Playing with Constructions – Class 6 Maths Chapter 8 Notes & NCERT PDF

Class 6MathematicsChapter 8NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 8 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 8: Playing with Constructions

Introduction

Chapter 8 turns the geometry notebook into an art studio. It starts with some pictures (a person, a wavy wave, a pair of eyes). Children first try them freehand, then pick up a ruler and a compass and learn to draw them exactly. On the way they rediscover the circle, write down what makes a rectangle and a square, and learn a planning habit mathematicians use: draw a rough diagram first, then construct.

Key Ideas

8.1 Artwork: the compass and the circle

Mark a point P and many points 4 cm away from it in all directions. Joined up, they make a circle.

To draw it, open the compass against a ruler so the gap between the metal tip and the pencil is 4 cm. Keep the tip fixed on P and turn only the pencil.

All the points of a circle are at the same distance from its centre. This distance is called the radius.

Wavy Wave: take a central line AB = 8 cm. The first wave is a half circle over half the line, so AX = 4 cm and the compass radius is 2 cm.

A Person and Eyes: the hard part is choosing where to place the compass tip and what radius to use for a curve. Estimate, try and adjust.

8.2 Squares and rectangles

In rectangle ABCD, A, B, C, D are the corners, AB, BC, CD, DA are the sides, and AB and CD are opposite sides.

Rectangle: R1) opposite sides are equal; R2) all angles are 90°.

Square: S1) all sides are equal; S2) all angles are 90°.

Naming: a valid name lists the corners in order of travel around the figure, starting from any corner. ABCD, BCDA, ADCB and DCBA are fine; ABDC and ACBD are not. For the square in the book, PQSR is not a valid name.

Rotated squares and rectangles: turning a square does not change its lengths or angles, so a tilted square is still a square. The same goes for a rectangle. On a dot grid you can even check this without measuring, just from where the corners sit on the dots.

8.3 Constructing a square (side 6 cm)

Draw PQ = 6 cm.

Draw a perpendicular to PQ through P.

Mark S on it so that PS = 6 cm, with a ruler (Method 1) or a compass (Method 2).

Draw a perpendicular to PQ through Q.

Mark R on it so that QR = 6 cm. A compass already set to 6 cm makes this quick.

Join SR. Check that RS = 6 cm and that ∠R = ∠S = 90°.

8.4 An exploration in rectangles

Rectangle ABCD has AB = 7 cm and BC = 4 cm. Point X slides along AD and point Y slides along BC.

When X and Y are the same distance from A and B, XY = AB = 7 cm and ABYX is a rectangle. This is also the shortest XY can be.

XY is longest when it is a diagonal (AC or BD).

Record trials in a table: Distance of X from A | Distance of Y from B | Length of XY.

Breaking rectangles: for a rectangle made of 2 identical squares, the long side is twice the short side. For 3 squares, it is three times. Draw a rough figure and mark equal sides with a small tick (|). You can transfer a length with a compass without measuring it.

8.5 Diagonals of rectangles and squares

The two diagonals of a rectangle are equal.

A diagonal splits each opposite angle into two parts. These parts are equal (45° and 45°) only when the rectangle is a square.

Construction with angles: to make a rectangle whose diagonal splits the corner angles into 60° and 30°: (1) draw AB of any length; (2) draw a perpendicular at B and a line from A making 60° with AB — they meet at C, so diagonal AC splits ∠A into 60° and 30°; (3) draw a perpendicular to AB through A, since D lies on it; (4) find D either with a perpendicular to BC at C ("all angles are 90°") or by marking AD = BC with a compass ("opposite sides are equal").

Given a side and a diagonal (5 cm and 7 cm): draw DC = 5 cm and a perpendicular line l at C. B lies on l and 7 cm from D. So draw an arc of radius 7 cm centred at D. Where it cuts l is B. Then draw perpendiculars at D and B to get A.

8.6 Points equidistant from two given points

The House: every border line is 5 cm. The roof tip A must be 5 cm from both B and C. Draw an arc of radius 5 cm from B and another from C. Where they cross is A, with no trial and error. The curved roof is then an arc of radius 5 cm centred at A.

Worked Examples

1. Wavy Wave. AB = 8 cm, split into two half-circle waves. Each wave sits on 4 cm, so AX = 4 cm and the compass radius = 4 ÷ 2 = 2 cm.

2. Two-square rectangle. If AF = 4 cm, then AB = AF = 4 cm and BC = 4 cm, so AC = 8 cm.

3. Rectangle with a 5 cm side and a 7 cm diagonal

Draw DC = 5 cm.

Draw line l perpendicular to DC at C.

With centre D and radius 7 cm, draw an arc cutting l at B.

Draw perpendiculars to DC at D and to BC at B. They meet at A.

Check R1 and R2.

4. Diagonal splitting 50° and 40°. 50° + 40° = 90°, so the pieces fit in a right angle. Build the 50° angle at one corner, then complete the rectangle as in the 60°/30° example.

5. Diagonal splitting 45° and 45°. The sides come out equal, so the rectangle is a square.

Figure it Out: Hints

Can a 4-sided figure have all angles 90° but opposite sides unequal? No. Try it and the sides refuse to close.

A rectangle that cannot be split into two identical squares: for example, 4 cm × 2.5 cm (the long side isn't twice the short side).

Square with a hole: the circle's centre is where the two diagonals of the square meet.

A 4-sided figure with all sides equal that is not a square: it exists. Use the House idea: find the fourth point D, 5 cm from both B and C, with two arcs.

Maths Words

Compass (परकार), Ruler (पैमाना)

Circle (वृत्त), Centre (केंद्र), Radius (त्रिज्या)

Arc (चाप), Curve (वक्र)

Rectangle (आयत), Square (वर्ग)

Corner (कोना/शीर्ष), Opposite sides (सम्मुख भुजाएँ)

Perpendicular (लम्ब), Right angle (समकोण)

Diagonal (विकर्ण)

Rough diagram (कच्चा चित्र)

Equidistant (समदूरस्थ)

Common Mistakes

Letting the compass tip slip, or its opening change, while drawing a circle.

Naming a rectangle "ABDC": the letters must follow the boundary.

Thinking a tilted square is a "diamond" and not a square. Check S1 and S2!

Using trial and error with a ruler when one arc would locate the point exactly.

Assuming every rectangle's diagonal splits the corner angles equally. That only happens in a square.

Skipping the rough diagram, then not knowing which line to draw first.

Maths in Daily Life

Carpenters, tailors, architects and rangoli artists all use these ideas. A mason marks a right angle, a gardener draws a circular flower bed with a rope and a peg, and an architect plans a house front before building it.

💡 Key Learning Points

✓A circle is the set of all points at the same distance (radius) from its centre; a compass draws it exactly.
✓A rectangle has opposite sides equal and all angles 90°; a square has all sides equal and all angles 90° — rotating them does not change this.
✓Squares and rectangles can be constructed step by step using perpendiculars and a compass to transfer lengths.
✓A rectangle can be constructed from its two sides, or from one side and a diagonal, by using an arc to locate the missing corner.
✓A point at given distances from two points is found where two arcs intersect — making a rough diagram first helps plan the construction.

👨‍🏫 Teaching Tips

→Begin with the Artwork page: let children draw freehand first, then with compass — discuss which looked better and why.
→Practise safe compass handling: set the radius against a ruler, keep the tip fixed, rotate only the pencil.
→Use dot paper for rotated squares so children reason about equal sides and right angles from dot positions.
→Run the X–Y sliding exploration in pairs with a recording table; ask them to predict before measuring.
→Insist on a labelled rough diagram with equal-side ticks before every construction; ask 'Which line can be drawn first?'.
→Demonstrate the 5 cm side + 7 cm diagonal rectangle live, contrasting ruler trial-and-error with the single arc.
→End with the House: let each child build a 7 cm house and exhibit the constructions on the class wall.

📋 Assessment Questions

Can the student draw a circle of a given radius and explain what 'radius' means?
Can the student state properties R1, R2 and S1, S2 and identify valid names like ABCD vs ABDC?
Can the student construct a square of side 6 cm and a rectangle given a side and a diagonal, listing the steps?
Can the student locate a point equidistant from two given points using two arcs?
Can the student explain when a diagonal splits the corner angles equally (only in a square)?

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