Papers 2 and 3 are both calculator papers. That changes how certain topics are tested but it does not mean everything is provided. This is the companion to our Paper 1 reference sheet and covers everything from that post that still applies here, plus the additional topics that are either specific to the calculator papers or commonly tested across all three. Some of the formulas below appear on the formula sheet depending on your board. Where that is the case it is noted, but it is always worth knowing them regardless so you can apply them quickly and confidently.
Formula sheets vary by board. AQA and Edexcel provide one; OCR does not. Even where a sheet is given, several things in this guide will not be on it. The notes below flag where something is typically provided so you know what you still need to have ready from memory.
Volume and surface area of prisms
Pyramids, spheres and cones have formulas that are usually given on the formula sheet. Prisms are different. The method for prisms is not given as a single formula because it applies the same principle to every shape in the family, and knowing that principle is what is tested.
Prisms include cuboids, cubes, cylinders, triangular prisms, and any other 3D shape with a consistent cross-section running from one end to the other.
Volume of any prism:
The cross-section is the face that stays the same all the way through the shape. For a cylinder it is a circle. For a triangular prism it is a triangle. For a cuboid it is a rectangle. Work out the area of that face first using the appropriate area formula, then multiply by the length of the prism.
Surface area of any prism:
There is no single formula for surface area because every prism has a different net. The method is always the same: think about the net of the shape, work out the area of each individual face, then add them all together. For a cylinder, the net gives two circles and one rectangle. For a triangular prism, it gives two triangles and three rectangles. Sketch the net if it helps you identify every face before starting.
Students often treat the cylinder separately and forget that volume = area of circle x length applies directly. The curved surface area of a cylinder is \(2\pi r h\) and is not always given, so it is worth memorising. The total surface area is \(2\pi r h + 2\pi r^2\) (curved surface plus both circular ends).
Sine rule
The sine rule is provided on most formula sheets. However, students regularly lose marks because they use the wrong version or do not recognise when to flip it. Both versions are equivalent but one is faster for finding a length and the other for finding an angle.
Use this version when finding a length:
Use this version when finding an angle:
Either version can technically be used for both. The reason for flipping is practical: when finding an angle, it is faster to have sin A as the numerator so you can multiply across directly without an extra rearrangement step.
Use it when you have a matching pair of side and opposite angle, plus one other piece of information. If you have two sides and the angle between them, that is the cosine rule instead.
Cosine rule
The cosine rule is provided on most formula sheets in one form. There are three length versions and three angle versions, one for each side or angle in the triangle. Knowing all six means you never need to relabel your triangle to match the formula sheet version.
Three versions for finding a length:
Three versions for finding an angle:
If you prefer, just memorise the three length versions and rearrange after substituting your values. The angle versions above are simply those rearrangements done in advance. Either approach works, and having all values substituted before rearranging reduces the chance of an algebra error.
This is often on the formula sheet but worth knowing. Area = \(\frac{1}{2}ab\sin C\), where a and b are two sides and C is the angle between them. Use it when you have two sides and the included angle and are asked for the area rather than a missing side.
Describing transformations
When a question asks you to describe a transformation fully, each type has specific components that must all be present to get full marks. Naming the transformation alone will not score more than one mark. The information required for each type is below.
Name: Translation
Required: the column vector \(\dbinom{x}{y}\) where \(x\) is the horizontal movement (positive right, negative left) and \(y\) is the vertical movement (positive up, negative down)
Name: Rotation
Required: the angle of rotation, the direction (clockwise or anticlockwise), and the centre of rotation as a coordinate
Name: Reflection
Required: the equation of the mirror line in the form \(y = \ldots\), \(x = \ldots\), \(y = x\) or \(y = -x\)
Name: Enlargement
Required: the scale factor (can be fractional for a reduction, negative for an enlargement on the opposite side) and the centre of enlargement as a coordinate
A common error is writing "rotation 90 degrees" and stopping. Without the direction and the centre of rotation, that is an incomplete answer. Every type has its required components and they must all be present for full marks.
Circles and sectors
Area of a circle and circumference (check whether these are given on your sheet):
Area of a sector and arc length (these are almost never provided):
Where \(\theta\) is the angle of the sector in degrees and \(r\) is the radius. Both use the same fraction of the full circle.
Interior angles of polygons
Where \(n\) is the number of sides. For a regular polygon, divide by \(n\) to get each interior angle.
Straight line graphs
Gradient:
Midpoint of a line segment:
Distance between two points:
- Parallel lines have the same gradient
- Perpendicular lines have negative reciprocal gradients: if one gradient is \(m\), the perpendicular is \(-\dfrac{1}{m}\)
Completing the square
For a quadratic \(x^2 + bx + c\), the completed square form is:
The turning point is read directly as \(\left(-\dfrac{b}{2},\ c - \left(\dfrac{b}{2}\right)^2\right)\). The x-coordinate of the turning point carries a sign change.
Compound interest, growth and depreciation
Check your formula sheet. If it is not provided:
- \(A\) = final amount, \(P\) = starting amount, \(r\) = percentage rate, \(n\) = number of years
Graph transformations
Think about what happens to individual coordinates rather than the whole graph shifting.
Point \((x,\ y)\) moves to \((x + a,\ y)\)
Translates right by \(a\)
Point \((x,\ y)\) moves to \((x - a,\ y)\)
Translates left by \(a\)
Point \((x,\ y)\) moves to \((x,\ y + a)\)
Translates up by \(a\)
Point \((x,\ y)\) moves to \((x,\ y - a)\)
Translates down by \(a\)
Point \((x,\ y)\) moves to \((x,\ ay)\)
Vertical stretch, scale factor \(a\)
Point \((x,\ y)\) moves to \(\left(\dfrac{x}{a},\ y\right)\)
Horizontal stretch, scale factor \(\dfrac{1}{a}\)
Recurring decimals to fractions
Let \(x\) equal the recurring decimal. Multiply to shift one full cycle, subtract to eliminate the recurring part, then solve for \(x\). For a two-digit recurring block, multiply by 100. For a single digit, multiply by 10.
Probability rules
- All probabilities in a sample space sum to 1
- AND (both events occur): multiply the probabilities
- OR (at least one occurs, mutually exclusive): add the probabilities
- For tree diagrams: multiply along branches, add between branches when combining outcomes
Index laws
- \(a^m \times a^n = a^{m+n}\)
- \(a^m \div a^n = a^{m-n}\)
- \((a^m)^n = a^{mn}\)
- \(a^0 = 1\) for any non-zero \(a\)
- \(a^{-n} = \dfrac{1}{a^n}\)
- \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
- \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m\), root first then power
Surds
- \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
- \(\sqrt{a} \div \sqrt{b} = \sqrt{\dfrac{a}{b}}\)
- To rationalise a denominator \(\sqrt{b}\): multiply numerator and denominator by \(\sqrt{b}\)
- To rationalise \(a + \sqrt{b}\): multiply by the conjugate \(a - \sqrt{b}\)
Final exam reminders
- Show all working. A wrong final answer with correct method still earns marks.
- Do not leave blanks. An attempt is always better than nothing.
- Check negative signs carefully, particularly in cosine rule calculations.
- Re-read the question before writing your final answer to confirm you have answered what was asked.
- One mark can be the difference between grade boundaries.
Time management in the exam
Spend roughly 30 seconds reading each question before writing anything. If the method is clear, begin and keep one eye on the time relative to the marks available. If the question does not feel accessible in that first glance, mark it and move on. Come back once the rest of the paper is done.
If you have started a question and spent more time than the mark allocation suggests with no clear progress, leave it and return later. A fresh look after a break in the session often unlocks something that felt stuck the first time.
The goal is always to secure every mark you can across the whole paper, not to spend all the available time on the questions that feel hardest.
