Scottish examsNational 5

National 5 Maths formula sheet: what the SQA formulae list gives you

Published 12 Nov 20266 min read

National 5 Maths

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Formula sheet

The card below opens today's National 5 Maths paper with this list in the drawer. The contents follow in our own layout. The list was the same in the 2025 and 2026 papers. Check it against the list in a recent paper, such as the 2026 Paper 1, before you rely on this page.

Formula sheet

National 5 Maths

Open formula sheet

Quadratics

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}the roots of ax² + bx + c = 0

Triangles

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}sine rule

a^2=b^2+c^2-2bc\cos Acosine rule, to find a side

\cos A=\frac{b^2+c^2-a^2}{2bc}cosine rule, to find an angle

A=\frac{1}{2}ab\sin Carea of a triangle

Volumes

V=\frac{4}{3}\pi r^3volume of a sphere

V=\frac{1}{3}\pi r^2hvolume of a cone

V=\frac{1}{3}Ahvolume of a pyramid

Standard deviation

s=\sqrt{\frac{\sum(x-\overline{x})^2}{n-1}}sample standard deviation, n is the sample size

s=\sqrt{\frac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1}}the same, in the form that avoids the mean

In the standard deviation formulae, x is a data value, the mean is written with a bar over x, and n is the number of values. Use either form: they give the same answer.

What is not on the National 5 Maths formula sheet

The list is short, and a lot of what you use every day in the course is missing from it:

Circles. Circumference, area, arc length and sector area are not printed, and neither is the volume of a cylinder or a prism.
Pythagoras and trigonometry in right angles. a² + b² = c² and sin, cos and tan as ratios are not on it. The sine and cosine rules are, but only for triangles without a right angle.
Straight lines. y = mx + c, the gradient formula and the equation of a line.
Algebra and indices. The laws of indices, how to rationalise a denominator, and the completed square and turning point of a quadratic.
Statistics and money. The mean, median and quartiles, the interquartile range, compound percentage change and the formula for the nth term of a sequence.

The National 5 Applications of Mathematics list is the other way round: it does print circumference, circle area, Pythagoras, and the volumes of a cylinder and a prism, but not the sine rule, the cosine rule or the quadratic formula.

How the papers use the list

National 5 Mathematics is two question papers and no assignment, so the list is printed in both. The course specification gives 40 marks and 1 hour for Paper 1, where you may not use a calculator, and 50 marks and 1 hour 30 minutes for Paper 2, where you may. In the 2027 timetable both are on Monday 17 May: Paper 1 from 09:00 to 10:00 and Paper 2 from 10:30 to 12:00 (Exam Timetable 2027, version 02).

Paper	Marks	Time	Calculator
Paper 1	40	1 hour	No
Paper 2	50	1 hour 30 minutes	Yes

Our National 5 Maths guide covers the course, the grades and the boundaries.

Using the formulae list under exam pressure

Decide the topic first. A triangle with no right angle points to the sine or cosine rule.
Write the formula down, then substitute. The method mark is for the substitution.
Work in full on your calculator and round only at the end.
Check the units and the size. A side of a triangle cannot be longer than the other two added together.

Quadratic formula

Solve x² − 4x − 3 = 0, giving answers to 1 decimal place. Here a = 1, b = −4 and c = −3, so

x=\frac{4\pm\sqrt{16+12}}{2}=\frac{4\pm\sqrt{28}}{2}

That gives x = 4.6 and x = −0.6. Note that −b is +4 when b is −4, and that the term under the root is 16 + 12, because −4ac = +12.

Try the quadratic formula from memory in the Algebra course, then press Mark.

Press Mark to have your answer marked in the Algebra course.

Cosine rule

Two sides of a triangle are 6 cm and 8 cm, with an angle of 50° between them. Find the third side. The angle is between b and c, so it is the angle A opposite the side we want:

a^2=6^2+8^2-2\times6\times8\times\cos50^\circ=100-96\cos50^\circ=38.29

So a = 6.19 cm, to 2 decimal places. If your answer comes out as 100 − 96 = 4, you subtracted before multiplying: work out 96 cos 50° first.

Standard deviation

Find the standard deviation of 3, 5, 8 and 12. The mean is 28 ÷ 4 = 7. The squared differences from the mean are 16, 4, 1 and 25, which add up to 46, so

s=\sqrt{\frac{46}{4-1}}=\sqrt{15.33}=3.92

Check the sum in the Standard Deviation Calculator. Divide by n − 1, not n.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the list open beside you, mark it, and the worked answers show which formula each question wanted. We do not publish official or past paper answers.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper for National 5 Maths. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full SQA paper and not set by SQA. Many schools set a longer prelim in winter: see our prelims guide. Open the sheet, set a timer, and write the line you use at the start of each answer.

Common mistakes

Rounding too early. Keep the full calculator value through the cosine rule and round only the final answer.
Using the wrong angle in the cosine rule. The angle is opposite the side on the left of the equation.
Dividing by n. The list divides by n − 1 for the sample standard deviation.
Sign slips in the quadratic formula. Put brackets round a negative b, and remember that −4ac changes sign when c is negative.
Searching the list for Pythagoras. It is not there. Learn it.

Common questions

Do you get a formula sheet in National 5 Maths?

Yes. SQA prints a formulae list at the front of both National 5 Mathematics papers.

What is on the National 5 Maths formulae list?

The quadratic formula, the sine rule, the cosine rule, the area of a triangle, the volumes of a sphere, a cone and a pyramid, and the formulae for standard deviation.

Is Pythagoras on the National 5 Maths formula sheet?

No. Pythagoras, the circle formulae and the volume of a cylinder are not on the National 5 Maths list, so you must know them.

Is the quadratic formula given in National 5 Maths?

Yes, it is the first item on the list. It is not printed at Higher.

Is the formula sheet the same in Paper 1 and Paper 2?

Yes, the same list is printed in both papers.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

Solve it
Standard Deviation Calculator
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