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Quantitative Reasoning›Geometry

Geometry

Tackle UCAT geometry questions with common shapes and their formulae—plus strategies for working with 2D and 3D shapes.
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You may or may not see geometry in Quantitative Reasoning questions on Test Day. It appears occasionally in official practice, so you will want to practise geometry to ensure you can answer with speed and accuracy.

As you revise, you will see that the UCAT does not include geometry formulae for the most common shapes. Thus, you must ensure you have a solid grasp of these shapes, along with any geometry concepts that will help you work with complex shapes in case these appear on Test Day.

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As you revise geometry, it helps to think in terms of two-dimensional and three-dimensional shapes.

The same common shapes have appeared again and again in UCAT practice questions over the years, so these common shapes are most likely to appear on Test Day.

You can normally break down complex shapes into simpler, common shapes. This means that you do not need to memorise more obscure formulae. Most likely:

  • The UCAT will provide the formula as part of the QR data.
  • You can use common shapes to work with components of a complex shape.
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Circles are a simple and powerful shape because the common formulae are all defined in terms of the radius.

The image titled 'Circle Formulae' explains how circle formulas are built from the radius. In the center, a large circle is depicted with a radius labeled 'r' in red, and a diameter labeled 'd' in purple. On the right, three smaller circles illustrate different aspects of circle measurements: 1. Circumference labeled 'C = πd = 2πr' in orange; 2. Area labeled 'A = πr²' in blue; 3. Diameter labeled 'd = 2r' in purple. The text at the bottom reads 'Know one measurement -> work out the others.'
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The UCAT may or may not tell you the value of pi (π).

  • If the value of pi is given, use that value. Most commonly, it will be 3.14, though it could have additional digits or be given as a fraction, such as
  • If the value of pi is not provided, use 3.14 and this will lead you to the correct answer. Further digits are unlikely to be significant in distinguishing among the answer choices.
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Do not worry about arcs or chords, as these have never appeared in official UCAT practice materials and are unlikely to appear in the actual UCAT.

The same applies for segments of a circle, though you may need to work with portions of a pie chart. In a pie chart, the relative area of each portion corresponds to its proportional share of the whole amount.

Thus, you can think of each pie chart segment as a partial percentage (out of 100%) or a fractional share (out of 1).

When the UCAT presents a pie chart labelled with degrees (e.g. 72°) and calls it a circle graph, the whole amount of the pie chart is 360°. 

The image shows a pie chart divided into six segments with the following angles labeled inside each segment: 72°, 108°, 90°, 36°, 54°, and 36°. The chart demonstrates that a full circle is 360°, as annotated with a blue arrow and '360°' along the circle's perimeter. The background text reads 'CIRCLE GRAPHS USE 360°' at the top, and below it says 'If a pie chart is labelled in degrees, the full circle is 360°.' At the bottom, the text reads 'Degree labels tell you to work from 360°.'
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Rectangles are the most common quadrilateral you may see in Quantitative Reasoning.

Remember that a square is a type of rectangle as it fits the definition. However, a square’s length and width are equal.

The image titled 'Rectangle Formulae' shows a blue rectangle with labels. The rectangle has four right angles marked at each corner. The length is labeled 'l' with a purple double-headed arrow along the bottom, and the width is labeled 'w' with a red double-headed arrow along the side. Text above the rectangle states, 'A rectangle has 4 right angles.' To the right, there are two smaller diagrams. The top one shows a rectangle with an orange outline labeled 'Perimeter = 2l + 2w.' The bottom one shows a blue rectangle labeled 'Area = l × w.' Below the main rectangle, text reads, 'Find length and width before substituting.' The image is © Medify.
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Walkthrough question

Circles and rectangles

Try and tackle this geometry question involving common shapes.

Triangles are another simple and powerful shape as they can unlock geometry questions whether as an individual shape or part of a larger shape.

You should not need trigonometry on Test Day as it has never appeared in official UCAT practice questions.

The image features three triangles illustrating common triangle formulae. The first triangle shows a right angle at the base with the height 'h' marked in red and the base 'b' in purple. The formula for the area is shown as Area = 1/2 × b × h in blue. The second triangle is labeled with sides 'a', 'b', and 'c'. The perimeter formula is shown as Perimeter = a + b + c in orange. The third triangle is a right triangle with sides 'a', 'b', and hypotenuse 'c', with the right angle marked. Pythagoras’ theorem is displayed as a² + b² = c² in green. The title reads 'COMMON TRIANGLE FORMULAE' at the top. Below the triangles, the text advises, 'Check which formula you need before substituting.' The image is credited to Medify.
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Use Pythagoras’ theorem (a2 + b2 = c2) to find any unknown side of a right-angled triangle when you know two other sides. In this theorem, c is the hypotenuse – the longest side. 

Some more challenging questions may require you to use both the area formula and Pythagoras’ theorem to solve for an unknown.

The image illustrates Pythagoras' Theorem with a right triangle labeled with sides 'a', 'b', and hypotenuse 'c'. The equation a^2 + b^2 = c^2 is shown. The hypotenuse 'c' is labeled as 'c = hypotenuse'. To the right, the text 'Rearrange to solve for any side:' is followed by equations: c = √(a^2 + b^2), a = √(c^2 - b^2), and b = √(c^2 - a^2). Below, the phrase 'Know two sides → find the third.' is written.
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You can divide any quadrilateral into two triangles by drawing a straight line to connect two non-adjacent vertices (or ‘corners’).

This approach can turn any shape with four straight sides into a triangle question, allowing you to use all triangle properties and formulae that might be relevant.

An educational graphic illustrating that a quadrilateral can be divided into two triangles. The image shows four quadrilaterals, each with a dashed diagonal line indicating the division into two triangles.
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Any rectangle divides into two equal triangles with the same area and same perimeter.

When you divide a rectangle into two triangles:

  • Area of the rectangle = l × w
  • Area of two equal triangles =
  • Each triangle is one-half the rectangle’s area. You have two ways to solve:

Watch out for hidden rectangles and triangles which might be described in text in data or questions without a diagram.

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Walkthrough question

Triangles

Let's practice a geometry question using challenging data.

You are less likely to see unusual two-dimensional shapes in Quantitative Reasoning. Should other polygons appear on Test Day, remember these key facts:

  • The perimeter of a polygon is the sum of the lengths of its sides.
  • The area of a polygon equals the area of smaller shapes it is made of.
  • It will generally be quicker and simpler to break a complex figure into smaller shapes, then calculate the areas of the smaller shapes. The UCAT will usually allow you to divide complex shapes into triangles, rectangles, squares or circles.
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If you see a complex shape or an unusual formula that might require two-dimensional geometry, ask yourself:

  • What common shape could I use to solve? Think in terms of circles, squares, rectangles or triangles.
  • How can I use geometry formulae or common shape properties (like area, perimeter/circumference, sides of a shape or hypotenuse of a triangle) to find the path to the correct answer?
  • Do I have to work backwards or reverse the normal logic to solve?
An illustration demonstrating how to break down complex shapes into simpler ones, featuring a large semicircle and a smaller semicircle combined with a triangle, all in blue.
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Three-dimensional shapes are far less frequent than two-dimensional shapes in Quantitative Reasoning.

Unusual 3D shapes are highly unlikely to appear, though in theory the UCAT could provide a formula to use with an unconventional shape.

 You will not need to use calculus or complete advanced maths involving 3D shapes in the UCAT. Only fairly basic concepts are involved.

The image is titled '3D SHAPES IN QR' and shows two columns of 3D shapes. The left column is labeled 'More likely in QR' and includes three shapes: a green Cube at the top, a green Cuboid in the middle, and a green Cylinder at the bottom. The right column is labeled 'Less likely in QR' and includes two shapes: a red Cone at the top and a red Sphere at the bottom. At the bottom of the image, there is a caption that reads, 'In QR, you may need to work out the volume or surface area of these shapes.' The image is credited to Medify.
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The volume of a three-dimensional shape is simple to calculate if the 3D shape is a prism.

A prism is a solid shape with a cross-section consistent in size and shape throughout the solid’s height. This cross-section is therefore the prism’s base.

To find the volume of a prism, multiply the area of the base by the height.

This rule is handy for conventional prisms like cubes, cuboids and cylinders. It can also help with unusual prisms, such as triangular prisms. Note that with a triangular prism, the longer side is called the length to distinguish from the base and height of the triangular cross-section.

An educational diagram illustrating the volume formulas for three types of prisms: a cuboid, a cylinder, and a triangular prism. Each shape is labeled with its dimensions and corresponding volume formula: Cuboid (length × width × height), Cylinder (π × r² × height), and Triangular Prism (1/2 × base × height × length).
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You can save time if you memorise the volume formulae:

  • Volume of a cube
  • Volume of a cuboid
  • Volume of a cylinder
  • Volume of a sphere
  • Volume of a cone

We think it is unlikely that the UCAT would expect you to know the last two. QR data will tend to include the sphere and cone formulae, but you might learn them as a backup.

A shortcut to remember the cone volume formula: You can fit three cones into one cylinder of the same radius and height. Thus, the cone’s volume is one-third of the cylinder’s volume.

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The surface area of a three-dimensional shape is simple to calculate if the 3D shape is a prism.

A prism is a solid shape with a cross-section consistent in size and shape throughout the solid’s height. This cross-section is therefore the prism’s base.

To find the surface area of a prism:

  • Use the formula: surface area = 2 × base area + (base perimeter × height)
  • Remember to distinguish the bases and double the base area if both bases are included.
  • The final part treats the lateral faces of the prism as a single shape that runs the height of the prism.

This approach works for cubes, cuboids and cylinders, as well as any prism with a polygonal base.

We expect the UCAT will tell you the formula for surface area of a sphere or cone as part of the data if you will need it in Quantitative Reasoning.

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You could use a shortcut to find the surface area of cubes or cuboids.

In practice, you might calculate the surface area of a cuboid simply by finding the surface area of the three different rectangular faces, then summing and doubling it. This is more of a quick, manual process rather than writing out a formula.

The image titled 'Surface Area Shortcuts' shows two diagrams: a cube and a cuboid. The cube is labeled with 's' for the side length, and an arrow indicates the surface area formula '6s^2' with the annotation '6 identical faces.' The cuboid is labeled with dimensions 'l' (length), 'w' (width), and 'h' (height), with arrows indicating each dimension. The surface area formula '2(lw + wh + lh)' is shown with the annotation '2 of each.' Below the diagrams, the text reads 'Use shortcuts when you can to work faster.' The image is © Medify.
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Make sure to set up the steps mentally in a geometry question before you calculate. This will help to save time and minimise errors.

If a geometry question seems time-consuming, decide whether to tackle the question straightaway or flag for review without attempting. Sometimes, if running out of time, you must make a quick guess, perhaps by eyeballing or doing a quick estimation, and click an answer. It’s better than leaving a question blank.

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