GCSE maths formulae

x = (−b ± √(b² − 4ac)) / 2a

a, b, c

the coefficients of the quadratic written as ax² + bx + c = 0

b² − 4ac

the discriminant — positive gives two roots, zero gives one, negative gives none

Not on the sheet. Rearrange to = 0 before reading off a, b and c, and divide the whole numerator by 2a, not just the square root.

Practise: A18 Solving quadratic equations

C = 2πr = πd

C

circumference, the distance around the circle

r

radius

d

diameter, which is 2r

Not on the sheet, and needed on both tiers. Check which of r and d the question gives you before substituting.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones · G18 Arc lengths and sector areas

A = πr²

A

area, in square units

r

radius

Not on the sheet. Putting the diameter in where the radius belongs makes the answer four times too big.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones · G18 Arc lengths and sector areas

a² + b² = c²

c

the hypotenuse — the side opposite the right angle, always the longest

a, b

the two shorter sides

Not on the sheet. To find a shorter side, subtract: a² = c² − b². Only valid in a right-angled triangle.

Practise: G20 Pythagoras’ theorem and trigonometric ratios · G6 Geometric reasoning and simple proofs

sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj

θ

the angle you are working from

opp

the side opposite θ

adj

the side next to θ that is not the hypotenuse

hyp

the hypotenuse, opposite the right angle

Not on the sheet. Label the three sides relative to θ first, then pick the ratio that contains the two sides in play.

Practise: G20 Pythagoras’ theorem and trigonometric ratios · G21 Exact trigonometric values

a / sin A = b / sin B = c / sin C

a, b, c

the side lengths

A, B, C

the angles opposite those sides — a is opposite A

Not on the sheet. Use it when you have a side paired with its opposite angle. Watch for the second, obtuse solution when finding an angle.

Practise: G22 The sine rule and the cosine rule

a² = b² + c² − 2bc cos A

a

the side you want, opposite angle A

b, c

the two sides enclosing angle A

A

the angle between b and c

Not on the sheet. This is the SAS rule — two sides and the angle between them — and it rearranges to cos A = (b² + c² − a²) / 2bc for finding an angle from three sides.

Practise: G22 The sine rule and the cosine rule

Area = ½ ab sin C

a, b

two sides of the triangle

C

the angle between those two sides

Not on the sheet. C must be the enclosed angle; using any other angle gives a wrong area that still looks reasonable.

Practise: G23 Area of a triangle: ½ab sin C

A = ½ (a + b) h

a, b

the two parallel sides

h

the perpendicular distance between them, not a slanted side

Rebuild it by cutting the trapezium into two triangles, or by picturing the average of the two parallel sides times the height.

Practise: G16 Area of 2D shapes and volume of prisms

V = area of cross-section × length

V

volume, in cubic units

cross-section

the shape that stays the same all the way through the solid

length

the distance the cross-section is extended, measured perpendicular to it

A cylinder is a prism with a circular cross-section, so its volume is πr²h. Multiplying by the perimeter instead of the area is the standard slip.

Practise: G16 Area of 2D shapes and volume of prisms

Final = P × (1 + r/100)ⁿ

P

the starting amount (the principal)

r

the percentage rate per period — negative for decay or depreciation

n

the number of periods

It is repeated multiplication by the same multiplier, which is why it is a power. A 15% fall is × 0.85 per year, not × 0.15.

Practise: R16 Growth and decay, compound interest · R9 Percentages and percentage change

P(A or B) = P(A) + P(B) for mutually exclusive events

P(A)

the probability that A happens

mutually exclusive

A and B cannot both happen

If the events can overlap, adding double-counts the overlap. On a Venn diagram, count the region once.

Practise: P4 Probabilities of exhaustive events sum to one · P6 Sets, Venn diagrams and tree diagrams

P(A and B) = P(A) × P(B) for independent events

independent

the first outcome does not change the probability of the second

dependent

it does — as in drawing without replacement, where the second probability shifts

Multiply along the branches of a tree diagram. Without replacement, both the numerator and the denominator change on the second branch.

Practise: P8 Independent and dependent combined events · P9 Conditional probability

A = πrl

r

the radius of the circular base

l

the slant height, from the apex down the sloping side — not the perpendicular height

Given in the exam. For the total surface area, add the base circle πr². Substituting the perpendicular height for l is the usual error.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones

A = 4πr²

r

the radius of the sphere

Given in the exam. A hemisphere's curved surface is half of this, but its total surface also needs the flat circle.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones

V = 4/3 πr³

r

the radius of the sphere

Given in the exam. The cube on r is what makes volume scale by the cube of the scale factor.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones

V = 1/3 πr²h

r

the radius of the base

h

the perpendicular height, from the apex straight down to the base

Given in the exam. Note that this one takes the perpendicular height, while the curved surface area takes the slant height.

Practise: G17 Circles, composite shapes, spheres, pyramids and cones

v = u + at

u

initial velocity

v

final velocity

a

constant acceleration

t

time taken

Given in the exam. All the kinematics formulae assume the acceleration is constant; check the question says so.

Practise: A14 Real-life graphs and kinematics · A15 Gradients and areas under graphs

s = ut + ½at²

s

displacement

u

initial velocity

a

constant acceleration

t

time taken

Given in the exam. Use it when you know the time; use the next one when you do not.

Practise: A14 Real-life graphs and kinematics · A15 Gradients and areas under graphs

v² = u² + 2as

v

final velocity

u

initial velocity

a

constant acceleration

s

displacement

Given in the exam. This is the one with no t in it, which is exactly when to reach for it.

Practise: A14 Real-life graphs and kinematics · A15 Gradients and areas under graphs