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NCEA Level 1 Maths formula sheet (91947 resource booklet)

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Formula sheet

The card below opens today's Level 1 Maths paper with the sheet in the drawer, so you can use it for practice. The contents follow in our own layout. This is the 2025 sheet for 91947, the latest NZQA has published. The other Level 1 standard, 91946, has its own resource booklet of graphs, tables and articles to interpret, and no formulae.

Formula sheet

NCEA Level 1 Maths

Open formula sheet

Measurement

Areas and volumes, printed with labelled diagrams

A=\frac{1}{2}\times\text{base}\times\text{height}Area of a triangle
A=\text{base}\times\text{height}Area of a parallelogram
A=\frac{1}{2}(a+b)hArea of a trapezium
C=\pi d=2\pi rCircumference of a circle
A=\pi r^2Area of a circle
V=\text{area of cross section}\times\text{length}Volume of a prism
V=\pi r^2hVolume of a cylinder
A=2\pi rh+2\pi r^2Surface area of a cylinder
V=\frac{4}{3}\pi r^3Volume of a sphere
A=4\pi r^2Surface area of a sphere
V=\frac{1}{3}\times\text{area of base}\times\text{height}Volume of a pyramid
V=\frac{1}{3}\pi r^2hVolume of a cone
A=\pi rL+\pi r^2Surface area of a cone, slant height L
V=\text{height}\times\text{width}\times\text{length}Volume of a cuboid

Units, speed and angles

1\ \text{hectare}=10\ 000\ \text{m}^2Hectare
1\ \text{litre}=1000\ \text{cm}^3Litre
1\ \text{ml}=1\ \text{cm}^3Millilitre
1\ \text{tonne}=1000\ \text{kg}Tonne
\text{average speed}=\frac{\text{total distance}}{\text{total time}}Average speed
\text{sum}=(n-2)\times180^\circInterior angles of an n-sided polygon

Right-angled triangles

a^2+b^2=c^2Pythagoras' theorem
\sin x=\frac{\text{opposite}}{\text{hypotenuse}}Sine ratio
\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}Cosine ratio
\tan x=\frac{\text{opposite}}{\text{adjacent}}Tangent ratio
\text{opposite}=\text{hypotenuse}\times\sin xOpposite side
\text{adjacent}=\text{hypotenuse}\times\cos xAdjacent side
\text{opposite}=\text{adjacent}\times\tan xOpposite side from the adjacent

The sheet also gives the definition of a bearing, "measured from North, in a clockwise direction", and says to take π as 3.14 to 2 decimal places.

What is not on the sheet

  • Algebra. Expanding, factorising, solving linear and quadratic equations and simultaneous equations are all unaided. So are the gradient of a line and rate of change.
  • Surface areas of prisms and pyramids. NZQA lists them in the 2026 specification, but the sheet gives surface areas only for the cylinder, sphere and cone. Add up the faces yourself.
  • Similar shapes. NZQA lists properties of similar shapes. No scale factor rule is printed.
  • Statistics. Mean, median, quartiles and the like are not given.
  • Which formula to use. The sheet does not tell you when a question wants Pythagoras and when it wants a trigonometric ratio.

NZQA's 2026 specification says to bring a ruler and an approved calculator, and that 91947 questions give you several problems and ask you to explain, with mathematical reasoning, how to work through them. The sheet helps with the arithmetic. The reasoning is yours.

How to use the sheet in the exam

  1. Draw a diagram and label what you know.
  2. Find the formula on the sheet and write it out before you substitute.
  3. Use π = 3.14, as the sheet says, unless the question says otherwise.
  4. Write a sentence that explains what you did and why, because 91947 marks the reasoning.

Volume and units

A cylindrical water tank has a radius of 0.40 m and a height of 1.2 m. How many litres does it hold? Use the cylinder line with π = 3.14:

V=\pi r^2h=3.14\times0.40^2\times1.2=0.603\ \text{m}^3

The sheet says 1 litre is 1000 cm³, so 1 m³ is 1000 litres. The tank holds 0.603 × 1000 = 603 litres. The common slip is to forget that 0.40 m is squared before it is multiplied.

Right-angled triangles

A wheelchair ramp is 6.0 m long and rises at an angle of 8° to the ground. How high does it rise? The ramp is the hypotenuse and the rise is the side opposite the angle, so use opposite = hypotenuse × sin x:

\text{rise}=6.0\times\sin8^\circ=0.835\ \text{m}

The ramp covers 6.0 × cos 8° = 5.94 m along the ground. Check by Pythagoras: 0.835² + 5.94² is about 36.0, and 6.0² = 36. Practise rearranging a formula in our Algebra course, then press Mark to check your answer there:

Rearrange a formula for the named variable.

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

The Level 1 sheet and the others

The Level 2 Maths sheet goes the other way: the quadratic formula, logarithms, two calculus lines and a normal table, with no areas or volumes. For how Level 1 works, see the NCEA Level 1 Maths guide.

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. Every question on the daily paper is our own.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper on the NCEA Level 1 Maths track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Open the sheet from the paper itself and practise choosing the formula quickly.

Common mistakes with the Level 1 Maths sheet

  • Using the diameter as the radius. The circle formulae use r, and d is double it.
  • Mixing up the volume and surface area of a cylinder. They are different lines.
  • Forgetting the units conversion. 1 litre is 1000 cm³ and 1 hectare is 10 000 m².
  • Measuring a bearing anticlockwise. The sheet says clockwise from North.

Common questions

Do you get a formula sheet in NCEA Level 1 Maths?
Yes, for 91947. NZQA's 2026 specification says a formula sheet will be provided. The other standard, 91946, has a resource booklet of data and no formulae.
What value of pi does NCEA Level 1 Maths use?
The formula sheet says π = 3.14 to 2 decimal places.
When was the 2026 Level 1 Maths exam?
Tuesday 10 November 2026, in the morning session, which starts at 9.30 am.
Are trigonometry formulae on the Level 1 Maths sheet?
Yes. It gives Pythagoras' theorem and the sine, cosine and tangent ratios for right-angled triangles.

Revision cards

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

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