25 cardsUK GCSE

UK GCSE Mathematics Higher: Key Formulas, Rules & Circle Theorems

Essential formulas, circle theorems, surds rules, algebra tools, and trigonometry identities for GCSE Mathematics Higher Tier (Edexcel, AQA, OCR).

Start studying Play Match

PDF study sheet

Free account needed to save, copy or download. We bring you back here.

1 / 25

0 known

TermClick or press Space to flip

Rules of Surds Multiplication & Division

GetMyMockTestQuestion side

Full list

Terms in this deck (25)

UK GCSE mock tests →

Rules of Surds Multiplication & Division

ab = a b

;

\frac{a}{b} = \frac{a}{b}

;

a + b \neq = a + b

Rationalising the Denominator: Conjugate Surds

To rationalise

\frac{c}{a + b}

, multiply numerator and denominator by the conjugate

(a - b)

\frac{c ( a - b )}{( a + b ) ( a - b )} = \frac{c ( a - b )}{a ^{2} - b}

Fractional and Negative Indices

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

;

a^{1/ n} = n a

;

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

;

a^{0} = 1

(

a \neq = 0

Upper and Lower Bounds Calculations

Max of

\frac{A}{B} = \frac{A _{upper}}{B _{lower}}

; Min of

\frac{A}{B} = \frac{A _{lower}}{B _{upper}}

; Max of

A - B = A_{upper} - B_{lower}

Quadratic Formula

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

for

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

Completing the Square Form

x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c

. The turning point (vertex) of the parabola is

(- \frac{b}{2}, c - (\frac{b}{2})^{2})

Difference of Two Squares Factorisation

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

Quadratic Sequences (

n^{th}

Term)

T_{n} = a n^{2} + bn + c

. Constant second difference

= 2 a

. Find

a

, then subtract

a n^{2}

to find linear sequence

bn + c

Circle Theorem: Angle at the Centre

The angle subtended by an arc at the centre of a circle is twice the angle subtended by it at any point on the circumference:

∠ A O C = 2∠ A B C

Circle Theorem: Angles in the Same Segment

Angles subtended by the same arc in the same segment of a circle are equal.

Circle Theorem: Angle in a Semicircle

The angle subtended by a diameter at the circumference is always a right angle (

9 0^{\circ}

Circle Theorem: Cyclic Quadrilateral Opposite Angles

The opposite angles of a cyclic quadrilateral inscribed in a circle sum to

18 0^{\circ}

Circle Theorem: Tangent and Radius

The angle between a tangent to a circle and the radius drawn to the point of contact is

9 0^{\circ}

Circle Theorem: Two Tangents from an External Point

Tangents drawn to a circle from the same external point are equal in length and form congruent right-angled triangles with the radius.

Circle Theorem: Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

Trigonometric Area of a Triangle

Area = \frac{1}{2} ab sin C

The Sine Rule

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}

(use to find sides) or

\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c}

(use to find angles).

The Cosine Rule

Side formula:

a^{2} = b^{2} + c^{2} - 2 b c cos A

. Angle formula:

cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

Similar Shapes: Length, Area and Volume Ratios

If linear scale factor is

k

: Area ratio

= k^{2}

, Volume ratio

= k^{3}

Direct vs Inverse Proportion Equations

Direct:

y = k x

y = k x^{2}

. Inverse:

y = \frac{k}{x}

y = \frac{k}{x ^{2}}

. Find constant

k

from initial conditions.

Compound Percentage Multipliers

Increase by

r % ⟹ \times (1 + \frac{r}{100})

. Decrease by

r % ⟹ \times (1 - \frac{r}{100})

. Value after

n

periods

= P_{0} (1 \pm \frac{r}{100})^{n}

Conditional Probability Formula

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

. For independent events:

P (A \cap B) = P (A) \times P (B)

and

P (A ∣ B) = P (A)

Probability of Mutually Exclusive vs Non-Exclusive Events

Mutually exclusive:

P (A \cup B) = P (A) + P (B)

. Non-exclusive:

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Histogram Frequency Density Formula

Frequency Density = \frac{Frequency}{Class Width}

. The area of each bar represents the frequency.

Equation of a Circle with Centre at Origin

x^{2} + y^{2} = r^{2}

where

r

is the radius. The tangent at

(x_{1}, y_{1})

has gradient

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

(perpendicular to radius).

UK GCSE in the United Kingdom

In the United Kingdom, UK GCSE is part of: 11+, GCSE and A-Level.

Students on the same pathway also practise:

More UK GCSE practice

UK GCSE mock tests

All UK GCSE practice →