Geometry and Measurement

Geometry and measurement questions test your understanding of shapes, angles, lengths, areas, and volumes. In the CEM 11+ exam, these questions may involve diagrams, or they may describe shapes in words and expect you to work without a picture. Being able to visualise shapes and recall key formulas is essential.

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Angles

Types of Angles

Type	Size	Example
Acute	Less than 90°	45°
Right	Exactly 90°	90°
Obtuse	Between 90° and 180°	120°
Straight	Exactly 180°	180°
Reflex	Between 180° and 360°	270°

Key Angle Facts

Rule	Total
Angles on a straight line	180°
Angles at a point	360°
Angles in a triangle	180°
Angles in a quadrilateral	360°
Interior angles of a polygon (n sides)	(n - 2) × 180°

Worked Example

Two angles on a straight line are 115° and x°. Find x.

x = 180 - 115 = 65°

Angles in Triangles

An isosceles triangle has a top angle of 40°. What is each base angle?

• Sum = 180°
• 180 - 40 = 140
• Each base angle = 140 ÷ 2 = 70°

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2D Shapes: Properties

Triangles

Type	Properties
Equilateral	All sides equal, all angles 60°
Isosceles	Two sides equal, two base angles equal
Scalene	All sides different, all angles different
Right-angled	One angle is exactly 90°

Quadrilaterals

Shape	Sides	Angles	Symmetry
Square	All equal	All 90°	4 lines
Rectangle	Opposite equal	All 90°	2 lines
Rhombus	All equal	Opposite angles equal	2 lines
Parallelogram	Opposite equal	Opposite angles equal	0 lines
Trapezium	One pair parallel	Varies	0 or 1 line
Kite	Two pairs of adjacent sides equal	One pair of equal angles	1 line

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Perimeter

The perimeter is the total distance around the outside of a shape.

Rectangle: Perimeter = 2 × (length + width)

Worked Example: A rectangle is 12 cm long and 7 cm wide. Perimeter = 2 × (12 + 7) = 2 × 19 = 38 cm

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Area

Shape	Formula
Rectangle	length × width
Triangle	1/2 × base × height
Parallelogram	base × perpendicular height
Trapezium	1/2 × (a + b) × height
Circle	pi × radius²

Worked Examples

Area of a triangle with base 10 cm and height 6 cm: = 1/2 × 10 × 6 = 30 cm²