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UCAT Quantitative Reasoning
UCAT Quantitative Reasoning Formulae and Cheat Sheet (2026)
I am Dr Akash, and after coaching thousands of students through the UCAT, the request I hear most often before Quantitative Reasoning is simple: just give me the formulae. So here they are. Every formula, unit conversion and percentage value you need for UCAT Quantitative Reasoning, on one page.
The exam is set by the UCAT Consortium, and as the official UCAT test information confirms, no formula sheet is given on screen. Every formula below has to be automatic, not looked up. Bookmark this page or print it, then drill each one until it is instant.
An honest word before the tables: you will not need every formula here on the day, and some rarely show up. The trouble is that nobody can tell you in advance which will appear, so learning them all simply means you walk in with every base covered and nothing left to chance. If you are short on time, master the high-yield core first, which is percentages, the areas of common shapes, and speed, distance and time. Treat the rest as insurance and quiet confidence.
How to use this sheet: learn the high-yield tables first, then prove you know them by writing the sheet from memory. In the exam you recall these in about 43 seconds per question, so reflex speed matters more than deep understanding. The rarer formulae are worth a look so nothing surprises you, but do not lose sleep over them.
UCAT QR formulae: 2D shapes (area and perimeter)
The circle is the most tested shape, so know it cold. Full method and worked examples are in our geometry questions guide.
Shape | Perimeter | Area |
|---|---|---|
Square | 4 × side | side² |
Rectangle | 2 × (length + width) | length × width |
Parallelogram | 2 × (side a + side b) | base × perpendicular height |
Triangle | side a + side b + side c | ½ × base × height |
Trapezium | add all four sides | ½ × (a + b) × height |
Circle | 2 × π × r (circumference) | π × r² |
UCAT QR formulae: 3D shapes (volume and surface area)
Cuboids and cylinders are the ones that actually turn up. Spheres and cones are rare, so learn them last, but do learn them, so a stray question never throws you.
Shape | Volume | Surface area |
|---|---|---|
Cube | side³ | 6 × side² |
Cuboid | length × width × height | 2 × (lw + lh + wh) |
Cylinder | π × r² × height | 2πr² + 2πrh |
Sphere | (4 ÷ 3) × π × r³ | 4 × π × r² |
Cone | (1 ÷ 3) × π × r² × height | πr² + πrl (l = slant height) |
Circle formulae for the UCAT
- Diameter = 2 × radius
- Circumference = π × diameter = 2 × π × radius
- Area = π × radius²
The on-screen calculator has no π key. Type 3.14 (or use 22/7 for a quick fraction). This is the number one reason students freeze on circle questions.
Pythagoras' theorem
In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). Rearranged, c = √(a² + b²). Use it whenever a right angle and two sides appear.
Speed, distance and time formulae for the UCAT
- Speed = distance ÷ time
- Distance = speed × time
- Time = distance ÷ speed
- Average speed = total distance ÷ total time
Average speed is total distance over total time. It is never the average of the two speeds. A trip at 40 then 60 is not 50. To switch units: 1 m/s = 3.6 km/h (× 3.6 to go from m/s to km/h, ÷ 3.6 to reverse).
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UCAT percentage formulae and shortcuts
- Percentage of a number = (percentage ÷ 100) × number
- One number as a percentage of another = part ÷ whole × 100
- Percentage change = (new − old) ÷ old × 100
- Increase by r% = × (1 + r ÷ 100). Decrease by r% = × (1 − r ÷ 100)
- Reverse percentage (find the original) = final ÷ (1 ± r ÷ 100)
Mental shortcuts: these save you seconds on every question. More in our percentage shortcuts guide.
- 10% = ÷ 10 | 1% = ÷ 100 | 5% = half of 10%
- 20% = ÷ 5 | 25% = ÷ 4 | 50% = ÷ 2
- 15% = 10% + 5%. Build any percentage from 10%, 5% and 1%.
- Percentages flip: 8% of 50 = 50% of 8 = 4
Fractions, decimals and percentages
Memorise these so a fraction, its decimal and its percentage are one instant thought.
Fraction | Decimal | Percentage |
|---|---|---|
1/2 | 0.5 | 50% |
1/3 | 0.333… | 33.3% |
2/3 | 0.667 | 66.7% |
1/4 | 0.25 | 25% |
3/4 | 0.75 | 75% |
1/5 | 0.2 | 20% |
2/5 | 0.4 | 40% |
1/6 | 0.167 | 16.7% |
1/8 | 0.125 | 12.5% |
3/8 | 0.375 | 37.5% |
1/9 | 0.111 | 11.1% |
1/10 | 0.1 | 10% |
1/12 | 0.083 | 8.3% |
1/20 | 0.05 | 5% |
1/100 | 0.01 | 1% |
Ratios and proportion
- To share an amount in the ratio a : b, one part = amount ÷ (a + b), then multiply out
- Scale a ratio by multiplying or dividing every part by the same number
- A ratio as a fraction: a : b means a ÷ (a + b) of the whole
- Direct proportion: y ÷ x stays constant (the unit rate)
- Best value: compare cost ÷ quantity (price per unit)
Techniques and traps are covered in our ratio and proportion guide.
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Averages
- Mean = sum of values ÷ number of values
- Weighted mean = Σ(value × weight) ÷ Σ(weights)
- Median = the middle value when ordered (the mean of the two middle values if the count is even)
- Mode = the most frequent value
- Range = largest − smallest
Financial maths formulae for the UCAT
- Simple interest = principal × rate × time (rate as a decimal, so 5% = 0.05)
- Total with simple interest = principal × (1 + rate × time)
- Compound interest total = principal × (1 + rate)^number of periods
- Interest earned = total − principal
- Profit = selling price − cost price. Profit % = profit ÷ cost price × 100
- Add VAT at 20% = × 1.2. Remove VAT = ÷ 1.2
Tax and tiered bills are progressive. Apply each band's rate only to the income that falls inside that band, then add the bands. Never apply one rate to the whole amount. Full worked examples are in the tax and financial maths guide.
See the tax and financial maths guide for tiered tax, compound growth and profit questions worked end to end.
Unit conversions for UCAT QR
Unit slips lose more marks in QR than hard maths does. Learn these until they are reflex.
Measure | Key conversions |
|---|---|
Length | 1 km = 1000 m |
Mass | 1 tonne = 1000 kg |
Volume (capacity) | 1 litre = 1000 ml |
Area | 1 m² = 10,000 cm² |
Volume (cubic) | 1 m³ = 1,000,000 cm³ |
Time | 1 minute = 60 s |
Speed | 1 m/s = 3.6 km/h |
Currency | value in B = value in A × exchange rate (A to B) |
Probability formulae for the UCAT
- Probability = favourable outcomes ÷ total outcomes (always between 0 and 1)
- P(not A) = 1 − P(A)
- Independent events: P(A and B) = P(A) × P(B)
- Mutually exclusive events: P(A or B) = P(A) + P(B)
- Convert to a percentage by × 100
Drug dose calculations in the UCAT
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These appear more often in recent cycles, so do not skip them.
- Dose to give = (dose prescribed ÷ stock strength) × stock volume
- Number of tablets = total dose prescribed ÷ strength per tablet
- Weight-based dose = dose per kg × patient weight in kg
- Infusion rate (ml/hour) = total volume ÷ time in hours
Always match units first (mg against microgram, ml against litre) before you divide. A units mismatch is the classic drug-dose trap.
What you do not need to know for UCAT QR
It helps to know where the maths stops, so you do not waste revision over-preparing.
- No trigonometry: sine, cosine and tangent are not tested
- No calculus: no differentiation or integration
- No advanced algebra: you rearrange simple formulae, but never solve quadratics or simultaneous equations under exam pressure
- No statistics beyond the mean, median, mode and range above (no standard deviation)
Everything in QR is GCSE level or below. The skill being tested is speed and accuracy under time pressure, not the difficulty of the maths.
How to memorise the UCAT QR formulae
- Active recall: write this whole sheet from memory each morning until it is perfect
- Drill to reflex speed with our free mental maths trainer and unit conversions trainer
- Apply them under time pressure in full QR practice questions
- Go deeper on the tricky topics: geometry, percentages, ratio and proportion, tax and financial maths, and the on-screen calculator
If you want a tutor to turn these formulae into speed, our 1-to-1 UCAT tutoring builds a QR method around exactly where you lose time.
Your printable UCAT QR formula sheet
This whole page is the sheet. Everything above fits the exam and nothing here is padding.
Print or save it: press Ctrl+P (Cmd+P on a Mac) and choose Save as PDF. Stick it above your desk, or send it to a friend who is revising. Then test yourself by covering the answers and rebuilding each table from memory.
FAQs
Frequently asked questions
Does the UCAT give you a formula sheet?
No. The UCAT provides no formulae or reference sheet on screen. You must memorise every formula, from the area of a circle to compound interest, and recall it instantly under time pressure. That is exactly why we built this page.
Do I need to memorise all of these formulae for the UCAT?
Honestly, no. Not every formula appears in every exam, and some are rare. But you cannot predict which will come up, so learning them all means you are never caught out. If time is short, prioritise the high-yield core first: percentages, the areas of common shapes, and speed, distance and time. Treat the rest as confidence and insurance.
What formulae do I need to memorise for UCAT Quantitative Reasoning?
The essentials are: areas and perimeters of 2D shapes, volumes and surface areas of 3D shapes, the circle formulae, Pythagoras, speed distance time, percentages and percentage change, ratios, averages, simple and compound interest, and the common unit conversions. Every one is on this page.
Which UCAT QR formulas come up most often?
Percentages and percentage change, reading data from tables and graphs, ratios, averages and speed distance time appear most often. Geometry, interest and drug dose calculations appear regularly but less often. Learn the common ones to reflex speed first.
What is the formula for the area and circumference of a circle?
Area = π × radius², and circumference = 2 × π × radius (the same as π × diameter). Use π as 3.14, because the on-screen calculator has no π key.
What is the area of a triangle, and do I need the other shape formulae?
Area of a triangle = ½ × base × height. You should also know the rectangle, square, parallelogram and trapezium, but the circle and rectangle come up most. All of them are in the 2D shapes table above.
What is the speed, distance and time formula for the UCAT?
Speed = distance ÷ time, distance = speed × time, and time = distance ÷ speed. Remember the units, and that 1 m/s = 3.6 km/h.
How do I calculate average speed in the UCAT?
Average speed = total distance ÷ total time. It is never the average of the two speeds. A journey at 40 then 60 is not 50. This is the single most common trap in speed questions.
How do I convert a fraction to a percentage quickly?
Divide the top by the bottom to get a decimal, then multiply by 100. Faster still is to memorise the common ones: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/3 = 33.3%. The full conversion table is above.
What is the percentage change formula?
Percentage change = (new value − old value) ÷ old value × 100. A positive answer is an increase and a negative answer is a decrease. To reverse a percentage change, divide by (1 ± rate).
What unit conversions do I need to know for the UCAT?
The common ones are length (km, m, cm, mm), mass (kg, g, mg), volume (litres, ml, cm³), area, and time (hours, minutes, seconds). They are all in the conversions table above. Unit slips lose more marks in QR than hard maths does.
Do I need to know the volume of a sphere for the UCAT?
It is rare but it can appear, so learn it: volume = (4 ÷ 3) × π × radius³, and surface area = 4 × π × radius². Cuboids and cylinders come up far more often, so prioritise those and learn the sphere and cone last.
What is the compound interest formula?
Total = principal × (1 + rate)^number of periods, with the rate as a decimal. The interest earned is the total minus the principal. Do not fall back on simple interest (principal × rate × time) when the question says compound.
Do I need trigonometry, calculus or algebra for UCAT QR?
No. There is no trigonometry, no calculus, and no advanced algebra. You only rearrange simple formulae. Everything tested is GCSE level or below, so do not waste revision on higher maths.
Can I use a calculator in the UCAT Quantitative Reasoning section?
Yes, there is a basic on-screen calculator, but it has no π key, so type 3.14. For round percentages and simple sums, mental maths is faster than reaching for it, so do not rely on it for everything.
Is GCSE maths enough for UCAT Quantitative Reasoning?
Yes. Every formula on this page is GCSE level or below. The difficulty is not the maths, it is applying it in about 43 seconds per question. Speed and accuracy, not advanced content, are what you are training.
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