Select Medium / माध्यम चुनें:

CBSE • Class 6 • Mathematics • Ch 14

Estimated Time: 45 Mins

Study Progress: In Progress

Practical Geometry

In CBSE Class 6 Mathematics, "Practical Geometry" focuses on precise ruler and compass geometric constructions: drawing circles of given radius, copying line segments, perpendicular bisectors, constructing perpendiculars from points on/off a line, angle bisectors, and special angles (60°, 120°, 90°, 45°, 30°) without a protractor.

Take Chapter CBT Test View All Chapters

Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

📏 Have You Ever Wondered?

How did ancient Greek architects construct a perfect 90° right angle or bisect angles with just an unmarked straightedge and a pair of compasses?

Compass constructions are pure geometric logic! Master Euclidean tools to create mathematically flawless constructions.

Why This Chapter Matters

Before You Begin (Prerequisites)

Using a compass and ruler
Concept of radius and angles

What You Will Learn (Core Objectives)

Draw a circle of given radius using compass.
Construct a copy of a given line segment.
Construct the perpendicular bisector of a line segment.
Construct the bisector of a given angle.
Construct special angles ($60^\circ, 120^\circ, 90^\circ, 45^\circ, 30^\circ$) using ruler and compass.

Chapter Roadmap & Progression

1 1. Construction of Circles & Line S...

2 2. Perpendicular Bisector & Angle B...

3 3. Constructing 60°, 120°, 90°, 45°...

Complete Concept Guide (100% Curriculum Coverage)

1. Construction of Circles & Line Segments

Circle: Set compass span to radius $r$ on ruler, fix metallic pointer at center $O$, and rotate pencil $360^\circ$.
Copying a Line Segment: Never measure with a ruler directly; open compass span to match endpoints of $\overline{AB}$, place pointer at new point $C$ on a line, and draw an arc cutting at $D$. Then $\overline{CD} = \overline{AB}$.

2. Perpendicular Bisector & Angle Bisector

Perpendicular Bisector of $\overline{AB}$:

With center $A$ and radius more than half of $AB$, draw arcs above and below the segment.
With same radius and center $B$, draw arcs intersecting earlier arcs at $P$ and $Q$.
Join $PQ$. $PQ$ is perpendicular to $AB$ and bisects it into two equal halves.

Angle Bisector: Draw an arc cutting arms at $P$ and $Q$. With centers $P$ and $Q$ and equal radius, draw intersecting arcs inside the angle at $R$. Ray $OR$ bisects the angle into two equal parts.

3. Constructing 60°, 120°, 90°, 45° & 30° Angles

$60^\circ$ Angle: Draw a ray $OA$. With center $O$ and any radius, draw an arc cutting $OA$ at $B$. With center $B$ and the same radius, draw an arc intersecting the first arc at $C$. Join $OC$. $\angle AOC = 60^\circ$.
$120^\circ$ Angle: From $C$ with same radius, cut another arc at $D$. Join $OD$. $\angle AOD = 120^\circ$.
$90^\circ$ Angle: Bisect the $60^\circ$ arc between $C$ and $D$ ($60^\circ + 30^\circ = 90^\circ$).
$30^\circ$ Angle: Bisect the $60^\circ$ angle.
$45^\circ$ Angle: Bisect the $90^\circ$ angle.

Key Formulas, Identities & Theorems

Angle Bisector Law

$$\angle AOC = \angle BOC = \frac{1}{2}\angle AOB$$

Ruler-compass bisector property.

Conceptual Solved Examples & Case Studies

Example 1

Why must the radius of the compass be greater than half of AB when constructing a perpendicular bisector?

Step-by-Step Solution:

If the radius is less than or equal to half of $AB$, the arcs drawn from $A$ and $B$ will either never touch or touch at only one midpoint, failing to create two intersecting reference points ($P$ and $Q$) needed to determine a line.

Common Misconceptions & Examiner Traps

Common Misconception

Changing compass width while drawing the first and second arcs for a 60° angle.

Scientific Reality & Correction

The compass radius MUST remain identical to create the equilateral triangle relationship that defines 60°.

Visual Learning & Conceptual Map

Ruler-Compass Angle Hierarchy

Constructions from Base 60° Arc

60°

First arc on base.

120°

Second arc on base.

90°

Bisect 60° and 120°.

45°

Bisect 90°.

30°

Bisect 60°.

Chapter Summary & 10 Key Takeaways

Takeaway 1

Compass draws circles and replicates segments.

Takeaway 2

Perpendicular bisector divides a segment in half at 90° using radius > half length.

Takeaway 3

Special angles (60°, 120°, 90°, 45°, 30°) are constructed without protractors using arcs and bisectors.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

How can you construct an angle of 15° using ruler and compass?

Reveal Answer & Explanation

Answer: Construct a 60° angle, bisect it to get 30°, then bisect the 30° angle to obtain 15°.
Half of 30 degrees.

Finished Studying This Chapter?

READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test

15 MCQs • Core Concepts

Advanced Test

15 MCQs • HOTS & Traps