Practical Geometry — Class 6 Mathematics

'Geometry is not just thinking — it is doing. The ruler and compass are your tools for bringing shapes to life.'

1. Introduction

Practical geometry is about constructing geometric figures using tools. The two most important tools are the ruler (for straight lines and measuring) and the compass (for circles and transferring lengths).

Tools of Geometry

Why Construct, Not Just Draw?

Construction is PRECISE. Drawing freehand may be approximate, but construction with tools guarantees accuracy. 'In construction, a millimetre matters — especially in engineering and architecture.'

2. Drawing a Circle

A circle is a closed curve where every point is at the same distance (radius) from a fixed point (centre).

Steps to Draw a Circle

1. Set the radius: Place compass tip at 0 cm mark on ruler. Open compass to the desired radius (e.g., 4 cm)
2. Mark the centre: Put a dot on the paper labelled 'O'
3. Draw the circle: Place compass tip exactly on O. Hold the hinge firmly. Rotate the compass 360°

Worked Example

Draw a circle of radius 4 cm. Mark its centre, radius, and diameter.

Steps:

1. Set compass to 4 cm on ruler
2. Mark centre O
3. Draw circle with compass at O
4. Draw radius OA (O to any point A on circle)
5. Draw diameter BOC (line through O touching circle at B and C)

Check Your Work

• All points on the circle should be exactly 4 cm from O
• Measure radius in different directions — all should be 4 cm
• Diameter should be exactly 8 cm (2 × 4)

3. Constructing Line Segments

Constructing a Given Length

To construct a line segment of exactly 5.3 cm:

Method 1: Using a Ruler Only

1. Mark point A on paper
2. Place ruler with 0 at A
3. Mark point B at 5.3 cm
4. Join A and B

Method 2: Using Ruler and Compass (More Accurate)

1. Draw a line l. Mark point A on it
2. Set compass to 5.3 cm on ruler
3. With compass tip at A, draw an arc cutting line l
4. Mark the intersection as B
5. AB = 5.3 cm

Constructing a Line Segment Equal to Another

To copy segment PQ to create RS of same length:

1. Draw a line. Mark point R on it
2. Set compass to length PQ (place tip at P, pencil at Q)
3. Without changing compass opening, place tip at R
4. Draw an arc cutting the line — that's S
5. RS = PQ

4. Constructing Perpendicular Lines

A perpendicular line makes an angle of 90° with another line.

Method 1: Using a Protractor

1. Draw a line AB. Mark point P on it
2. Place protractor centre at P, base along AB
3. Mark 90° on the protractor
4. Draw line from P through the 90° mark
5. This line ⟂ AB

Method 2: Using Ruler and Compass

To construct perpendicular at point P on line l:

1. With compass at P, draw an arc cutting l on both sides at A and B
2. With same radius (or larger), draw arcs from A and B above l
3. Let the arcs meet at C
4. Join PC. PC ⟂ l.

Method 3: Perpendicular Through a Point Not on the Line

1. Place compass at P, draw an arc cutting l at A and B
2. From A and B, draw arcs below l meeting at Q
3. Join PQ. PQ ⟂ l.

Checking Perpendicularity

Use a protractor or set square to verify the angle is 90°.

5. Constructing Angle Bisectors

An angle bisector is a ray that divides an angle into two EQUAL parts.

Steps to Bisect an Angle

1. With vertex O, draw an arc cutting both arms of the angle at A and B
2. From A, draw an arc inside the angle
3. From B, draw another arc of the same radius, meeting the first arc at C
4. Join OC. OC is the angle bisector

Check Your Work

• ∠AOC should equal ∠COB
• Measure both angles with a protractor to verify

Application

Bisecting 60° gives two 30° angles. Bisecting 90° gives two 45° angles.

6. Constructing Angles

Constructing 60° Angle

1. Draw a ray AB
2. With compass at A, draw an arc cutting AB at C
3. With same radius, place compass at C and draw an arc cutting the first arc at D
4. Join AD. ∠DAB = 60°

Justification: Triangle ACD is equilateral → all angles 60°.

Constructing 120° Angle

1. Construct 60° as above
2. Place compass at D (the 60° mark), draw another arc cutting the original arc at E
3. Join AE. ∠EAB = 120°

Constructing 30° Angle

1. Construct 60°
2. Bisect 60° → two 30° angles

Constructing 90° Angle

Method 1: Perpendicular to a line (Use perpendicular construction — already learned)

Method 2: Using compass

1. Draw ray AB
2. With compass at A, draw arc cutting AB at C
3. With same radius, from C draw arc cutting the first at D
4. From D, with same radius, draw arc cutting the first at E
5. From D and E, draw arcs meeting above at F
6. Join AF. ∠FAB = 90°

Constructing 45° Angle

1. Construct 90°
2. Bisect 90° → two 45° angles

Summary of Constructible Angles

'Once you can construct 60° and bisect angles, you can construct many other angles: 15°, 30°, 45°, 75°, 90°, 105°, 120°, 135°, 150°.'

7. Worked Problems

Problem 1

Draw a circle of radius 3.5 cm. Construct a chord AB of length 5 cm.

Solution:

1. Draw circle with radius 3.5 cm, centre O
2. Mark point A on the circle
3. Set compass to 5 cm
4. Place compass at A, draw arc cutting circle at B
5. AB is the chord of 5 cm

Problem 2

Construct a 75° angle.

Solution:

1. Construct 60° (∠DAB)
2. Construct 90° (∠FAB) — on the same side
3. Bisect the angle between 60° and 90°
4. 60 + (90-60)/2 = 60 + 15 = 75°

Problem 3

Construct a square of side 5 cm.

Solution:

1. Draw base AB = 5 cm
2. At A, construct a perpendicular (90°)
3. On the perpendicular, mark AD = 5 cm
4. At B, construct a perpendicular
5. On it, mark BC = 5 cm
6. Join CD. ABCD is the required square.

8. Common Mistakes — Fix Them Now

9. Exam Focus

Marks Blueprint

Quick Self-Test (5 Questions)

Q1: What is the first step to construct a 60° angle?

Q2: To draw a circle of radius 5 cm, what opening of the compass is needed?

Q3: What is an angle bisector?

Q4: How do you construct a 30° angle?

Q5: What tools are needed for constructing perpendicular lines?

10. Chapter Summary

• Circle: set compass to radius, place tip at centre, rotate
• Line segment: use ruler or compass to copy length
• Perpendicular lines: construct using compass arcs meeting above/below
• Angle bisector: arcs from both arms, join vertex to intersection
• Constructible angles: 60°, 120°, 30° (bisect 60°), 90°, 45° (bisect 90°)
• Key principle: compass transfers length accurately; arcs create intersection points

'Practical geometry is where mathematics meets craftsmanship. With practice, your constructions will be as precise as they are beautiful.'