GCSE Maths Formulas for the Foundation Paper

Resitting GCSE Maths can create a peculiar kind of pressure. You recognise many of the questions, yet the right formula sometimes disappears just when you need it. The encouraging truth is that the list of GCSE maths formulas for foundation is manageable. Some formulas will be provided, while others are worth learning until they feel automatic.

Under current Ofqual arrangements, a formulae and equations sheet will be provided in the November 2026 GCSE Maths series. This applies to the GCSE Maths support arrangements announced for exams in 2025, 2026 and 2027. You should receive a clean copy in each relevant exam. Your exam board will publish the precise version, so check the final sheet from Edexcel, AQA, OCR or Eduqas before sitting the paper.

The sheet reduces the burden of recall. It does not remove the need to recognise a formula, substitute correctly or use your calculator accurately.

Your foundation formula checklist

A strong resit plan should cover these areas:

• perimeter, area and volume
• circles
• Pythagoras and basic trigonometry
• speed, density and pressure
• percentages and compound growth
• averages and probability
• straight-line graphs
• unit conversions

You can review these topics through the free GCSE Maths revision lessons, then use the GCSE past papers and mark schemes to see how formulas appear in complete papers.

What the GCSE formula sheet does and does not do

A formula sheet is useful in the same way that a map is useful. It shows possible routes, but it cannot decide which road the question requires.

The GCSE formula sheet typically gives important geometry relationships, including formulas involving circles, prisms, Pythagoras and trigonometry. The exact content and presentation should always be checked against the sheet published for your exam board and series.

Even when a formula is printed, you must still know:

• what each letter represents
• which measurements belong in the formula
• whether units must be converted first
• how to rearrange the relationship when necessary
• how to round the final answer appropriately

Do not treat the sheet as something you will inspect for the first time in November. Download your board's version during revision and keep it beside you when completing practice papers. Familiarity saves time.

Area and perimeter formulas

Perimeter means the total distance around the outside of a shape. There is no single universal perimeter formula. Add every outside length, making sure the measurements use the same unit.

For a rectangle with length $l$ and width $w$:

$$P=2l+2w$$

The area of a rectangle is:

$$A=lw$$

For a triangle with perpendicular height $h$ and base $b$:

$$A=\frac{1}{2}bh$$

The height must meet the base at a right angle. A sloping side is not automatically the perpendicular height.

For a parallelogram:

$$A=bh$$

For a trapezium whose parallel sides have lengths $a$ and $b$:

$$A=\frac{1}{2}(a+b)h$$

A useful way to remember the trapezium formula is to find the mean length of the parallel sides, then multiply by the perpendicular height.

For example, if $a=6$, $b=10$ and $h=5$:

$$A=\frac{1}{2}(6+10)\times 5=40$$

Area answers require square units, such as ${cm}^2$. The grade 2 revision topics and grade 3 revision topics provide a useful route through core shape skills.

Circle formulas

For a circle of radius $r$ and diameter $d$:

$$d=2r$$

The circumference is:

$$C=2\pi r$$

or equivalently:

$$C=\pi d$$

The area is:

$$A=\pi r^2$$

Circumference is a length, so its answer has ordinary units. Area uses square units. Unless a question requests an answer in terms of $\pi$, use the calculator's $\pi$ button and round only at the end.

If the diameter is $12cm$, the radius is $6cm$, so the area is:

$$A=\pi \times 6^2=36\pi {cm}^2$$

The most common error is putting the diameter into a formula that requires the radius.

Volume formulas

For a cuboid:

$$V=lwh$$

For any prism:

$$V=area\; of\; cross-section\times length$$

A cylinder is a prism with a circular cross-section, so:

$$V=\pi r^{2}h$$

Volume uses cubic units, such as ${cm}^3$. Read the diagram carefully because the prism's length is not always drawn horizontally. It is the distance through which the constant cross-section continues.

Capacity questions may require the conversion:

$$1litre=1000{cm}^3$$

Pythagoras and trigonometry

For a right-angled triangle, Pythagoras' theorem is:

$$a^2+b^2=c^2$$

Here, $c$ must be the hypotenuse, the longest side opposite the right angle. Use Pythagoras when the question involves side lengths and gives enough information to identify two of the three sides.

For right-angled trigonometry:

$$\sin \theta =\frac{opposite}{hypotenuse}$$ $$\cos \theta =\frac{adjacent}{hypotenuse}$$ $$\tan \theta =\frac{opposite}{adjacent}$$

Label the sides relative to the angle $\theta$ before selecting a ratio. When finding an angle, use the inverse trigonometric function on your calculator. Check that the calculator is in degree mode.

These topics often appear towards the upper end of foundation papers, making the grade 4 revision resources and grade 5 revision resources particularly valuable for students aiming to secure a standard pass or higher foundation mark.

Formulas to memorise for number and measures

These relationships may not be the formulas you instinctively look for on a geometry-focused sheet. They are therefore important formulas to memorise and practise.

For speed:

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

For density:

$$density=\frac{mass}{volume}$$

For pressure:

$$pressure=\frac{force}{area}$$

The structure is similar in each case. A total quantity is divided by the relevant time, volume or area. Rather than relying only on formula triangles, practise rearranging.

From the density formula, for example:

$$mass=density\times volume$$

Unit consistency matters. A speed question using kilometres and minutes may require conversion before the final answer can be stated in kilometres per hour.

Percentages and repeated change

To find a percentage of an amount:

$$percentage\; amount=\frac{p}{100}\times original\; amount$$

For a percentage increase of $p\%$, use the multiplier:

$$1+\frac{p}{100}$$

For a decrease of $p\%$, use:

$$1-\frac{p}{100}$$

Repeated percentage change is calculated by applying the multiplier more than once. After $n$ periods:

$$final\; amount=initial\; amount\times {\left(1+\frac{p}{100}\right)}^n$$

For depreciation, replace the addition with subtraction. A frequent mistake is finding the change but forgetting to add it to, or subtract it from, the original amount.

Averages, probability and straight lines

The mean is:

$$mean=\frac{total\; of\; values}{number\; of\; values}$$

For a frequency table:

$$mean=\frac{\sum fx}{\sum f}$$

Basic probability can be written as:

$$P(A)=\frac{number\; of\; favourable\; outcomes}{total\; number\; of\; equally\; likely\; outcomes}$$

Probabilities range from $0$ to $1$, and the probability of an event not happening is:

$$P(notA)=1-P(A)$$

The standard form of a straight-line equation is:

$$y=mx+c$$

Here, $m$ is the gradient and $c$ is the $y$-intercept. Recognising this structure is often more important than simply recalling it.

Common formula mistakes in foundation papers

Most lost marks do not come from forgetting everything. They come from one small decision made too quickly.

Substituting before converting units

Convert measurements first. Mixing metres and centimetres in one formula produces an answer with no consistent meaning.

Confusing radius and diameter

Underline which measurement has been provided. If it is the diameter and the formula requires $r$, divide by $2$ first.

Using the wrong type of unit

Lengths use units such as $cm$, areas use ${cm}^2$, and volumes use ${cm}^3$.

Rounding during the calculation

Keep the calculator's full value until the final line. Early rounding can move an answer outside the accepted range.

Copying without understanding

A formula copied from the sheet earns little by itself. Write the formula, substitute the known values, calculate, and state the unit. This clear structure can also help you collect method marks when the final answer is incorrect.

Turn the formula list into marks

Knowing formulas is not a memory competition. It is a recognition skill built through repeated use. Choose one group, complete several focused questions, mark them carefully, and record the exact reason for each error.

Then move into timed papers. MathsGenie's predicted GCSE papers can help you practise current-style questions, while the grade 1 foundation skills provide a sensible starting point if basic calculations or units still feel uncertain.

For November 2026, take confidence from knowing that the formulae and equations sheet is part of the current arrangements. But prepare as though every formula still needs a decision from you. Use MathsGenie's free lessons, practice questions, mini tests, video solutions, past papers and mark schemes to make those decisions familiar. Begin with one formula group today, then test it in a real exam question.