Cambridge IGCSE Maths 0580 formulas you must learn

Q: Which Cambridge IGCSE Maths 0580 formulas are not in the formula booklet?

The ones to learn by heart are: Laws of indices; Compound interest and growth; Reverse percentages; Upper and lower bounds; Speed, distance and time; nth term of a linear sequence; Gradient and equation of a line; Perpendicular gradients; Length and midpoint of a line segment; Differentiation and turning points; Pythagoras' theorem; Right-angled trigonometry; Areas of quadrilaterals; Arc length and sector area (degrees); Similar shapes: area and volume; Magnitude of a vector; Frequency density. Each one is on this page with a worked example.

Q: Do I get a formula booklet in the exam?

Yes, a short one. Cambridge prints a List of formulas on page 2 of every 0580 paper; the Extended list adds the quadratic formula and the sine and cosine rules. Everything on this page is not on it.

Q: What is the best way to memorise them?

Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.

Q: Is this page a copy of the official booklet?

No. We link to the official Cambridge IGCSE Mathematics 0580 List of formulas; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.

The exam gives you a list of formulas, but it leaves these 17 out. Learn them by heart. Each has a short worked example: tap “Worked example” to open it.

Checked against the Core and Extended Lists of formulas in Cambridge's 0580 syllabus for 2025 to 2027. The list is printed on page 2 of every paper. Checked against Cambridge's 0580 syllabus (PDF) (syllabus for 2025, 2026 and 2027) on 5 October 2026.

Test yourself with flashcards One-page formula sheet

Ext = Extended tier only (Papers 2 and 4). Core students can skip those.

Number

Laws of indices Ext

$\displaystyle a^ma^n=a^{m+n},$ $\displaystyle \frac{a^m}{a^n}=a^{m-n},$ $\displaystyle (a^m)^n=a^{mn},$ $\displaystyle a^{-n}=\frac1{a^n},$ $\displaystyle a^{\frac mn}=\left(\sqrt[n]a\right)^m$

Use it for: Simplifying powers and roots without a calculator, and as the first step of many log and calculus questions.

Worked example

Work out $8^{\frac23}\times4^{-\frac12}$ without a calculator.

$8^{\frac23}=\left(\sqrt[3]8\right)^2=2^2=4$
$4^{-\frac12}$${}=\frac1{\sqrt4}$${}=\frac12$
$4\times\frac12=2$

Answer: $2$

Compound interest and growth

$\displaystyle \text{amount}=P\Big(1+\frac r{100}\Big)^n$ $\displaystyle \Big(\text{decay: }1-\frac r{100}\Big)$

Use it for: Savings, depreciation and population questions.

Worked example

\$2500 is invested at 3.2% compound interest per year. Find its value after 5 years.

Multiplier $1.032$
$2500\times1.032^5$

Answer: \$2926.43

Reverse percentages Ext

$\displaystyle \text{original}$$\displaystyle {}=\frac{\text{new value}}{\text{multiplier}}$

Use it for: Finding the price before a rise or a discount.

Worked example

After a 15% reduction a jacket costs \$68. Find the original price.

Multiplier $0.85$
$68\div0.85=80$

Answer: \$80

Upper and lower bounds Ext

$\displaystyle x\text{ to the nearest }u:$ \(\displaystyle x-\tfrac u2\le\text{value}

Use it for: Rounding and accuracy questions.

Worked example

A square has side 7.4 cm to 1 decimal place. Find the bounds of its perimeter.

Side: $7.35\le s<7.45$
Perimeter $=4s$: $4\times7.35=29.4$, $4\times7.45=29.8$

Answer: $29.4\le P<29.8$ cm

Speed, distance and time

$\displaystyle \text{speed}$$\displaystyle {}=\frac{\text{distance}}{\text{time}}$

Use it for: Travel graphs and average speed.

Worked example

A cyclist rides 54 km in 4 hours 30 minutes. Find the average speed.

$4\text{ h }30\text{ min}$${}=4.5$ h
$54\div4.5=12$

Answer: 12 km/h

Algebra and graphs

nth term of a linear sequence

$\displaystyle u_n=a+(n-1)d$ $\displaystyle (\text{or }dn+(a-d))$

Use it for: Finding a rule and a term of a sequence.

Worked example

Find the $n$th term of 5, 9, 13, 17, … and the 50th term.

$d=4$: $u_n=5+4(n-1)=4n+1$
$u_{50}=201$

Answer: $4n+1$; 201

Gradient and equation of a line Ext

$\displaystyle m$$\displaystyle {}=\frac{y_2-y_1}{x_2-x_1},$ $\displaystyle y=mx+c$

Use it for: Straight-line graphs.

Worked example

A straight line passes through $(-2,1)$ and $(4,19)$. Write its equation in the form $y=mx+c$.

$m=\frac{19-1}{4+2}=3$
$1=3(-2)+c\Rightarrow c=7$

Answer: $y=3x+7$

Perpendicular gradients Ext

$\displaystyle m_1m_2=-1$ $\displaystyle \Big(m_2$$\displaystyle {}=-\frac1{m_1}\Big)$

Use it for: Finding a perpendicular line, a normal, or a perpendicular bisector.

Worked example

Line $L$ passes through $(2,1)$ and meets $y=3x-4$ at right angles. Give the equation of $L$.

The given gradient is $3$, so the perpendicular gradient is $-\frac13$.
$y-1=-\frac13(x-2)$
$y=-\frac13x+\frac53$

Answer: $y=-\frac13x+\frac53$ (or $x+3y-5=0$)

Length and midpoint of a line segment Ext

$\displaystyle \text{length}$$\displaystyle {}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},$ $\displaystyle \text{midpoint}$$\displaystyle {}=\Big(\frac{x_1+x_2}2,\frac{y_1+y_2}2\Big)$

Use it for: Coordinate geometry.

Worked example

$A(-2,3)$ and $B(4,11)$. Find the length of $AB$ and its midpoint.

$\sqrt{6^2+8^2}=10$
$\big(\frac{-2+4}2,\frac{3+11}2\big)$${}=(1,7)$

Answer: Length 10, midpoint $(1,7)$

Differentiation and turning points Ext

$\displaystyle y=ax^n$$\displaystyle {}\Rightarrow\frac{dy}{dx}=nax^{n-1};$ $\displaystyle \text{turning points where }\frac{dy}{dx}$$\displaystyle {}=0$

Use it for: Gradients of curves and finding maximum and minimum points.

Worked example

Find the turning points of $y=x^3-6x^2+5$.

$\frac{dy}{dx}$${}=3x^2-12x$${}=3x(x-4)$${}=0$${}\Rightarrow x$${}=0,\ 4$
$y(0)=5$, $y(4)=64-96+5=-27$
The cubic is positive-led, so $(0,5)$ is the maximum.

Answer: Maximum $(0,5)$, minimum $(4,-27)$

Shape and space

Pythagoras' theorem

$\displaystyle a^2+b^2=c^2$ $\displaystyle (c\text{ the hypotenuse})$

Use it for: Any right-angled triangle, in 2D or 3D.

Worked example

A rope 8.5 m long runs in a straight line from the top of a vertical pole to a peg 4 m from its base. How tall is the pole?

$h^2$${}=8.5^2-4^2$${}=72.25-16$${}=56.25$
$h=\sqrt{56.25}$

Answer: 7.5 m

Right-angled trigonometry

$\displaystyle \sin\theta$$\displaystyle {}=\frac{\text{opp}}{\text{hyp}},$ $\displaystyle \cos\theta$$\displaystyle {}=\frac{\text{adj}}{\text{hyp}},$ $\displaystyle \tan\theta$$\displaystyle {}=\frac{\text{opp}}{\text{adj}}$

Use it for: Sides and angles in right-angled triangles, bearings, and elevation.

Worked example

A ramp makes an angle of $38^\circ$ with level ground and its sloping surface is 13 m long. How high does it rise?

The slope is the hypotenuse; the rise is opposite the $38^\circ$ angle.
$\text{rise}$${}=13\sin38^\circ$

Answer: 8.00 m (3 s.f.)

Areas of quadrilaterals

$\displaystyle \text{trapezium}$$\displaystyle {}=\tfrac12(a+b)h,$ $\displaystyle \text{parallelogram}=bh$

Use it for: Area and composite-shape questions (the 0580 list gives only the triangle).

The 0580 list gives the area of a triangle, not of quadrilaterals.

Worked example

A trapezium has parallel sides 7 cm and 11 cm, 4 cm apart. Find its area.

$\tfrac12(7+11)\times4$

Answer: 36 cm²

Arc length and sector area (degrees)

$\displaystyle \text{arc}$$\displaystyle {}=\frac\theta{360}\times2\pi r,$ $\displaystyle \text{sector}$$\displaystyle {}=\frac\theta{360}\times\pi r^2$

Use it for: Sectors with the angle in degrees.

The list gives the area and circumference of a whole circle; take the fraction $\frac\theta{360}$ of them.

Worked example

A slice of pizza is a sector of radius 12 cm with angle $75^\circ$. Find the length of its curved edge and its area, in terms of $\pi$.

Arc $=\frac{75}{360}\times24\pi$${}=5\pi$
Area $=\frac{75}{360}\times144\pi$${}=30\pi$

Answer: Arc $5\pi\approx15.7$ cm, area $30\pi\approx94.2$ cm²

Similar shapes: area and volume Ext

$\displaystyle \text{length ratio }k\ $$\displaystyle {}\Rightarrow\ \text{area ratio }k^2,$ $\displaystyle \text{volume ratio }k^3$

Use it for: Scaling areas and volumes of similar shapes and solids.

Worked example

Two similar solids have heights 2 cm and 3 cm. The smaller has volume 160 cm³. Find the volume of the larger.

$k=\frac32$, so the volume ratio is $\frac{27}8$
$160\times\frac{27}8=540$

Answer: 540 cm³

Magnitude of a vector Ext

$\displaystyle \begin{pmatrix}x\\y\end{pmatrix}:$ $\displaystyle \text{magnitude}$$\displaystyle {}=\sqrt{x^2+y^2}$

Use it for: Vector questions on Papers 2 and 4.

Worked example

Find the magnitude of $\begin{pmatrix}-6\\8\end{pmatrix}$.

$\sqrt{36+64}$

Answer: 10

Statistics

Frequency density Ext

$\displaystyle \text{frequency density}$$\displaystyle {}=\frac{\text{frequency}}{\text{class width}}$

Use it for: Drawing and reading histograms.

Worked example

The class \(20

Class width $=15$
$24\div15=1.6$

Answer: Frequency density 1.6

Keep going

Questions

Which Cambridge IGCSE Maths 0580 formulas are not in the formula booklet?

The ones to learn by heart are: Laws of indices; Compound interest and growth; Reverse percentages; Upper and lower bounds; Speed, distance and time; nth term of a linear sequence; Gradient and equation of a line; Perpendicular gradients; Length and midpoint of a line segment; Differentiation and turning points; Pythagoras' theorem; Right-angled trigonometry; Areas of quadrilaterals; Arc length and sector area (degrees); Similar shapes: area and volume; Magnitude of a vector; Frequency density. Each one is on this page with a worked example.

Do I get a formula booklet in the exam?

Yes, a short one. Cambridge prints a List of formulas on page 2 of every 0580 paper; the Extended list adds the quadratic formula and the sine and cosine rules. Everything on this page is not on it.

What is the best way to memorise them?

Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.

Is this page a copy of the official booklet?

No. We link to the official Cambridge IGCSE Mathematics 0580 List of formulas; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.

Our own list, wording and examples, written and checked by IGCSE Math Revision. Not produced or endorsed by Pearson Edexcel.