Cambridge IGCSE Mathematics 0580: Geometrical terms, constructions and scale drawings – Study Guide

This study guide teaches subtopics 4.1 Geometrical terms, 4.2 Geometrical constructions and 4.3 Scale drawings of the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027. The wording is almost the same at both tiers (C4.1–C4.3 Core, E4.1–E4.3 Extended). The Extended list adds the terms plane, perpendicular bisector, hemisphere, frustum, and major and minor arc, marked Extended only below. Core content is examined on Papers 1 (non-calculator) and 3 (calculator); Extended on Papers 2 (non-calculator) and 4 (calculator).

Use it with the 0580 course hub and the printable 0580 checklist. The rest of topic 4 is in the Geometry study guide; recap with the Geometry revision notes and test yourself with the Geometry practice questions, which use different numbers from this page. To find weak spots, take the free Core diagnostic or Extended diagnostic.

What these subtopics cover

0580 reference	What you must be able to do	Tier
4.1 (1)	Use and interpret basic geometrical terms (listed below)	Both; plane and perpendicular bisector Extended only
4.1 (2)	Vocabulary of triangles, special quadrilaterals, polygons, nets and solids	Both; hemisphere and frustum Extended only
4.1 (3)	Vocabulary of a circle	Both; major and minor arc Extended only
4.2 (1)–(3)	Measure and draw lines and angles; construct a triangle from three sides with ruler and compasses only; draw, use and interpret nets	Both
4.3 (1)–(2)	Draw and interpret scale drawings; use and interpret three-figure bearings	Both

The syllabus notes say you are not expected to show that two shapes are congruent, and that constructing perpendicular bisectors and angle bisectors is not required. For every paper you need a pair of compasses, a protractor and a ruler.

4.1 Geometrical terms

Points, lines and angles

• A point marks a position. A vertex is a corner where lines or edges meet.
• A line is straight. A plane is a flat surface (Extended only).
• Parallel lines never meet. Perpendicular lines meet at a right angle (90°).
• The perpendicular bisector of a segment AB cuts AB in half at right angles (Extended only, as a term).

Angle type	Size
Acute	less than 90°
Right angle	exactly 90°
Obtuse	between 90° and 180°
Reflex	between 180° and 360°

An interior angle of a polygon is inside the shape at a vertex. An exterior angle is between one side and the next side extended outwards. At each vertex they add up to 180°, because they lie on a straight line. Calculating them is subtopic 4.6, in the Geometry study guide.

Similar, congruent and scale factor

Congruent shapes have the same shape and size. Similar shapes have the same shape, but one is an enlargement of the other: angles are equal and corresponding lengths are in the same ratio. That ratio is the scale factor (new length ÷ original length). A 3 cm by 5 cm rectangle and a 6 cm by 10 cm rectangle are similar with scale factor 2, but not congruent.

Triangles, quadrilaterals and polygons

Triangle	Property
Equilateral	three equal sides, three 60° angles
Isosceles	two equal sides, two equal angles
Scalene	no equal sides
Right-angled	one 90° angle

Quadrilateral	Key properties
Square	four equal sides, four right angles
Rectangle	opposite sides equal, four right angles
Rhombus	four equal sides, opposite sides parallel
Parallelogram	opposite sides equal and parallel
Trapezium	exactly one pair of parallel sides
Kite	two pairs of equal adjacent sides

A square is also a rectangle and a rhombus, so when asked to name a shape give the most specific name that fits.

A polygon is a closed shape with straight sides. It is regular if all sides and all angles are equal, otherwise irregular. Learn pentagon (5 sides), hexagon (6), octagon (8) and decagon (10).

Solids

A face is a flat surface of a solid, an edge is where two faces meet, and the surface is the whole outside. A net is a flat shape that folds up into a solid.

• Cube, cuboid: six square or rectangular faces.
• Prism: the same cross-section all the way through. Cylinder: a circular cross-section.
• Pyramid: a polygon base with triangular faces meeting at a vertex. Cone: a circular base and a curved surface meeting at a vertex.
• Sphere: a ball. Hemisphere (half a sphere) and frustum (a cone or pyramid with the top cut off parallel to the base) are Extended only.

Worked example 1. Count the faces, edges and vertices of a pentagonal prism and of a square-based pyramid.

Pentagonal prism: 2 pentagons + 5 rectangles = 7 faces
                  5 + 5 + 5 joining edges = 15 edges;  10 vertices
Square-based pyramid: 1 square + 4 triangles = 5 faces
                  4 base + 4 sloping = 8 edges;  4 + 1 apex = 5 vertices

Circle vocabulary

Term	Meaning
Centre, radius (plural radii)	the middle point; the distance from it to the circle
Diameter	a chord through the centre; twice the radius
Circumference	the distance all the way round
Semicircle	half a circle, cut off by a diameter
Chord	a straight line joining two points on the circle
Tangent	a straight line touching the circle at one point
Arc	part of the circumference; minor arc shorter, major arc longer (Extended only)
Sector	region between two radii and an arc
Segment	region between a chord and an arc

A sector is bounded by two radii, a segment by a chord. They are the pair most often confused.

4.2 Geometrical constructions

Measuring and drawing

Use a ruler for every straight edge and measure to the nearest millimetre. With a protractor, put the centre on the vertex, line up the zero line with one arm, and count from the zero on that arm. If an angle looks acute, your reading must be under 90°.

Constructing a triangle from three sides

Use only a ruler and compasses, and show the construction arcs.

Worked example 2. Construct triangle ABC with AB = 9 cm, AC = 7.5 cm and BC = 6 cm.

1. Draw AB = 9 cm (start with the longest side).
2. Compasses at 7.5 cm, point on A: draw an arc above AB.
3. Compasses at 6 cm, point on B: draw an arc crossing the first.
4. The crossing point is C. Join AC and BC with a ruler.
5. Leave both arcs showing.

Check: angle BAC should measure about 41°, angle ABC about 56° and angle ACB about 83°. If one side is as long as the other two together, or longer, the arcs do not cross and no triangle exists.

Constructing a rhombus from two triangles

The syllabus gives this as an example. A rhombus is two congruent isosceles triangles sharing a diagonal.

Worked example 3. Construct rhombus PQRS with sides 5 cm and diagonal PR = 8 cm.

1. Draw PR = 8 cm.
2. Compasses at 5 cm: arcs from P and R above PR meet at Q.
3. Same radius: arcs from P and R below PR meet at S.
4. Join PQ, QR, RS and SP. Keep all four arcs.

Check: diagonal QS should measure 6 cm, angles P and R about 74°, angles Q and S about 106°.

Nets: drawing, using and interpreting

Sketch the solid first, then lay out every face so that edges which meet in the solid are equal in the net. Count faces at the end: a cuboid net has six rectangles; a triangular prism net has two triangles and three rectangles.

Surface area is the total area of the net. The volume formulas for a prism (V = Al) and a pyramid (V = (1/3)Ah) are on the formula list printed in every paper.

Worked example 4. A net has two right-angled triangles with sides 3 cm, 4 cm and 5 cm, and three rectangles 10 cm long with widths 3 cm, 4 cm and 5 cm. Find the surface area and volume of the solid.

Solid: triangular prism, length 10 cm
Triangles: 2 x (1/2) x 3 x 4 = 12 cm^2
Rectangles: (3 + 4 + 5) x 10 = 120 cm^2
Surface area = 132 cm^2
Volume = 6 x 10 = 60 cm^3

Worked example 5. A net is a 6 cm square with an isosceles triangle on each side; each triangle is 5 cm high from the middle of its base. Find the surface area and volume of the pyramid.

Surface area = 6 x 6 + 4 x (1/2) x 6 x 5 = 36 + 60 = 96 cm^2
5 cm is the slant height. Vertical height h (Pythagoras,
hypotenuse 5, base 3 = half of 6): h = sqrt(25 - 9) = 4 cm
Volume = (1/3) x 36 x 4 = 48 cm^3

More on this in the Mensuration study guide.

4.3 Scale drawings

Using a scale

A scale can be a ratio, such as 1 : 50 000 (both parts in the same unit), or a statement, such as “1 cm represents 2 km”. To turn a statement into a ratio, use one unit: 2 km = 200 000 cm, so 1 : 200 000.

Worked example 6. A map has scale 1 : 50 000.

(a) Two farms are 7.3 cm apart on the map.
    7.3 x 50 000 = 365 000 cm = 3650 m = 3.65 km
(b) A lake is 2.4 km long. Map length?
    2.4 km = 240 000 cm;  240 000 / 50 000 = 4.8 cm

Drawing to real: multiply. Real to drawing: divide. On Papers 1 and 2 there is no calculator, so practise multiplying and dividing by powers of 10. A scale drawing of a room works the same way: at 1 : 50, a 4.6 m wall is 460 ÷ 50 = 9.2 cm long.

Three-figure bearings

A bearing is an angle measured clockwise from north, written with three figures from 000° to 360°: 45° is 045°. North is 000°, east 090°, south 180°, west 270°. “D is due east of C” means the bearing of D from C is 090°.

The point after from is where you stand. “The bearing of B from A” means: stand at A, face north, turn clockwise until you face B. Draw a north line at A before measuring.

Reverse bearings. North lines are parallel, so the two bearings between a pair of points differ by 180°.

Less than 180°: add 180°.   180° or more: subtract 180°.
Bearing of Q from P = 063°  ->  P from Q = 243°
Bearing of T from S = 290°  ->  S from T = 110°

Scale drawing with bearings

Worked example 7. A boat sails 12 km from harbour H on a bearing of 070° to buoy A, then 8 km on a bearing of 160° to buoy B. Using 1 cm to 2 km, draw the route and find the distance and bearing of B from H.

Drawing lengths: 12 / 2 = 6 cm;  8 / 2 = 4 cm
1. Draw a north line at H. Measure 70° clockwise; draw HA = 6 cm.
2. Draw a north line at A. Measure 160° clockwise; draw AB = 4 cm.
3. Measure HB: about 7.2 cm, so about 7.2 x 2 = 14.4 km.
4. Measure clockwise from north at H to HB: about 104°.

Checks: the bearing of H from A is 250°, so angle HAB = 250° − 160° = 90°. Pythagoras’ theorem and trigonometry give 14.4 km and 103.7°, matching the drawing. The bearing of H from B is about 104° + 180° = 284°. The trigonometry subtopics say knowledge of bearings may be required; see the Trigonometry study guide.

Common errors

• Rubbing out construction arcs, or drawing edges freehand.
• Reading the wrong protractor scale, so 70° becomes 110°.
• Measuring a bearing anticlockwise or from the wrong point.
• Writing 70° instead of 070°, or adding 180° to a bearing already over 180°.
• Converting units twice in a scale question.
• Using the slant height as the vertical height in a pyramid.
• Calling a segment a sector, a trapezium a parallelogram, or a 200° angle obtuse.

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027 (Version 3), Subject content: C4.1–C4.3 and E4.1–E4.3, Geometry.