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Junior Cycle Maths: Formulae & Rules

52 flashcards on area, volume, algebra, geometry, trigonometry, coordinate geometry, statistics and probability for Junior Cycle Mathematics.

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Area of a rectangle and a triangle

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Area of a rectangle and a triangle

Rectangle: length

\times

width. Triangle:

\frac{1}{2} \times

base

\times

height.

Area and circumference of a circle

A = π r^{2}

;

C = 2 π r

Volume of a cylinder

π r^{2} h

Volume of a cone and a sphere

Cone:

\frac{1}{3} π r^{2} h

. Sphere:

\frac{4}{3} π r^{3}

Theorem of Pythagoras

a^{2} + b^{2} = c^{2}

in a right-angled triangle, with

c

the hypotenuse.

Sine, cosine and tangent

sin A = \frac{o pp}{h y p}

cos A = \frac{a d j}{h y p}

tan A = \frac{o pp}{a d j}

Sum of angles in a triangle and in a polygon

Triangle:

18 0^{\circ}

. Polygon with

n

sides:

(n - 2) \times 18 0^{\circ}

Slope of a line

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Equation of a line

y - y_{1} = m (x - x_{1})

Distance between two points

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Midpoint of a line segment

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Mean, median and mode

Mean: sum divided by the number of values. Median: the middle value when ordered. Mode: the most frequent value.

Probability of an event

\frac{number of favourable outcomes}{total number of outcomes}

Percentage change

\frac{new - old}{old} \times 100%

Simple and compound interest

Simple:

I = P r t

. Compound:

A = P (1 + r)^{t}

Laws of indices

a^{m} a^{n} = a^{m + n}

;

\frac{a ^{m}}{a ^{n}} = a^{m - n}

;

(a^{m})^{n} = a^{mn}

;

a^{0} = 1

Difference of two squares

a^{2} - b^{2} = (a - b) (a + b)

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Arithmetic sequence

T_{n} = a + (n - 1) d

Ratio sharing

Add the parts, divide the total by the sum of the parts and multiply by each part.

Speed, distance and time

speed = \frac{distance}{time}

Scale factors for similar shapes

Lengths scale by

k

, areas by

k^{2}

and volumes by

k^{3}

Angle at the centre of a circle

It is twice the angle at the circumference standing on the same arc.

Angle in a semicircle

It is a right angle.

Perimeter of a rectangle

2 (length + width)

Distributive law of multiplication

a (b + c) = ab + a c

; expands a bracket across terms.

Expanding

(a + b)^{2}

and

(a - b)^{2}

(a + b)^{2} = a^{2} + 2 ab + b^{2}

;

(a - b)^{2} = a^{2} - 2 ab + b^{2}

Quadratic equation standard form

a x^{2} + b x + c = 0

, where

a \neq = 0

; solved by factorising, completing the square or using the quadratic formula.

Solving linear inequalities and negative division rule

When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must reverse (e.g.

- 2 x < 6 ⟹ x > - 3

Laws of indices: multiplication and division

a^{m} \times a^{n} = a^{m + n}

;

a^{m} \div a^{n} = a^{m - n}

(

a \neq = 0

Zero and negative index rules

a^{0} = 1

(for

a \neq = 0

);

a^{- n} = \frac{1}{a ^{n}}

Fractional index rule

a^{1/2} = a

;

a^{1/ n} = n a

;

a^{m / n} = n a^{m}

Scientific notation (Standard form)

A number written in the form

a \times 1 0^{n}

, where

1 \leq a < 10

and

n

is an integer.

Percentage profit or loss formula

Percentage profit/loss = \frac{Actual profit or loss}{Cost price} \times 100%

Simple interest formula

I = \frac{P \times R \times T}{100}

, where

P

is principal,

R

is annual rate, and

T

is time in years.

Compound interest formula

A = P (1 + i)^{t}

, where

P

is principal,

i

is interest rate per period expressed as a decimal, and

t

is number of periods.

Speed, distance and time relations

Distance = Speed \times Time

;

Speed = \frac{Distance}{Time}

;

Time = \frac{Distance}{Speed}

Dividing an amount into a given ratio

Add the ratio parts to find total shares, divide the total quantity by total shares to find 1 share, then multiply by each ratio part.

Slopes of parallel lines

Two lines are parallel if and only if their slopes are equal:

m_{1} = m_{2}

Slopes of perpendicular lines

Two non-vertical lines are perpendicular if and only if

m_{1} \times m_{2} = - 1

Slope-intercept form of a linear equation

y = m x + c

, where

m

is the slope and

c

is the

y

-intercept

(0, c)

Vertically opposite angles theorem

When two straight lines intersect, the vertically opposite angles are equal.

Alternate and corresponding angles on parallel lines

Alternate angles (Z-shape) are equal; corresponding angles (F-shape) are equal.

Co-interior angles on parallel lines

Interior angles on the same side of a transversal add up to

18 0^{\circ}

(supplementary).

Angle in a semicircle theorem

An angle subtended by the diameter at any point on the circumference of a circle is always a right angle (

9 0^{\circ}

Area of a parallelogram

Area = base \times perpendicular height

Area of a trapezium

Area = \frac{a + b}{2} \times h

, where

a

and

b

are parallel sides and

h

is vertical height.

Total surface area of a closed cylinder

A = 2 π r h + 2 π r^{2}

(

2 π r h

curved surface plus two circular ends).

Range and interquartile range

Range = Maximum value

-

Minimum value. Interquartile range (IQR) =

Q_{3} - Q_{1}

(middle 50% spread).

Fundamental Principle of Counting

If one operation can be performed in

m

ways and a second independent operation in

n

ways, together they can be performed in

m \times n

ways.

Complementary event probability

P (E^{'}) = 1 - P (E)

; the probabilities of all mutually exclusive and exhaustive outcomes sum to 1.

Probability with tree diagrams

Multiply probabilities along branches for combined sequential events; add probabilities across distinct terminal branches.

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