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JAC • Class 6 • Mathematics • Ch 16

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Practical Geometry

In JAC Board 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.

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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 Syllabus and book alignment

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

3 2. Perpendicular Bisector & Angle B...

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

Complete Concept Guide (100% Curriculum Coverage)

Syllabus and book alignment

JCERT 2026–27: Regular schools (Non-SOE) — Chapter 16 · Practical Geometry. Syllabus pages 236–253. These notes follow this topic. Official syllabus. Official JCERT Hindi textbook. Some source PDFs are damaged. The supplementary reference uses different chapter numbering.

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.

Chapter practice

CBT tests for this chapter are not published yet. Solve the self-check questions above and compare your answers.

All Class 6 Mathematics Chapters

Ch 1: Knowing Our Numbers Ch 2: Whole Numbers Ch 3: Integers Ch 4: Playing with Numbers Ch 5: Fractions Ch 6: Decimals Ch 7: Ratio and Proportion Ch 8: Algebra Ch 9: Line, Line Segment and Ray Ch 10: Pairs of Lines Ch 11: Angles Ch 12: Polygons Ch 13: Circle Ch 14: Three-Dimensional Shapes Ch 15: Symmetry Ch 16: Practical Geometry Ch 17: Perimeter and Area Ch 18: Data Handling

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