Revision reference
Every formula in the CBSE Class 8 syllabus, unit by unit — and next to each one, the short reason it is true. Formulas you understand are the ones you still remember in the exam hall.
Rational numbers behave exactly like fractions — every rule is the fraction rule wearing a new name.
a/b + c/b = (a + c)/b
Same denominator: just add the numerators.
a/b + c/d = (ad + bc)/bd
Give both fractions the same bottom (bd) first.
a/b × c/d = ac/bd
Multiply tops together and bottoms together — no common denominator needed.
a/b ÷ c/d = a/b × d/c
Dividing = flipping the second fraction and multiplying.
Additive inverse of a/b is −a/b
a/b + (−a/b) = 0 — they cancel to nothing.
Reciprocal of a/b is b/a
a/b × b/a = 1 — the number that undoes it.
a(b + c) = ab + ac
Distributive law — b of one thing plus c of another, a times each.
Between two rationals lies another rational
Their average (a + b)/2 always fits in between — so they never run out.
An equation is a balance scale — whatever you do to one side, do to the other, and it stays true.
ax + b = c → x = (c − b)/a
Undo in reverse order: shift b, then divide by a.
Moving a term across = changing its sign
Transposition: +5 on the left becomes −5 on the right.
a/x = b → x = b/a ... check: ax = b → x = b/a
Cross-multiplication clears the fraction in one step.
Multiply both sides by the LCM to clear denominators
The balance survives, but the fractions disappear.
Equations with brackets: expand first, then collect x terms
Same unknowns to one side, numbers to the other.
Always verify by substituting the answer back
Both sides must give the same value — your built-in error check.
Every polygon is a set of triangles glued together — the angle rules all come from the triangle's 180°.
Sum of interior angles of an n-gon = (n − 2) × 180°
n − 2 triangles fit inside any polygon.
Each exterior angle of a regular n-gon = 360°/n
Walking the boundary, you turn a full circle: 360°.
Each interior angle of a regular n-gon = 180° − 360°/n
Interior + exterior at a vertex make a straight line.
Sum of exterior angles of any polygon = 360°
One full turn, whatever the shape.
Parallelogram: opposite sides equal and parallel
The diagonal splits it into congruent triangles.
Parallelogram: diagonals bisect each other
The two crossing triangles are identical copies.
Rhombus: all sides equal, diagonals bisect at 90°
A pushed-over square — symmetry forces the right angle.
Rectangle: all angles 90°, diagonals equal
Right angles make the two diagonals congruent.
Kite: two pairs of adjacent equal sides, one diagonal bisects the other
Fold it along the long diagonal — the halves match.
Trapezium: exactly one pair of parallel sides
The parallel pair is what separates it from general quadrilaterals.
Probability measures how much of the whole 'pie' of outcomes your event takes up.
Probability = favourable outcomes / total outcomes
Your share of all equally likely results.
0 ≤ P(E) ≤ 1
You can't have less than none or more than everything.
P(not E) = 1 − P(E)
Either it happens or it doesn't — the two shares add to 1.
Central angle of a pie slice = (value/total) × 360°
A full circle is 360°, shared out proportionally.
A square number counts dots arranged in a perfect square — its properties fall out of that picture.
n² = n × n
The area of a square with side n.
Square numbers end in 0, 1, 4, 5, 6 or 9 only
Square the last digit and see — 2, 3, 7, 8 never appear.
Numbers ending in 2, 3, 7, 8 are never perfect squares
The same fact read backwards — a quick rejection test.
√(a × b) = √a × √b
Pairing factors: whatever pairs up completely comes out of the root.
√a × √a = a
The square root is the power that undoes squaring.
Pythagorean triplet: 2m, m² − 1, m² + 1
Pick any m > 1 and these three always satisfy a² + b² = c².
Sum of first n odd numbers = n²
L-shaped strips of 1, 3, 5, 7... build up a square dot by dot.
Between n² and (n+1)² there are 2n non-square numbers
(n+1)² − n² − 1 = 2n.
A cube number counts blocks stacked into an actual cube — you need 27 of them to make 3×3×3.
n³ = n × n × n
The volume of a cube with side n.
Cubes of even numbers are even; of odd numbers, odd
n × n × n keeps whatever parity n had.
Perfect cubes end in any digit (unlike squares)
8 ends in 8, 27 ends in 7 — no forbidden endings.
³√(a × b) = ³√a × ³√b
Grouping prime factors in triples: triples come out, leftovers stay in.
A number is a perfect cube ⟺ every prime factor appears a multiple of 3 times
72 = 2³ × 3² — the lone 3² stops it from being one.
1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²
The sum of cubes is the square of the triangular sum.
Percent, profit and interest are all one idea: a fraction of something, times 100 for display.
Percentage = (part/whole) × 100%
How many parts out of every hundred.
Profit % = (profit/cost price) × 100
Always measured on what you paid, not what you sold for.
Loss % = (loss/cost price) × 100
Same base — the cost price.
Discount = marked price − sale price
Discount % is on the marked price.
SP = MP × (100 − discount%)/100
Paying, say, 80% of the ticket after a 20% cut.
Simple Interest = P × R × T / 100
Interest recalculated on the original principal every year.
Amount (CI) = P(1 + R/100)ⁿ
Each year's interest joins the principal — growth on growth.
CI = P(1 + R/100)ⁿ − P
The compound amount minus what you started with.
Half-yearly compounding: R/2 per period, 2n periods
The rate halves, the count doubles.
Growth/decay: new value = old × (1 ± R/100)ⁿ
Population and depreciation follow the same compounding rule.
Identities are true for every value of the letters — they come from cutting and rearranging areas, not from expanding.
(a + b)² = a² + 2ab + b²
A square of side a + b splits into four regions.
(a − b)² = a² − 2ab + b²
Same picture, but the ab strip overlaps and is counted twice.
a² − b² = (a + b)(a − b)
Cut the b-square out of the a-square and rearrange into a rectangle.
(x + a)(x + b) = x² + (a + b)x + ab
The middle coefficient is the sum, the constant is the product.
(a + b)³ = a³ + b³ + 3ab(a + b)
A cube of side a + b cut into pieces.
(a − b)³ = a³ − b³ − 3ab(a − b)
Replace b with −b in the expansion above.
(x + a)(x + b) × (x + c): add pairwise products for the x² term
Each pair of constants multiplies to build one middle term.
Evaluating: substitute, then respect the order of operations
Brackets, powers, multiply/divide, add/subtract.
Surface area counts the wrapping; volume counts the filling. Every formula is one of those two jobs.
Trapezium: A = ½ × (a + b) × h
Average of the parallel sides, times the distance between them.
Rhombus: A = ½ × d₁ × d₂
The diagonals cut it into four right triangles — two fill each half.
General quadrilateral: A = ½ × d × (h₁ + h₂)
One diagonal, with the two heights hanging off it.
Cuboid: LSA = 2h(l + b), TSA = 2(lb + bh + hl), V = lbh
Four wall rectangles; six faces in matching pairs; length × breadth × height.
Cube: LSA = 4a², TSA = 6a², V = a³
A cuboid with l = b = h = a.
Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
A rolled-up rectangle is the curved wall; two circles cap the ends.
1 m³ = 1,000,000 cm³; 1 L = 1000 cm³ = 1 dm³
Cubing the conversion factor of 100.
A negative exponent doesn't make the number negative — it makes it a fraction. Small minus, small value.
aᵐ × aⁿ = aᵐ⁺ⁿ
Counting the factors: m of them plus n more.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
m factors, then n of them cancel out.
(aᵐ)ⁿ = aᵐⁿ
n groups of m factors each.
aⁿ × bⁿ = (ab)ⁿ
n copies of a next to n copies of b pair up.
a⁻ᵐ = 1/aᵐ
A negative power moves the number below the fraction bar.
(a/b)⁻ⁿ = (b/a)ⁿ
A negative power flips the fraction.
a⁰ = 1 (a ≠ 0)
aᵐ ÷ aᵐ = 1, and the subtract rule says it equals a⁰.
Small numbers: 0.0000456 = 4.56 × 10⁻⁵
Count how far the decimal point travels to standard form.
Two quantities are linked when one is a constant multiple (direct) or a constant product (inverse) of the other.
Direct: x₁/y₁ = x₂/y₂
Double one, the other doubles — the ratio stays fixed.
Inverse: x₁y₁ = x₂y₂
Double one, the other halves — the product stays fixed.
Direct check: y/x is constant
Divide at every step; the same answer means direct proportion.
Inverse check: xy is constant
Multiply at every step; the same answer means inverse proportion.
Speed–time (fixed distance) is inverse
Same journey: faster speed, proportionally less time.
Workers–days (fixed work) is inverse
Same job: more workers, proportionally fewer days.
Factorisation is multiplication played in reverse — you are asking what was multiplied to build this expression.
ab + ac = a(b + c)
Pull out the common factor — the reverse of the distributive law.
x² + (a + b)x + ab = (x + a)(x + b)
Split the middle term into two numbers whose sum is a + b and product is ab.
a² − b² = (a + b)(a − b)
Difference of squares always factors into a sum and a difference.
Grouping: ab + bc + ad + cd... pair terms with common factors
(ab + bc) + (ad + cd) = b(a + c) + d(a + c) = (a + c)(b + d).
Division of a polynomial by a monomial: divide each term
Split it into terms and cancel each one.
Division by a polynomial: factorise both, cancel common factors
Cancellation only ever works across factors, never across + and −.
A graph turns a table of numbers into a picture — patterns that hide in rows of values become visible slopes.
Point = (x, y)
x = horizontal distance from the origin, y = vertical.
x-axis: the horizontal number line (y = 0)
Points on it have no vertical distance.
y-axis: the vertical number line (x = 0)
Points on it have no horizontal distance.
A linear graph is a straight line
Two points fix it; a third point confirms the relation is linear.
Direct proportion graphs pass through the origin
Zero of one quantity means zero of the other.
Steeper line = faster change
The slope shows how quickly y grows with x.
Each Class 8 chapter has an animated lesson, a worked example and a quiz that explains every answer.