Geometry and Spatial Calculations

Geometry questions in the UCAT Quantitative Reasoning subtest test your ability to calculate areas, perimeters, volumes, and surface areas. You may also encounter questions involving scale drawings, map calculations, and spatial reasoning. The formulas you need will usually be provided — the challenge is applying them correctly under time pressure.

---

Areas and Perimeters of Common Shapes

Rectangle

Measure	Formula
Area	length × width
Perimeter	2 × (length + width)

Triangle

Measure	Formula
Area	(1/2) × base × height
Perimeter	sum of all three sides

Important: The "height" must be the perpendicular height — the vertical distance from the base to the opposite vertex. It is NOT always one of the sides of the triangle.

Circle

Measure	Formula
Area	pi × r²
Circumference	2 × pi × r (or pi × d)

Where r = radius, d = diameter = 2r, and pi ≈ 3.14159

Trapezium

Measure	Formula
Area	(1/2) × (a + b) × h

Where a and b are the parallel sides and h is the perpendicular height between them.

Parallelogram

Measure	Formula
Area	base × perpendicular height
Perimeter	2 × (side₁ + side₂)

---

Worked Examples: Area

Example 1: Rectangle

Question: A ward has dimensions 12 m × 8 m. What is its area?

Solution: 12 × 8 = 96 m²

Example 2: Triangle

Question: A triangular garden has a base of 15 m and a perpendicular height of 9 m. What is its area?

Solution: (1/2) × 15 × 9 = 67.5 m²

Example 3: Circle

Question: A circular pond has a radius of 4 m. What is its area?

Solution: pi × 4² = pi × 16 = 50.27 m² (to 2 d.p.)

Example 4: Composite Shape

Question: A room is rectangular (10 m × 6 m) with a semicircular extension on one of the 6 m walls. The semicircle has a diameter of 6 m. What is the total floor area?

Solution:

• Rectangle: 10 × 6 = 60 m²
• Semicircle: (1/2) × pi × 3² = (1/2) × pi × 9 = 14.14 m²
• Total: 60 + 14.14 = 74.14 m²

---

Volumes

Cuboid (Rectangular Box)

Volume = length × width × height

Cylinder

Volume = pi × r² × h

Triangular Prism

Volume = area of triangle × length

Sphere

Volume = (4/3) × pi × r³

Cone

Volume = (1/3) × pi × r² × h

Note: Sphere and cone formulas would be given in a UCAT question. You do not need to memorise them, but being familiar with them saves reading time.

---

Worked Examples: Volume

Example 1: Cuboid

Question: A storage container is 2.5 m long, 1.8 m wide, and 2 m tall. What is its volume?

Solution: 2.5 × 1.8 × 2 = 9 m³

Example 2: Cylinder

Question: A cylindrical water tank has radius 0.5 m and height 1.2 m. What is its volume?

Solution: pi × 0.5² × 1.2 = pi × 0.25 × 1.2 = pi × 0.3 = 0.942 m³ (to 3 d.p.)

In litres: 0.942 × 1,000 = 942 litres

Example 3: Volume and Capacity

Question: A rectangular fish tank is 80 cm × 40 cm × 50 cm. How many litres of water can it hold?

Solution:

• Volume = 80 × 40 × 50 = 160,000 cm³
• Since 1 litre = 1,000 cm³: 160,000 / 1,000 = 160 litres

---

Surface Areas

Cuboid

Surface area = 2(lw + lh + wh)

Where l = length, w = width, h = height.

Cylinder