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National 5 Applications of Maths formula sheet: the SQA formulae list

Published 6 min read

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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

Circles

C=\pi dcircumference of a circle
A=\pi r^2area of a circle

Right-angled triangles

a^2+b^2=c^2theorem of Pythagoras
\text{gradient}=\frac{\text{vertical height}}{\text{horizontal distance}}gradient

Volumes

V=\pi r^2hvolume of a cylinder
V=Ahvolume of a prism
V=\frac{1}{3}\pi r^2hvolume of a cone
V=\frac{4}{3}\pi r^3volume of a sphere

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

The gradient line is printed as a word equation, so write it out in words in your working. In the standard deviation formulae, x is a data value, n is the number of values, and the mean has a bar over the x.

What is not on the National 5 Applications of Maths formula sheet

  • Area of a triangle, rectangle and composite shapes. The specification expects you to find the areas of triangles and composite shapes, but no area formula for a triangle is printed.
  • Speed, distance and time. The specification asks you to calculate all three, and the list does not print the relationship.
  • Finance. Compound percentage increase and decrease is one of the skills the specification lists, and no formula for it is on the list.
  • Trigonometry. There are no trig ratios, and no sine or cosine rule.
  • Averages and probability. The mean, median, range and the probability formula are yours to learn.
  • Ratio and proportion. Direct and indirect proportion and scale are not on it.

The course specification lists these skills: see the Skills, knowledge and understanding section.

How the papers use the list

National 5 Applications of Mathematics is two question papers and no assignment. The course specification gives 35 marks and 50 minutes for Paper 1, where you may not use a calculator, and 55 marks and 1 hour 40 minutes for Paper 2, where you may. Most questions are set in real contexts. In the 2027 timetable both are on Monday 26 April: Paper 1 from 09:00 to 09:50 and Paper 2 from 10:30 to 12:10 (Exam Timetable 2027, version 02). The list is printed in both.

PaperMarksTimeCalculator
Paper 13550 minutesNo
Paper 2551 hour 40 minutesYes

Our National 5 Applications of Maths guide covers the course, the grades and the 2026 boundaries.

Using the formulae list under exam pressure

Volume of a cone

A cone has radius 6 cm and height 10 cm. Find its volume to 3 significant figures.

V=\frac{1}{3}\pi r^2h=\frac{1}{3}\times\pi\times6^2\times10=120\pi=377\ \text{cm}^3

Use the vertical height, 10 cm, and not the slant height.

Gradient

A ramp rises 0.9 m over a horizontal distance of 7.5 m. What is its gradient?

\text{gradient}=\frac{\text{vertical height}}{\text{horizontal distance}}=\frac{0.9}{7.5}=0.12

As a percentage, that is 12%. Keep both measurements in the same unit before you divide.

Standard deviation

The times, in minutes, that five pupils took to finish a task were 12, 15, 9, 14 and 10. The mean is 12. The squared differences are 0, 9, 9, 4 and 4, which add up to 26, so

s=\sqrt{\frac{26}{5-1}}=\sqrt{6.5}=2.55\ \text{minutes}

Then say what it means: the times are spread about 2.55 minutes either side of the mean. A smaller standard deviation would mean the pupils were more consistent. Check it in the Standard Deviation Calculator.

Practise reading and comparing spreads in the Probability course, then press Mark.

Pick the empirical-rule percentage for a shaded region of a normal curve.

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

How the Applications list compares with National 5 Maths

National 5 Applications of MathsNational 5 Maths
CirclesCircumference and areaNot printed
Right-angled trianglesPythagoras and gradientNot printed
Other trianglesNot printedSine rule, cosine rule, area of a triangle
QuadraticsNot printedQuadratic formula
VolumesCylinder, prism, cone, sphereSphere, cone, pyramid
Standard deviationBoth formsBoth forms

See the National 5 Maths formula sheet for that list in full.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the sheet open beside you, mark it, and the worked answers show which line 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, which is the closest track to National 5 Applications of Mathematics. 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

  • Using the slant height. The cone and cylinder formulae want the vertical height.
  • Mixing units in a gradient. Change centimetres to metres, or the reverse, before you divide.
  • Dividing by n. The list divides by n − 1 for the sample standard deviation.
  • Quoting a standard deviation with no meaning. The marks are for saying what it shows about the data.
  • Looking for the triangle area formula. It is not on this list.

Common questions

Do you get a formula sheet in National 5 Applications of Maths?
Yes. SQA prints a formulae list at the front of both National 5 Applications of Mathematics papers.
What is on the National 5 Applications of Maths formulae list?
The circumference and area of a circle, Pythagoras, gradient, the volumes of a cylinder, a prism, a cone and a sphere, and the formulae for standard deviation.
Is the quadratic formula on the Applications list?
No. The quadratic formula is on the National 5 Maths list, not this one.
Is the area of a triangle on the Applications list?
No. You must know it.
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

    Find the standard deviation and variance of any data set

    Open the tool
  • Learn it

    Probability

    Distributions, expected values and the maths behind uncertainty

    Start the course
  • Test it

    National 5 Maths

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
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