Geometry

Area, perimeter, and volume questions make up roughly 8% of QR. They almost always involve basic shapes - circles, rectangles, triangles - sometimes combined into more complex forms. The UCAT never tests trigonometry, angles beyond basic reasoning, or anything past GCSE.

The core skill is decomposition: break any complex shape into shapes you already know.

These are the only formulas you need. Every QR geometry question uses one or more of them. Know them cold. (`π = 3.14159`, use `3.14` on the calculator. `r` = radius, `d` = diameter = `2r`.)

2D shapes

3D shapes

A cylinder is a circle extended into 3D. A cuboid is a rectangle extended into 3D. Every 3D prism = 2D base × height.

Complex shapes on the UCAT are always combinations of simple ones. Your job is to break them apart. Total area = sum of each piece's area. Each piece is a rectangle, triangle, or circle you can calculate individually.

Subtraction Method

Sometimes it's easier to calculate a larger shape minus a hole: Shaded area = outer − inner.

Worked Example: Decomposition

Question: A garden is L-shaped. The outer dimensions are 12 m × 8 m. A 4 m × 5 m rectangle is cut from the top-right corner. What is the area of the garden?

Subtraction method:
Full rectangle = 12 × 8 = 96 m²
Cut-out = 4 × 5 = 20 m²
Garden area = `96 − 20 = 76 m²`

Try subtraction first for L-shapes and shapes with cut-outs. It's usually one step fewer than building up from parts.

Question: A rectangular room is 6 m by 4.5 m. A semicircular bay window extends 1.5 m from one of the 4.5 m walls. What's the total floor area?

1. Rectangle area: 6 × 4.5 = 27 m²
2. Semicircle: bay extends 1.5 m, so radius = 1.5 m. Full circle = π × 1.5² = 3.14159 × 2.25 = 7.069 m². Semicircle = 7.069 / 2 = 3.534 m².
3. Total = `27 + 3.534 = 30.53 m²`

Calculator: 1.5 * * = 2.25 * 3.14159 = 7.069 / 2 = 3.534, press P. 6 * 4.5 = 27 + C = 30.534. Time: ~25 sec.

This is decomposition in action: rectangle + semicircle. The key is recognising the bay window as a semicircle and knowing which dimension is the radius.

The most common geometry error in QR. `diameter = 2 × radius`.

If a question says "a circle with diameter 10": r = 5. Area = π × 5² = 78.5, not π × 10² = 314.

If a question says "a circle with radius 10": d = 20. Circumference = π × 20 = 62.8, not π × 10 = 31.4.

Always check: does the question give r or d?

Calculator Sequence for Circle Area

`Area = π × r²` on the calculator: type r, then * * = (squares it), then * 3.14159 =.

Example: r = 7. 7 * * = 49, then * 3.14159 = 153.94. Total keystrokes: 7 * * = * 3.14159 = (fast, no brackets needed).

For any prism (a 3D shape with identical top and bottom faces):

`SA = 2 × (base area) + (base perimeter) × height`

The first term is top + bottom. The second is the "curtain" wrapped around the sides.

Worked Example: Cylinder Surface Area

Imagine unwrapping the curved side of a cylinder like peeling a label off a can. It flattens into a rectangle.

Cylinder with radius 5 cm, height 10 cm:
Top + bottom = 2 × (π × 5²) = 2 × 78.54 = 157.08 cm²
Curved surface = circumference × height = (2 × π × 5) × 10 = 31.42 × 10 = 314.16 cm²
Total SA = `471.24 cm²`

The same logic works for any prism: rectangular, triangular, hexagonal. Same formula, different base shape.

`Volume = Base area × Height` covers:

• Cuboid: rectangle area × height = l × w × h
• Cylinder: circle area × height = π × r² × h
• Triangular prism: triangle area × length = (½ × b × h) × L

If you know the base area and the height, you know the volume.

Rare in QR (~1% of questions) but easy marks when it appears. For a right-angled triangle:

`a² + b² = c²`

c (hypotenuse) is always the longest side, opposite the right angle.
Finding c: c = √(a² + b²). Finding a: a = √(c² − b²).

Calculator Sequence

Find c when a = 3, b = 4:

3 * * = 9 (a²) → press P (store in memory)
4 * * = 16 (b²) → + C = 25 (add memory: 9 + 16)
√ = 5 (square root)

Trap 1: Radius vs diameter

"Diameter 14 cm" → r = 7, not 14 in π × r². Always halve the diameter before squaring.

Trap 2: Units in area and volume

Convert lengths before calculating area/volume, not after.

Trap 3: Missing a face in surface area

An open-top box has 5 faces, not 6. Read carefully: "open", "no lid", "one end sealed".

Trap 4: Confusing perimeter and area

If the answer has no "squared" unit, it's perimeter.

Next lesson: 3.6 Money, Tax & Conversions - tax brackets, currency, and unit conversions, covering about 13% of QR questions combined.