Chapter 8 — Playing with Constructions

Ganita Prakash · Grade 6 · Study Notes

This chapter teaches us how to draw neat shapes using two simple tools — a ruler and a compass. We learn to make circles, squares, rectangles, and fun pictures like a house, a person and eyes.

Table of Contents

8.1 Artwork

First we try drawing shapes made of circles just with our hand (freehand). Then we use a compass to make them look neat and exact.

What is a curve? A curve is any shape we can draw on paper with a pencil. This includes straight lines, circles and other wavy figures.

Making Points at a Fixed Distance

Mark a point P. Now mark many points that are all exactly 4 cm away from P, in every direction. When you join all these points, you get a smooth round shape — a circle!

All points 4 cm from P form a circle.

The Compass and the Circle

Open the compass against a ruler so the gap between the metal tip and the pencil is 4 cm. Keep the tip fixed and move only the pencil around — this draws a full circle.

Imp Definitions:

Centre — the fixed point P in the middle of the circle.
Radius — the distance from the centre to any point on the circle. Every point on the circle is the same distance (the radius) from the centre.

Construct — A Person

The figure of "A Person" has two parts: a circle (the head) and a square with a curved bottom (the body). The tricky part is the inward-bulging curve at the bottom. Fix a radius on the compass, then try placing the tip at different spots until the arc fits the bottom of the square neatly.

"A Person" — a circle head above a square body with a curved bottom.

Construct — Wavy Wave

Take a central line AB = 8 cm. The first wave is drawn as a half circle above the line, and the second wave as a half circle below it.

Wavy Wave — half circle above, then half circle below the central line (AB = 8 cm).

Figure it Out (Page 191)	Answer
Q1. Radius for the half circle? Length of AX?	Radius = 2 cm, and AX = 4 cm (AX is the diameter of each half circle).
Q2. Draw the wave on a central line of different length.	Choose any AB length, split it into equal parts, and draw half circles up and down turn by turn.
Q3. Make waves smaller than a half circle (like the neck of "A Person").	Draw shorter arcs (less than half circle) using a bigger radius but only a small part of the arc. Keep both waves identical by using the same radius and same tip position.

The same wave repeated along a longer central line.

Construct — Eyes

Each eye is made of two curved arcs (an upper curve and a lower curve) that meet at both ends, with a filled dark circle in the middle. Two light horizontal helper lines are drawn first to place the compass tip (points A and B) so the upper and lower curves are perfectly symmetrical. This may need a few trials.

An eye — two matching arcs meeting at both ends with a dark centre.

8.2 Squares and Rectangles

These are shapes with straight-line boundaries. Consider rectangle ABCD with corners A, B, C, D. Its sides are AB, BC, CD, DA and its angles are ∠A, ∠B, ∠C, ∠D.

Properties of a Rectangle:

R1) Opposite sides are equal in length.
R2) All angles are 90°.

Properties of a Square:

S1) All four sides are equal.
S2) All angles are 90°.

Naming a Rectangle Correctly

The rectangle ABCD can also be named BCDA, CDAB, DABC, ADCB, DCBA, CBAD, BADC. The rule is: the corners must be listed in the order you travel around the shape (any starting corner). Names like ABDC or ACBD are wrong because they jump across the shape.

Q. Which of these is NOT a name for the square (corners S, P at top; R, Q at bottom)?
1. PQSR 2. SPQR 3. RSPQ 4. QRSP

Answer: PQSR is not a valid name (its letters do not follow the travel order around the square).

Rotated Squares and Rectangles

When we turn (rotate) a square, its sides stay equal and its angles stay 90°. So a rotated square is still a square, and a rotated rectangle is still a rectangle. Turning a shape does not change its lengths or angles.

Figure it Out (Page 194)

Q1. Draw the rectangle-and-four-squares figure on dot paper.

Answer: Draw the rectangle first, then leave one dot distance diagonally at each side to place the four smaller squares symmetrically.

Q2. Identify the squares in the collection A, B, C, D.

Answer: Shape A is a square (equal sides and right angles). The others are not squares.

Think: Can we tell equal sides and right angles without measuring?

Answer: Yes. By looking at how many dots the corners move across and up/down on the grid, we can check equal sides and right angles without any instrument.

Q3. Draw 3 rotated squares/rectangles with corners on dots and check properties.

Answer: Draw tilted shapes with corners sitting exactly on dots. Count grid steps to confirm opposite sides equal and all angles 90°.

8.3 Constructing Squares and Rectangles

Let us construct a square PQRS of side 6 cm.

Step	What to do
Step 1	Draw side PQ = 6 cm using a ruler.
Step 2	Use a protractor to mark a 90° perpendicular line through P.
Step 3	On this perpendicular, mark point S so that PS = 6 cm (ruler) or with a compass set to 6 cm.
Step 4	Draw a perpendicular to PQ through Q as well.
Step 5	Using the compass (radius 6 cm) from Q, mark point R on that line.
Step 6	Join R and S. Now RS = 6 cm and ∠R = ∠S = 90°. Square is ready!

Square PQRS of side 6 cm.

Construct (Page 197)

Q1. Draw a rectangle with sides 4 cm and 6 cm and check both properties.

Answer: ∠A = ∠B = ∠C = ∠D = 90°; AB = CD = 4 cm and AD = BC = 6 cm. Both properties are satisfied.

Q2. Draw a rectangle with sides 2 cm and 10 cm and check both properties.

Answer: ∠P = ∠Q = ∠R = ∠S = 90°; PQ = SR = 10 cm and PS = QR = 2 cm. Both properties hold.

Q3. Can we make a 4-sided figure with all angles 90° but opposite sides NOT equal?

Answer: No. If all four angles are 90°, the opposite sides are always forced to be equal.

8.4 An Exploration in Rectangles

Make rectangle ABCD with AB = 7 cm and BC = 4 cm. Point X can slide along side AD and point Y can slide along side BC. We measure the length XY for different positions.

Imp finding: When X and Y are placed at the same distance from A and B, the line XY is straight across and its length equals AB (7 cm). In this case the shape ABYX is a rectangle.

Table of Observations (X and Y at equal distances)

Distance of X from A	Distance of Y from B	Length of XY
5 mm	5 mm	7 cm
1 cm	1 cm	7 cm
1 cm 5 mm	1 cm 5 mm	7 cm

Table of Observations (unequal distances)

Distance of X from A	Distance of Y from B	Length of XY
5 mm	3 cm	7.4 cm
1 cm	1 cm	7 cm
2 cm	4 cm	7.3 cm

Q. In each equal-distance case, how does XY compare to AB, and what shape is ABYX?

Answer: (i) XY = AB. (ii) ABYX is a rectangle.

Q. How does the farthest distance between X and Y compare with AC or BD?

Answer: The farthest distance between X and Y is equal to the diagonal AC or BD.

Construct — Breaking Rectangles

To make a rectangle that splits into 3 identical squares, its length must be 3 times its breadth. To split into 2 identical squares, length must be 2 times the breadth.

Rectangle divided into 3 identical squares.

Plan first: Draw AF for one side without measuring. Use a compass to copy that length onto the perpendicular line to fix the square's size. Mark points B and C the same way to complete the rectangle.

Q. Give side lengths of a rectangle that CANNOT be divided into two/three identical squares.

Answer: Cannot split into two squares → e.g. Length 4 cm, Breadth 2.5 cm. Cannot split into three squares → e.g. Length 7 cm, Breadth 2 cm. (The length is not an exact multiple of the breadth.)

More Construct Tasks (Pages 201–203)

A Square within a Rectangle: In an 8 cm × 4 cm rectangle, place a square so its centre matches the rectangle's centre. The square's side equals the shorter side (4 cm), and it must be centred.
Falling Squares: Draw squares of the given sizes (4 cm each, or 3 cm–5 cm–7 cm) aligned corner-to-corner in a stepped pattern.
Shadings: Draw one big square, split it into smaller squares, and shade half of each smaller square with slanting lines.

Q4. Square with a Hole — where is the circle's centre?

Answer: The centre of the circle is exactly at the centre of the square, i.e. where the two diagonals (lines joining opposite corners) meet.

Square with more Holes / Square with Curves: Split the square into 4 equal squares and place a circle in the centre of each. For curves, place the compass tip so that all four arcs bulge inward equally from each side.

8.5 Exploring Diagonals of Rectangles and Squares

In rectangle PQRS, the lines PR and QS are the diagonals. A diagonal cuts each corner's right angle into two smaller angles.

Imp observations:

Both diagonals of a rectangle are equal in length.
The two smaller angles made at each corner by a diagonal are equal only when the shape is a square.

Diagonals PR and QS of a rectangle.

Explore: How should the rectangle be built so the diagonal splits opposite angles into equal parts?

Answer: The two adjacent sides must be equal — that is, the rectangle must become a square.

Construct — Rectangle from Diagonal Angles

Example: Make a rectangle where a diagonal splits opposite angles into 60° and 30°.

Step	Action
1	Draw AB of any length; make a 90° line at B.
2	At A, draw a line making 60° with AB — this fixes the diagonal direction; it meets the line at C.
3	Draw a line through A perpendicular to AB (this is where D lies). The remaining angle at A becomes 30°.
4	Method 1: Draw a perpendicular to BC at C to get D. Method 2: With a compass, mark D so that AD = BC. Join CD.

Rectangle from a Side and a Diagonal

Example: Side = 5 cm, diagonal = 7 cm.

Draw base CD = 5 cm.
Draw a perpendicular line l at C.
With centre D and radius 7 cm, draw an arc (or circle). Where it cuts line l is point B (since the diagonal DB = 7 cm).
Draw perpendiculars at D and B; where they meet is point A. Check ABCD is a rectangle.

Construct (Page 211)

Q1. Rectangle whose diagonal splits opposite angles into 50° and 40°.

Answer: Draw AB, make 50° at A for the diagonal, complete with right angles. The two small angles at each corner are 50° and 40° (they add to 90°).

Q2. Rectangle where the diagonal splits angles into 45° and 45°. What about the sides?

Answer: All four sides become equal — the figure is a square.

Q3. Rectangle with one side 4 cm and diagonal 8 cm.

Answer: Draw PQ = 4 cm, a perpendicular at Q, then an arc of radius 8 cm from P cutting that line at R. Complete the rectangle PQRS.

Q4. Rectangle with one side 3 cm and diagonal 7 cm.

Answer: Draw AB = 3 cm, a perpendicular at B, then an arc of radius 7 cm from A cutting the line at C. Complete the rectangle ABCD.

8.6 Points Equidistant from Two Given Points

We make a House figure where every border line is 5 cm long. The main idea is to find a point that is the same distance from two other points, using a compass instead of guessing.

The House — every border side is 5 cm.

Steps to Build the House

Step	What to do
1	Draw the base DE = 5 cm, sides BD and CE = 5 cm each, and the small door (1 cm × 2 cm) in the middle of the base.
2	To find the roof top A (which is 5 cm from both B and C): draw an arc of radius 5 cm from B and another arc of radius 5 cm from C. Where the two arcs cross is point A.
3	Join A to B and A to C with straight lines (each 5 cm).
4	With radius 5 cm from A, draw the arc that touches B and C to complete the curved part. House is ready!

Imp idea for exams: To locate a point that is equal distance from two given points, draw arcs of the same radius from both points. Their crossing point is the point you need. This avoids all trial and error.

Construct (Page 215)

Q1. Construct a bigger house with all sides 7 cm.

Answer: Repeat the same steps but use 7 cm for every border side (base, walls, and roof lines). Use arcs of radius 7 cm to find the roof top and to draw the curve.

Q2. Recreate "A Person", "Wavy Wave" and "Eyes" using House ideas.

Answer: Use the arc-crossing method to place compass tips. For "A Person" draw a circle head over a square body; for "Wavy Wave" alternate half circles up and down; for "Eyes" draw two matching arcs meeting at both ends with a dark centre.

Q3. Is there a 4-sided figure with all sides equal that is NOT a square?

Answer: Yes — a rhombus. It has all four sides equal but its angles are not 90°. Construct it by drawing a slanting side, then using equal-radius arcs to place the other corners.

A rhombus — all sides equal, but not a square.

Hint B (Page 216) — 4-sided figure with all sides 5 cm

Start with side AB = 5 cm and side AC = 5 cm meeting at A. To get the fourth point D (which must be 5 cm from both B and C), use the same arc-crossing trick from the House problem: draw arcs of radius 5 cm from B and from C, and their meeting point is D.

Imp Points to Remember

All points of a circle are the same distance (the radius) from its centre.
A compass draws circles and their parts (arcs).
Draw a rough diagram first to plan any construction.
A rectangle can be constructed from the lengths of its sides, or from one side and a diagonal.
To find a point equal distance from two points, cross two equal-radius arcs.