 Solutions.

Chapter 12: Tales by Dots and Lines

• Mean = (sum of all values) / (number of values)
• Mean of first n natural numbers = (n+1)/2; Mean of first n odd numbers = n
• Median: middle value when data is sorted (odd count); average of two middle values (even count)
• Adding a value above the mean raises the mean; adding a value below the mean lowers it. Median shifts only if the new value affects the middle position.
• Combined mean of two groups is NOT simply the average of their means unless the groups are equal-sized — must weight by group size.
• Mean is sensitive to outliers; median is more robust to extreme values (e.g. skewed salary data).
• Dot plots and line graphs are used to visualise and interpret data trends.

See full Chapter 12 Solutions.

Chapter 13: Algebra Play

• “Think of a number” tricks: represent the number as x; multiply/add/divide steps cancel the x-term, leaving a fixed constant.
• 3-row pyramid (bottom a,b,c): top = a+2b+c. 4-row pyramid (bottom a,b,c,d): top = a+3b+3c+d (Pascal’s triangle coefficients).
• Virahanka-Fibonacci pyramid rule: if the first n Fibonacci numbers form the bottom row of an n-row pyramid, the top is the (2n−1)th Fibonacci number.
• 3×3 grid sum = 9 × centre value; 1×3 strip sum = 3 × middle value.
• Largest product (2-digit × 1-digit from 3 given digits): largest digit as the multiplier, remaining two digits in descending order.
• Divisibility tricks: 2-digit number minus reverse ÷ 9; plus reverse ÷ 11; 3-digit number + 2 cyclic rotations ÷ 37 (and 3); 3-digit number repeated as 6 digits ÷ 1001 (=7×11×13).

See full Chapter 13 Solutions.

Chapter 14: Area (final Class 8 Maths chapter)

• Rectangle: Area = l×b. Square: Area = side². Triangle: Area = ½×base×height.
• Parallelogram: Area = base × perpendicular height (not the slant side).
• Rhombus: Area = ½ × d1 × d2. Trapezium: Area = ½ × (sum of parallel sides) × height.
• Regular hexagon: Area = (3√3/2)×side². Equilateral triangle: Area = (√3/4)×side².
• Scaling rule: linear dimensions scaled by k → area scales by k².
• Path/border area = outer area − inner area; uniform border of width d around l×w: Area = 2d(l+w)+4d².
• Midpoint triangle theorem: joining midpoints of two sides creates a triangle with ¼ the area. Varignon parallelogram: midpoints of any quadrilateral form a parallelogram with half its area.
• Unit conversions: 1 in = 2.54 cm; 1 ft² = 144 in²; 1 km² = 1,000,000 m².

See full Chapter 14 Solutions.

This completes the Class 8 Maths Formulas Handbook (Chapters 1–14).

Written by Satish

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