CIE IGCSE Mathematics (0580) Geometrical constructions Study Notes - New Syllabus

CIE IGCSE Mathematics (0580) Geometrical constructions Study Notes

LEARNING OBJECTIVE

Geometrical constructions

Key Concepts:

Geometrical constructions

CIE iGCSE Maths (0580) Study Notes – All topics

Measuring and Drawing Lines and Angles

To construct accurate diagrams in geometry, you must be able to measure and draw straight lines and angles precisely using a ruler and protractor.

Key Tools:

Ruler – for drawing and measuring straight lines in centimeters or millimeters.
Protractor – for measuring and constructing angles in degrees.
Pencil – always draw constructions lightly first, then darken final lines.

Steps to Draw a Line and an Angle:

To draw a line: Place the ruler on paper, mark the start and end points, and join them using a pencil along the ruler.
To measure an angle: Place the midpoint of the protractor at the angle’s vertex, align one side with 0°, and read the value on the correct scale.
To draw an angle: Draw a base line, place the protractor’s center on one end, mark the required angle, and join the vertex to this point.

Example:

Construct a line segment with length 6 cm.. Then, from one end, construct a 60° angle.

▶️ Answer/Explanation

Use a ruler to draw a straight line 6 cm long. Label the ends as P and Q.

Place the center of the protractor at point P, with the baseline along PQ.

Mark a point at 60° on the scale.

Remove the protractor and draw a line from point A through the 60° mark.

This creates ∠QPR = 60°.

Constructing a Triangle Given All Three Sides (SSS)

To construct a triangle when all three side lengths are known, use only a ruler and a pair of compasses. This is called the Side–Side–Side (SSS) construction.

Steps to Construct the Triangle:

Draw the base of the triangle using a ruler.
Using a compass, draw arcs from each end of the base to locate the third vertex.
Join the intersection point of the arcs to the base ends to complete the triangle.

Example:

Construct triangle ABC with sides: AB = 4 cm, BC = 5 cm, and AC = 3 cm.

▶️ Answer/Explanation

Use a ruler to draw the base BC = 5 cm.

Open your compass to 4 cm. Place the compass at point B and draw an arc above the line.

Now open the compass to 3 cm. Place the compass at point C and draw another arc to intersect the previous arc.

Label the intersection point as A.

Join A to B and A to C using a ruler.

Triangle ABC is now constructed with the correct side lengths.

Tools: Only a ruler and compasses were used (no protractor).

Example:

Construct a rhombus PQRS where each side is 6 cm and diagonal PR is 8 cm. Use only a ruler and compasses. Show construction arcs.

▶️ Answer/Explanation

Draw diagonal PR = 8 cm using a ruler.
Open your compass to 6 cm (the side of the rhombus).
With P as center, draw an arc above and below PR.
With R as center, draw arcs of the same radius (6 cm) to intersect the previous arcs. Label the intersection points as Q and S.
Join PQ, QR, RS, and SP using a ruler.
Rhombus PQRS is now constructed by joining two congruent triangles △PQR and △PRS.
Note: All sides are equal, and the diagonals bisect each other at right angles.

Nets of 3D Shapes

A net is a two-dimensional pattern that can be folded to form a 3D shape. Nets are useful for calculating surface area and understanding the structure of solid objects.

Common Nets Include:

3D Shape	Net	Faces	Name
		4 rectangles (2 squares / rectangles)	Cuboid
		6 squares	Cube
		1 rectangle, 2 circles	Cylinder
		4 triangles	Triangular-based pyramid (Tetrahedron)
		2 triangles, 3 rectangles	Triangular Prism
		1 circle, 1 sector	Cone
		6 triangles, 1 hexagon	Hexagonal Pyramid