Geometrical terms
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Angles, Lines and Points
Geometry relies on precise vocabulary for its most basic building blocks - points, lines, and the angles between them.
- A point is an exact position, with no size, usually marked with a dot and a capital letter
- A vertex (plural: vertices) is the point where two lines or edges meet, e.g. the corner of a polygon or the point of an angle
- Parallel lines stay the same distance apart and never meet, however far they are extended (marked with matching arrows)
- Perpendicular lines meet at a right angle (90°)
- An angle is acute if it is between 0° and 90°, a right angle if it is exactly 90°, obtuse if it is between 90° and 180°, and reflex if it is between 180° and 360°
Worked Example: Naming Angle Types
- Question: State the type of angle for each of the following: 35°, 90°, 142°, 260°.
- Answer: 35° is acute; 90° is a right angle; 142° is obtuse; 260° is reflex
Common Mistakes
MistakeAssuming any angle that "looks big" is reflex, without checking whether it is actually over 180°
Fixa reflex angle must specifically be greater than 180° - an obtuse angle can still look fairly large but is always less than 180°
MistakeConfusing "perpendicular" with simply "crossing", when the lines don't actually meet at 90°
Fixperpendicular lines must meet at exactly a right angle - lines can cross at any other angle without being perpendicular
Triangles
Triangles are classified by their side lengths and angles, and a single triangle can belong to more than one category at once.
- A scalene triangle has all three sides (and all three angles) of different sizes
- An isosceles triangle has exactly two equal sides, and the two angles opposite those sides are also equal
- An equilateral triangle has all three sides equal, and all three angles equal to 60°
- A right-angled triangle has one angle of exactly 90°
Worked Example: Naming a Triangle from Its Properties
- Question: A triangle has side lengths 5 cm, 5 cm and 8 cm. Name this type of triangle.
- Answer: isosceles (exactly two equal sides)
Common Mistakes
MistakeAssuming a triangle with sides that look similar in a diagram must be equilateral, without checking the actual given lengths
Fixcheck the exact side lengths (or angle sizes) given, not just how the diagram appears - diagrams are often not drawn to scale
MistakeThinking a right-angled triangle cannot also be isosceles
Fixa triangle can belong to more than one category at once - a right-angled isosceles triangle has a 90° angle and two equal sides
Quadrilaterals
Quadrilaterals are named according to which sides are equal, which are parallel, and which angles are equal.
| Name | Key property |
|---|---|
| Square | four equal sides, four right angles |
| Rectangle | two pairs of equal opposite sides, four right angles |
| Parallelogram | two pairs of parallel and equal opposite sides, opposite angles equal |
| Rhombus | four equal sides, opposite sides parallel, opposite angles equal (no right angles required) |
| Trapezium | exactly one pair of parallel sides |
| Kite | two pairs of adjacent equal sides, one pair of opposite angles equal |
Worked Example: Naming a Quadrilateral from Its Properties
- Question: A quadrilateral has exactly one pair of parallel sides. Name this type of quadrilateral.
- Answer: trapezium
Common Mistakes
MistakeConfusing a rhombus with a square, assuming a rhombus must also have right angles
Fixa rhombus only requires four equal sides - it does not need right angles (a square is a special case of a rhombus that does have them)
MistakeConfusing a kite with a rhombus, since both shapes have some equal sides
Fixa kite has two separate pairs of adjacent equal sides, not all four sides equal like a rhombus
Polygons, Congruence and Similarity
Polygons are named by their number of sides, and shapes can be compared using the related ideas of congruence and similarity.
- A polygon is a 2D shape made of straight sides, named by its number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10)
- A regular polygon has all sides equal and all interior angles equal; an irregular polygon does not
- The interior angle is the angle inside the shape at each vertex; the exterior angle is the angle between a side and the extension of the next side
- Two shapes are congruent if they are exactly the same shape and size; two shapes are similar if they are the same shape but not necessarily the same size (one is an enlargement of the other)
Worked Example: Naming a Polygon
- Question: A prism has a cross-section that is a regular polygon with 5 equal sides. Name this polygon.
- Answer: pentagon
Common Mistakes
MistakeConfusing "regular" with simply "symmetrical", assuming any symmetrical shape must be regular
Fixa regular polygon needs both equal sides AND equal angles - some symmetrical shapes (like a kite) have neither
MistakeMixing up congruent and similar, describing two identically-shaped but different-sized shapes as congruent
Fixcongruent shapes must be exactly the same size; similar shapes are the same shape but can differ in size
Circles
A circle has its own set of vocabulary for describing its centre, its boundary, and the lines and regions associated with it.
- The centre is the fixed point at the middle of the circle; the radius is the distance from the centre to any point on the circumference
- The diameter is a straight line passing through the centre, joining two points on the circumference (twice the radius); the circumference is the distance all the way around the circle
- A chord is a straight line joining two points on the circumference that does not pass through the centre; an arc is a part of the circumference
- A sector is the region enclosed by two radii and an arc; a segment is the region enclosed by a chord and an arc
- A tangent is a straight line that touches the circumference at exactly one point, without crossing into the circle
Worked Example: Identifying Parts of a Circle
- Question: A circle has centre O. A straight line touches the circle at exactly one point, P, without crossing into the circle. Name this line.
- Answer: tangent
Common Mistakes
MistakeConfusing a chord with a diameter, forgetting that a diameter must pass through the centre
Fixevery diameter is a chord, but not every chord is a diameter - only a chord through the centre is a diameter
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