← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 4 Geometry · Chapter 22 · pp. 282–292

Geometrical vocabulary and construction

Geometry has its own words: acute, perpendicular, isosceles, chord, rhombus. This chapter collects them in one place, then shows how to draw shapes accurately with only a ruler and a pair of compasses, and how scale drawings turn a map distance into a real one.

Learning objectives

  1. Name angles by size: acute, right, obtuse and reflex.
  2. Use the words perpendicular, parallel and altitude (height).
  3. Name triangles by their sides and by their angles, and know when two triangles are congruent or similar.
  4. Name the parts of a circle and the properties of the special quadrilaterals.
  5. Name polygons and draw the nets of solids.
  6. Construct a triangle from its three sides using only a ruler and compasses.
  7. Read and draw scale drawings, and use map scales such as 1 : 10 000.

Book exercises: 22.1 to 22.5 and Student assessment 1 (pp. 282–292).

Types of angle

acute 0° to 90° right exactly 90° obtuse 90° to 180° reflex 180° to 360°
Acute is less than 90°, right is exactly 90°, obtuse is between 90° and 180°, and reflex is between 180° and 360°.
WordMeaning
Shortest distanceThe length of the straight line joining two points.
PerpendicularTwo lines that meet at a right angle (90°). Written CD ⊥ AB.
Altitude (height)The perpendicular line from a vertex of a triangle to the opposite side.
ParallelStraight lines that never meet, however far they go, like railway lines. Marked with matching arrows.
A B C D CD ⊥ AB · CD is the height of triangle ABC parallel: matching arrows, never meet
Left: CD meets AB at 90°, so they are perpendicular, and CD is the altitude of the dashed triangle. Right: parallel lines carry matching arrow marks.

📘 Example 1 · Name an angle of 235°

  1. Compare with the boundaries: 235° is more than 180°.
  2. It is less than 360°.
  3. Between 180° and 360° means reflex.

📘 Example 2 · The two angles on a straight line at a point are 64° and x°. Name both.

  1. Angles on a straight line add to 180°, so x = 180 − 64 = 116.
  2. 64° is between 0° and 90°: acute.
  3. 116° is between 90° and 180°: obtuse.
  4. Answer: 64° is acute and 116° is obtuse.

✏️ Try it

  1. Name the angles 89°, 91°, 180.5° and 300°.
  2. Two lines cross and one angle is 90°. What are the other three angles, and what word describes the lines?
  3. Can a triangle have a reflex angle? Why?
Answers

1. acute, obtuse, reflex, reflex · 2. All 90°; the lines are perpendicular · 3. No: the three angles add to 180°, so none can be more than 180°.

Next: triangles →

Types of triangle

acute-angled right-angled obtuse-angled isosceles equilateral scalene
Top row: named by the biggest angle. Bottom row: named by the sides; dashes mark equal sides. A triangle gets one name of each kind, for example "right-angled isosceles".
By anglesRuleBy sidesRule
Acute-angledall angles below 90°Isoscelestwo equal sides; the angles opposite them are equal
Right-angledone angle of 90°Equilateralthree equal sides and three equal angles (60° each)
Obtuse-angledone angle above 90°Scaleneall sides and all angles different

Congruent triangles

Congruent triangles are identical: matching sides are the same length and matching angles are equal. You only need to show one of these four tests:

SSS SAS ASA RHS Thick lines and arcs: the parts that must match
The four congruence tests. In SAS the angle must be the one between the two sides.
TestWhat must match
SSSall three sides
SAStwo sides and the included angle (the angle between them)
ASAtwo angles and the corresponding side
RHSa right angle, the hypotenuse and one other side

💡 Three equal angles (AAA) is not a congruence test: the triangles have the same shape but can be different sizes. That makes them similar.

Similar triangles

40° 60° 80° 40° 60° 80° A B C X Y Z
Triangles ABC and XYZ have the same three angles, so they are similar: every side of XYZ is the same multiple of the matching side of ABC. Chapter 23 covers this in detail.

📘 Example 3 · Triangle PQR has PQ = 5 cm, QR = 7 cm, ∠Q = 40°. Triangle XYZ has XY = 5 cm, YZ = 7 cm, ∠Y = 40°. Congruent?

  1. Match the sides: PQ = XY = 5 cm and QR = YZ = 7 cm.
  2. Find where the angle sits: ∠Q is between PQ and QR, and ∠Y is between XY and YZ. Both are included angles.
  3. Two sides and the included angle match: congruent (SAS).

📘 Example 4 · Two triangles both have angles 30°, 60°, 90°. The shortest sides are 3 cm and 6 cm.

  1. The angles all match, so the shapes are the same.
  2. The shortest sides differ (3 cm and 6 cm), so the sizes are different.
  3. Answer: similar but not congruent (scale factor 2).

✏️ Try it

  1. A triangle has angles 90°, 45° and 45°. Name it in two ways.
  2. Two right-angled triangles both have hypotenuse 13 cm and one side 5 cm. Congruent? Which test?
  3. Triangle 1 has sides 4, 6, 7 cm. Triangle 2 has sides 7, 4, 6 cm. Congruent?
Answers

1. right-angled isosceles · 2. Yes, RHS · 3. Yes, SSS (the same three lengths in a different order)

Next: circles and quadrilaterals →

Parts of a circle

segment: between a chord and the arc chord: joins two points on the circle arc: part of the circumference centre diameter: chord through the centre sector: a slice between two radii radius circumference
The circumference is the perimeter of the circle. A diameter is two radii long. Shaded: a segment at the top and a sector at the bottom.

Quadrilaterals

A quadrilateral is a flat shape with four sides and four angles. The dashed lines are the diagonals.

square rectangle rhombus parallelogram trapezium kite
The six special quadrilaterals with their diagonals dashed.
ShapeProperties (from the book)
Squaretwo pairs of parallel sides · all sides equal · all angles equal · diagonals cross at right angles
Rectangletwo pairs of parallel sides · opposite sides equal · all angles equal
Rhombustwo pairs of parallel sides · all sides equal · opposite angles equal · diagonals cross at right angles
Parallelogramtwo pairs of parallel sides · opposite sides equal · opposite angles equal
Trapeziumone pair of parallel sides (an isosceles trapezium also has its other two sides equal)
Kitetwo pairs of equal sides · one pair of equal angles · diagonals cross at right angles
Pairs of parallel sides? 0 1 2 Kite* Trapezium All sides equal? yes no Right angles? Right angles? yes no yes no Square Rhombus Rectangle Parallelogram *Kite: no parallel sides, but two pairs of equal sides next to each other.
Which quadrilateral? Follow the questions to the most exact name. A square passes every test, so it is also a rhombus, a rectangle and a parallelogram.

📘 Example 5 · A quadrilateral has all sides 5 cm and an angle of 70°. Name it.

  1. All sides equal: it is a square or a rhombus.
  2. A square has every angle 90°, but this one has 70°.
  3. Answer: rhombus.

✏️ Try it

  1. A circle has radius 4.5 cm. How long is its diameter?
  2. Which two special quadrilaterals have all four angles equal?
  3. Name a quadrilateral whose diagonals cross at right angles but which has no parallel sides.
Answers

1. 9 cm · 2. square and rectangle · 3. kite

Next: polygons and nets →

Polygons

  • A polygon is any flat closed shape made of straight lines.
  • A regular polygon has all its sides equal and all its interior angles equal.
  • Two polygons are similar if their angles are the same and their matching sides are in proportion.
Sides3456781012
Nametrianglequadrilateralpentagonhexagonheptagonoctagondecagondodecagon
regular hexagon irregular hexagon
Both have 6 sides, so both are hexagons. Only the left one has equal sides and equal angles, so only it is regular.

Nets

A net is a solid opened out flat. Fold the net up and you get the solid back.

cube: 6 squares triangular prism: 3 rectangles + 2 triangles
A net of a cube (one of 11 possible) and a net of a triangular prism. The two triangles fold up to make the ends.
SolidFaces in its net
Cube6 equal squares
Cuboid6 rectangles, in 3 matching pairs
Triangular prism2 triangles + 3 rectangles
Cylinder2 circles + 1 rectangle (its length is the circumference)
Square-based pyramid1 square + 4 triangles
Tetrahedron4 triangles

📘 Example 6 · A cylinder has radius 3 cm and height 10 cm. Describe its net.

  1. The two ends are circles of radius 3 cm.
  2. The curved side unrolls into a rectangle 10 cm high.
  3. The rectangle wraps once round the circle, so its length is the circumference: 2 × π × 3 = 18.8 cm (3 s.f.).
  4. Answer: two circles of radius 3 cm and a rectangle 18.8 cm by 10 cm.

✏️ Try it

  1. What is the name of a polygon with 8 sides? With 10?
  2. Is a rhombus a regular polygon? Explain.
  3. How many faces does the net of a square-based pyramid have?
Answers

1. octagon, decagon · 2. Not usually: its sides are equal but its angles are not (unless it is a square) · 3. 5 (1 square + 4 triangles)

Next: constructions →

Constructing a triangle from three sides

To construct means to draw accurately with a ruler and a pair of compasses only. Leave your arcs showing: they are the proof of the method.

  • Every point on an arc of radius 6 cm is exactly 6 cm from the compass point.
  • So where two arcs cross, the point is the right distance from both centres.
  • Three lengths make a triangle only if the two shorter ones add to more than the longest. Otherwise the arcs never meet.
A B C 8 cm 7 cm 6 cm 1 · Draw AB = 8 cm. 2 · Compasses 6 cm, centre B: arc. 3 · Compasses 7 cm, centre A: arc. 4 · Arcs cross at C. Join CA, CB.
The book's worked example: AB = 8 cm, BC = 6 cm, AC = 7 cm. C is 6 cm from B and 7 cm from A.

📘 Example 7 · Construct triangle ABC with AB = 8 cm, BC = 6 cm, AC = 7 cm

  1. Draw a line AB 8 cm long with a ruler.
  2. Open the compasses to 6 cm. Put the point on B and draw an arc above AB.
  3. Open the compasses to 7 cm. Put the point on A and draw an arc that crosses the first.
  4. Label the crossing point C: it is 6 cm from B and 7 cm from A.
  5. Join C to A and C to B. Triangle ABC is constructed.
  6. Check: measure CA and CB with the ruler (7 cm and 6 cm).

📘 Example 8 · Can you construct a triangle with sides 9 cm, 4 cm and 4 cm?

  1. Add the two shorter sides: 4 + 4 = 8 cm.
  2. Compare with the longest: 8 cm is less than 9 cm.
  3. Arcs of 4 cm from each end of a 9 cm line cannot reach each other.
  4. Answer: no, this triangle is impossible.

Two more ruler-and-compass constructions

These two are not worked in this chapter of the book, but you will need them for loci. Step through all three in the tool below.

ConstructionWhat it gives you
Perpendicular bisector of ABthe line through the midpoint of AB at 90°; every point on it is the same distance from A and B
Angle bisectorthe line that splits an angle into two equal halves; every point on it is the same distance from both arms

🧪 Try it: construction step-through

Newest marks are drawn thick. Arcs are drawn in the accent colour.

    ✏️ Try it

    1. Construct triangle PQR with PQ = 6 cm, QR = 5 cm, PR = 4 cm. Measure ∠P.
    2. Can a triangle have sides 3 cm, 5 cm and 8 cm? Explain.
    3. To construct an equilateral triangle of side 5 cm, what radius do both arcs use?
    Answers

    1. ∠P ≈ 56° (cos P = (36 + 16 − 25) ÷ 48) · 2. No: 3 + 5 = 8 is not more than 8, so the arcs only touch on the line and the "triangle" is flat · 3. 5 cm both times

    Next: scale drawings →

    Scale drawings

    • A scale drawing is an accurate diagram drawn in proportion: maps and plans are scale drawings.
    • A scale of 1 : 10 000 means 1 cm on the map is 10 000 cm in real life.
    • Map → real: multiply by the scale number, then change the units. Real → map: change to cm first, then divide.
    map (cm) real (cm) real (m) real (km) × n ÷ n ÷ 100 × 100 ÷ 1000 × 1000
    For a scale 1 : n. Top arrows go from the map to real life, bottom arrows come back.

    📘 Example 9 (book) · Scale 1 : 10 000. Two objects are 1 cm apart on the map. Real distance in metres?

    1. 1 cm on the map is 10 000 cm in real life.
    2. So 1 cm × 10 000 = 10 000 cm.
    3. Change to metres: 10 000 ÷ 100 = 100 m.

    📘 Example 10 (book) · A model boat is built to 1 : 50. The real boat is 12 m long. Model length in cm?

    1. Change the real length to cm: 12 m = 1200 cm.
    2. Every 50 cm of real boat is 1 cm of model, so divide: 1200 ÷ 50 = 24.
    3. Answer: 24 cm.

    📘 Example 11 · Scale 1 : 25 000. On the map two villages are 6.4 cm apart. Real distance?

    1. Multiply: 6.4 × 25 000 = 160 000 cm.
    2. Change to metres: 160 000 ÷ 100 = 1600 m.
    3. Change to km: 1600 ÷ 1000 = 1.6 km.

    Scale drawings of triangles

    A B C 6 cm 5 cm 4 cm 3.3 cm height (measured) = 3.3 cm area = ½ × base × height = 6 × 3.3 ÷ 2 = 9.9 cm²
    The book's worked example: construct the triangle, measure the perpendicular height from C to AB, then work out the area.

    📘 Example 12 (book) · Construct ABC with AB = 6 cm, AC = 5 cm, BC = 4 cm, then find its area

    1. Construct the triangle from three sides (scale 1 : 1), as in the Constructions tab.
    2. Measure the perpendicular from C to AB: 3.3 cm.
    3. Area = = .
    4. Answer: 9.9 cm².

    🧪 Try it: map scale converter

      ✏️ Try it

      1. Scale 1 : 20 000. Map distance 3.5 cm. Real distance in metres?
      2. Scale 1 : 50 000. Real distance 2.5 km. Map distance in cm?
      3. A room is 4.8 m long. How long is it on a plan drawn at 1 cm for every 2 m?
      Answers

      1. 70 000 cm = 700 m · 2. 250 000 cm ÷ 50 000 = 5 cm · 3. 4.8 ÷ 2 = 2.4 cm

      Next: summary and quiz →

      Chapter summary

      • Angles: acute < 90° · right = 90° · obtuse 90°–180° · reflex 180°–360°.
      • Perpendicular lines meet at 90°. Parallel lines never meet and carry matching arrows.
      • Triangles are named by angles (acute, right, obtuse) and by sides (isosceles, equilateral, scalene).
      • Congruent = identical. Tests: SSS, SAS (included angle), ASA, RHS. Same angles only = similar.
      • Circle: centre, radius, diameter, chord, arc, sector, segment, circumference.
      • Square, rectangle, rhombus, parallelogram, trapezium, kite: know their sides, angles and diagonals.
      • Polygons: pentagon 5, hexagon 6, heptagon 7, octagon 8, decagon 10, dodecagon 12. Regular = equal sides and angles.
      • A net folds up into a solid.
      • Triangle from 3 sides: draw one side, two arcs from its ends, join to the crossing. Shorter two sides must add to more than the longest.
      • Scale 1 : n: map → real multiply by n; real → map convert to cm, then divide by n.

      📄 Practice questions

      The book's Exercises 22.1 to 22.5 and Student assessment 1 are in the printable assignment for this chapter.

      Mini-quiz

      1. An angle measures 215°. What type of angle is it?
      2. Two lines meet at exactly 90°. What are the lines called?
      3. A triangle has sides 5 cm, 5 cm and 8 cm. What type is it?
      4. Which is NOT a test for congruent triangles?
      5. Which special quadrilateral has diagonals that cross at right angles and no parallel sides?
      6. A chord passes through the centre of a circle. What is it called?
      7. What is the name of a polygon with 8 sides?
      8. How many faces does the net of a square-based pyramid have?
      9. Which sides can form a triangle?
      10. Scale 1 : 20 000. Two places are 3 cm apart on the map. How far apart are they really?
      11. A model is built to scale 1 : 50. The real object is 6 m long. How long is the model?

      Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 22 (Geometrical vocabulary and construction). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.