National 5 Maths formula sheet (SQA)
In SQA National 5 Maths exams in Scotland you get a formulae list: the quadratic formula, the sine and cosine rules, the area of a triangle, the volumes of a sphere, cone and pyramid, and standard deviation. This free sheet shows all of it, plus the formulae you must learn: 29 formula groups on 4 A4 pages.
- SQA National 5 (Scotland)
- 4 A4 pages
- 21 to learn
- Checked 21 October 2026
PDF · 53 KB · prints at actual size on A4 · original revision aid, not the official SQA formulae list
What is on the SQA National 5 Maths formulae list?
SQA prints the same short list at the front of both National 5 Mathematics papers:
- Roots of a quadratic: The roots of ax2 + bx + c = 0 are x = (−b ± √(b2 − 4ac)) / 2a
- Volume of a sphere: V = (4/3)πr3
- Volume of a cone: V = (1/3)πr2h
- Volume of a pyramid: V = (1/3)Ah
- Sine rule: a/sin A = b/sin B = c/sin C
- Cosine rule: a2 = b2 + c2 − 2bc cos A or cos A = (b2 + c2 − a2) / 2bc
- Area of a triangle: A = ½ab sin C
- Standard deviation: s = √( Σ(x − x̄)2 / (n − 1) ) or s = √( (Σx2 − (Σx)2/n) / (n − 1) ), where n is the sample size
That is all. Surds and indices, completing the square, the discriminant, straight lines, circle arcs and sectors, Pythagoras, SOHCAHTOA, trig graphs and the semi-interquartile range are not on the list, so you must learn them.
How the labels work
- On the SQA listPrinted on the formulae list in both National 5 Mathematics question papers.
- Learn: not on the listNot printed in the exam. You are expected to know it.
Number and algebra
Surds
Learn: not on the list√ab = √a × √b; √(a/b) = √a / √b; rationalise: a/√b = a√b / b
Simplify by taking out square factors, for example √48 = √16 × √3 = 4√3.
Indices
Learn: not on the listam × an = am+n; am ÷ an = am−n; (am)n = amn; a0 = 1; a−n = 1/an; am/n = (ⁿ√a)m
Scientific notation
Learn: not on the lista × 10n, where 1 ≤ a < 10 and n is an integer
Factorising
Learn: not on the listCommon factor; difference of two squares a2 − b2 = (a + b)(a − b); trinomials such as x2 + 5x + 6 = (x + 2)(x + 3)
Completing the square
Learn: not on the listx2 + bx + c = (x + b/2)2 − (b/2)2 + c
Percentages
Learn: not on the listAppreciation or depreciation over n years: value = original × (1 ± r/100)n
For a reverse percentage, divide by the multiplier: original = new value ÷ (1 ± r/100).
Quadratics and graphs
Roots of a quadratic
On the SQA listThe roots of ax2 + bx + c = 0 are x = (−b ± √(b2 − 4ac)) / 2a
Discriminant and nature of roots
Learn: not on the listb2 − 4ac > 0: two real distinct roots; = 0: equal roots; < 0: no real roots
Turning point and axis of symmetry
Learn: not on the listy = (x + p)2 + q has turning point (−p, q) and axis of symmetry x = −p
For y = k(x + p)2 + q the turning point is still (−p, q); it is a minimum if k > 0 and a maximum if k < 0.
Straight lines
Learn: not on the listGradient m = (y2 − y1) / (x2 − x1); y = mx + c; y − b = m(x − a)
Parallel lines have equal gradients.
Geometry and measurement
Volume of a sphere
On the SQA listV = (4/3)πr3
Volume of a cone
On the SQA listV = (1/3)πr2h
Volume of a pyramid
On the SQA listV = (1/3)Ah
A is the area of the base and h the perpendicular height.
Prisms and cylinders
Learn: not on the listPrism: V = Ah; cylinder: V = πr2h
Circles: arcs and sectors
Learn: not on the listC = πd; A = πr2; arc length = (θ/360) × πd; sector area = (θ/360) × πr2
Pythagoras’ theorem and its converse
Learn: not on the lista2 + b2 = c2; if a2 + b2 = c2, the triangle is right-angled
Angles and circles
Learn: not on the listThe angle in a semicircle is 90°; a tangent meets the radius at 90°; interior angles of a polygon add up to (n − 2) × 180°
Similar shapes
Learn: not on the listLength scale factor k ⇒ area scale factor k2, volume scale factor k3
Vectors
Learn: not on the listMagnitude: |u| = √(a2 + b2) in 2D, √(a2 + b2 + c2) in 3D
Add vectors component by component to find a resultant.
Trigonometry
Sine rule
On the SQA lista/sin A = b/sin B = c/sin C
Side a is opposite angle A.
Cosine rule
On the SQA lista2 = b2 + c2 − 2bc cos A or cos A = (b2 + c2 − a2) / 2bc
Use the second form to find an angle.
Area of a triangle
On the SQA listA = ½ab sin C
Right-angled triangles (SOHCAHTOA)
Learn: not on the listsin x = opp/hyp; cos x = adj/hyp; tan x = opp/adj
Trigonometric identities
Learn: not on the listsin2x + cos2x = 1; tan x = sin x / cos x
Trig graphs
Learn: not on the listy = a sin bx + c: amplitude a, b cycles in 360°, period 360°/b, moved up by c
Statistics
Standard deviation
On the SQA lists = √( Σ(x − x̄)2 / (n − 1) ) or s = √( (Σx2 − (Σx)2/n) / (n − 1) ), where n is the sample size
Mean, median and quartiles
Learn: not on the listMean x̄ = Σx ÷ n; median = middle value; Q1, Q2, Q3 split ordered data into quarters
See the mean, median, mode and range guide.
Semi-interquartile range
Learn: not on the listSIQR = (Q3 − Q1) / 2
Compare data sets using an average (mean or median) and a measure of spread (standard deviation or SIQR).
Line of best fit
Learn: not on the listFind the gradient and y-intercept from two points on the line, then write y = mx + c
Formulae to learn
These are not on the SQA formulae list. The last page of the PDF has the same list as tick-boxes.
- Surds
- Indices
- Scientific notation
- Factorising
- Completing the square
- Percentages
- Discriminant and nature of roots
- Turning point and axis of symmetry
- Straight lines
- Prisms and cylinders
- Circles: arcs and sectors
- Pythagoras’ theorem and its converse
- Angles and circles
- Similar shapes
- Vectors
- Right-angled triangles (SOHCAHTOA)
- Trigonometric identities
- Trig graphs
- Mean, median and quartiles
- Semi-interquartile range
- Line of best fit
Official sources
Checked on 21 October 2026 against the formulae list SQA prints in National 5 Mathematics question papers. Always check a recent past paper.
- SQA: National 5 Mathematics Paper 1 formulae list (X847/75/01, 2022)
- SQA: National 5 Mathematics Paper 2 formulae list, 2026
- SQA: past papers and marking instructions
National 5 Maths past papers
SQA publishes National 5 Mathematics past papers and their marking instructions free on its past papers and marking instructions page. Print this sheet, work through a past paper without looking, then tick off any formula you needed but could not remember.
Practise the methods
MathsDaily follows the curriculum in England, so it has no National 5 level. These guides and the GCSE daily set cover many of the same skills:
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 question papers, the non-calculator Paper 1 and the calculator Paper 2.
What is on the National 5 Maths formulae list?
The quadratic formula, the sine rule, both forms of the cosine rule, the area of a triangle (½ab sin C), the volumes of a sphere, cone and pyramid, and both forms of the standard deviation formula.
Which National 5 Maths formulae do I need to learn?
21 of the 29 formula groups on this page are not on the list, including surds and indices, completing the square, the discriminant, the turning point of a parabola, gradient and straight lines, arc length and sector area, Pythagoras, SOHCAHTOA, trig graphs and the semi-interquartile range. The last page of the PDF is a tick-list.
Is National 5 the same as GCSE?
No. National 5 is the SQA qualification taken in Scotland, usually in S4. GCSE is taken in England. The content overlaps a lot, so the GCSE Maths formula sheet and topic guides on this site are useful practice too.
Is this the official SQA formulae list?
No. It is an original MathsDaily revision sheet that shows what the official list contains and what it leaves out. In the exam you use the list printed in your question paper.