ICSE Class 7 Maths All Formulas: Complete Selina Formula Sheet

Every year around exam week a parent messages me saying their child has forgotten which formula goes with which chapter. 'We have three days to the test and she does not know perimeter from area.' That moment is why this page exists. It is designed to be the single document you bookmark on your phone, the one you scroll through ten minutes before every class test, and the one your child can read during the school bus ride on exam day.

The Selina ICSE Class 7 syllabus covers roughly twenty chapters. Some are heavy on formulas, others are heavy on reasoning. This sheet lists every formula that is likely to be tested, along with a one line explanation for each so you understand what it does, not just what it looks like. Where a formula has a standard memory trick, I have included that too.

Use this page alongside the individual chapter guides. Read the guide first for deep understanding, then come back here for fast recall. A week of guide reading plus five minutes of formula scanning every morning is usually enough to feel confident on exam day.

The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$. The basic operation rules are the building blocks for everything that follows.

Addition rules
* Same signs: add absolute values, keep the sign.
* Different signs: subtract the smaller from the larger, keep the sign of the larger.

Multiplication and division sign rules

Same rules apply for division.

Properties
* Closure under addition, subtraction and multiplication.
* Commutative: $a + b = b + a$, $a \times b = b \times a$.
* Associative: $(a + b) + c = a + (b + c)$.
* Distributive: $a \times (b + c) = a \times b + a \times c$.
* Additive identity: $a + 0 = a$.
* Multiplicative identity: $a \times 1 = a$.
* Additive inverse of $a$ is $-a$.

Multiplication of fractions: multiply numerators and denominators separately.

Division of fractions: multiply by the reciprocal.

Reciprocal: the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ (assuming $a \ne 0$).

Decimal multiplication: multiply as whole numbers, then place the decimal by counting the total decimal places in the original numbers.

Decimal division: shift the decimal in both divisor and dividend until the divisor is a whole number, then divide normally.

Conversion: to convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a fraction, write it with a power of 10 in the denominator and simplify.

A rational number is any number of the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$.

Addition

Subtraction

Properties: rational numbers are closed under addition, subtraction and multiplication. They are not closed under division by zero. All the usual properties like commutativity and associativity hold.

Standard form: a rational number is in standard form when the numerator and denominator have no common factors other than 1 and the denominator is positive.

For any nonzero integer $a$ and positive integers $m$, $n$:

Negative exponents (brief mention in Class 7):

Standard form: a number expressed as $k \times 10^n$ where $1 \le k < 10$ and $n$ is an integer. For example, $45000 = 4.5 \times 10^4$.

An expression has terms, coefficients, variables and constants. Types by number of terms:
* Monomial: 1 term
* Binomial: 2 terms
* Trinomial: 3 terms
* Polynomial: general term for any expression with one or more terms

Rule for combining: only like terms (same variable part including powers) can be added or subtracted.

Multiplication rules for variables

Distributive law for multiplying: $a(b + c) = ab + ac$.

Removing brackets: when a plus sign is in front, signs inside stay the same. When a minus sign is in front, every sign inside flips.

Value of an expression: substitute the given values for each variable, then evaluate step by step. Always put negative substitutions in brackets to avoid sign errors.

A linear equation has the form $ax + b = c$ where $a$, $b$, $c$ are constants and $x$ is the variable.

Balance method: do the same operation to both sides to keep the equation balanced.

Transposition: a term crossing the equal sign flips its operation.
* $+$ becomes $-$
* $-$ becomes $+$
* $\times$ becomes $\div$
* $\div$ becomes $\times$

For fractions: multiply every term by the LCM of the denominators to clear fractions before solving.

Word problems four step method: read, name the unknown, write the equation, solve and answer in words.

A set is a collection of well defined, distinct objects.

Element notation: $x \in A$ means $x$ is an element of $A$.

Forms
* Roster form: list elements in curly braces.
* Set builder form: describe by a rule, e.g. $\{x : x \text{ is an even number}\}$.

Types of sets: empty, singleton, finite, infinite, equal, equivalent.

Cardinal number: $n(A)$ is the number of elements in $A$.

Subsets: $B \subseteq A$ if every element of $B$ is in $A$. Number of subsets of a set with $n$ elements is $2^n$.

Operations

Disjoint sets: $A \cap B = \emptyset$.

A ratio compares two quantities of the same kind: $a : b = \frac{a}{b}$.

Proportion: four numbers $a$, $b$, $c$, $d$ are in proportion if $a : b = c : d$, i.e. $\frac{a}{b} = \frac{c}{d}$.

Cross product rule: if $a : b = c : d$, then $a \times d = b \times c$.

Direct variation: $y = kx$ where $k$ is constant. Two quantities increase or decrease together.

Inverse variation: $xy = k$. As one increases, the other decreases proportionally.

Unitary method: find the value of one unit first, then multiply or divide by the required number of units.

Percent means per hundred. $x\% = \frac{x}{100}$.

Percent of a quantity

One as percent of another

Percentage increase

Percentage decrease

Common fraction equivalents: $\frac{1}{2} = 50\%$, $\frac{1}{4} = 25\%$, $\frac{1}{5} = 20\%$, $\frac{1}{8} = 12.5\%$, $\frac{1}{3} = 33\frac{1}{3}\%$.

* Profit $= SP - CP$ (when SP > CP)
* Loss $= CP - SP$ (when SP < CP)

Profit and loss percent (always based on CP)

Selling price formulas

Cost price from SP

Discount

Successive discount shortcut: for two discounts of $a\%$ and $b\%$, effective discount $= a + b - \frac{ab}{100}$.

Where $P$ is the principal, $R$ is the rate percent per annum, $T$ is the time in years, and $SI$ is the simple interest earned.

Amount: $A = P + SI$.

Rearranged formulas

Time unit conversions
* Months to years: divide by 12
* Days to years: divide by 365 (Selina convention)

Types of angles
* Acute: $0° < \theta < 90°$
* Right: $\theta = 90°$
* Obtuse: $90° < \theta < 180°$
* Straight: $\theta = 180°$
* Reflex: $180° < \theta < 360°$

Pair of angles
* Complementary angles: sum to $90°$.
* Supplementary angles: sum to $180°$.
* Linear pair: two adjacent angles on a straight line, sum to $180°$.
* Vertically opposite angles: equal.

When two parallel lines are cut by a transversal
* Corresponding angles are equal.
* Alternate interior angles are equal.
* Co interior angles sum to $180°$.

Angle sum property: in any triangle, $\angle A + \angle B + \angle C = 180°$.

Exterior angle theorem: exterior angle = sum of the two opposite interior angles.

Triangle inequality: the sum of any two sides is greater than the third side.

Classification by sides: equilateral (all equal), isosceles (two equal), scalene (all different).

Classification by angles: acute (all less than $90°$), right (one equals $90°$), obtuse (one greater than $90°$).

Isosceles triangle property: angles opposite to equal sides are equal.

Two triangles are congruent if they have the same shape and size. Symbol: $\cong$.

Four criteria
* SSS: three sides equal.
* SAS: two sides and the included angle equal.
* ASA: two angles and the included side equal.
* RHS: for right triangles, hypotenuse and one other side equal.

CPCT: Corresponding Parts of Congruent Triangles are equal. Use this after proving congruence to claim other equal parts.

In a right angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Or symbolically, if $c$ is the hypotenuse and $a$, $b$ are the other two sides:

Pythagorean triples: integer solutions to $a^2 + b^2 = c^2$. Common ones: $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$.

Angle sum property: the sum of interior angles of any quadrilateral is $360°$.

Types of quadrilaterals and properties
* Square: all sides equal, all angles $90°$, diagonals equal and perpendicular.
* Rectangle: opposite sides equal, all angles $90°$, diagonals equal.
* Rhombus: all sides equal, opposite angles equal, diagonals perpendicular.
* Parallelogram: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.
* Trapezium: exactly one pair of parallel sides.

Square: side $= s$

Rectangle: length $l$, breadth $b$

Triangle: base $b$, height $h$

Parallelogram: base $b$, height $h$

Circle: radius $r$

Use $\pi = \frac{22}{7}$ unless the question states otherwise.

Area of a path around a rectangle of outer dimensions $L \times B$ and inner $l \times b$:

Mean (average): $\text{mean} = \frac{\text{sum of observations}}{\text{number of observations}}$

Median: the middle value when data is arranged in order. For an even number of observations, the median is the average of the two middle values.

Mode: the value that occurs most frequently.

Range: maximum value minus minimum value.

Probability (introductory)

Probability lies between 0 and 1 inclusive.

Line of symmetry: a line that divides a figure into two identical halves.
* Square: 4 lines of symmetry
* Rectangle: 2 lines
* Equilateral triangle: 3 lines
* Isosceles triangle: 1 line
* Regular polygon of $n$ sides: $n$ lines

Rotational symmetry: order of rotational symmetry is the number of times a figure looks identical in one full rotation.

3D shapes and parts
* Cube: 6 faces, 12 edges, 8 vertices
* Cuboid: 6 faces, 12 edges, 8 vertices
* Cylinder: 2 circular faces, 1 curved surface
* Cone: 1 circular face, 1 curved surface, 1 vertex
* Sphere: 1 curved surface, no edges, no vertices