Playing with Constructions
Why This Matters
Have you ever drawn a circle freehand? It always comes out a little wobbly. One side is fat, the other side is flat. It is hard to make it perfect by hand.
Now think about the people who build things. A carpenter wants a perfectly square window. An engineer wants a perfectly round wheel. They cannot just guess. They need shapes that are exactly right.
This chapter teaches you how to draw such perfect shapes. You will use just two simple tools: a ruler and a compass. With these, you can draw a perfect circle, a straight line of an exact length, and neat squares and rectangles.
The best part is that it feels like play. You will draw shapes step by step, like following a recipe. And you will even make pretty artwork, like a flower, using only your compass. By the end, you will be able to build shapes that are correct down to the last millimetre.
The Big Idea
Two simple tools do all the work. A ruler draws straight lines and measures exact lengths. A compass keeps one fixed distance as it swings, so it draws perfect circles and arcs — and it can also carry a length from one place to another. A circle is the set of all points that are the same distance from one centre point; that fixed distance is its radius. Using these tools step by step, you can build any straight-edged shape — a line segment, a square, a rectangle, even a right angle — exactly, without guessing.
Let’s Break It Down
Your tools — ruler and compass
You only need two tools for this whole chapter. Let us meet them.
A ruler is the flat measuring stick you already use. It has a straight edge for drawing straight lines. It also has marks on it (in centimetres and millimetres) so you can measure how long something is.
A compass is the tool that has two legs joined at the top. One leg ends in a sharp metal point. The other leg holds a pencil. You open the legs apart, press the sharp point on the paper, and swing the pencil around. The pencil draws a curve.
Figure 8.1 below shows both tools and what each one is for.
Before we use these tools, let us quickly remember a few words from earlier classes, because we will use them a lot.
The circle — centre and radius
Now for the most fun shape: the circle.
Try this in your head first. Mark a point P on paper. Now mark a dot that is 4 cm away from P. Mark another dot 4 cm away, but in a different direction. Keep going. Mark many, many dots, all exactly 4 cm from P, in every direction.
What shape do all those dots make together? They make a circle! A circle is simply all the points that are the same distance from one centre point.
That centre point is called the centre of the circle. The fixed distance — from the centre to any point on the circle — is called the radius.
This is exactly why a compass draws a perfect circle. Look closely at how a compass works. You press the sharp tip down on the centre and keep it fixed there. The two legs stay open by the same amount the whole time. So the pencil is always the same distance from the tip. As you swing the pencil around, it traces every point that is that fixed distance from the centre. That is a circle.
Figure 8.2 below shows a circle with centre P. Notice the three radius lines drawn — they are all the same length, 3 cm.
This is the key idea you must remember: every point on a circle is the same distance from the centre. That distance is the radius.
A circle has centre O. Point A is on the circle. Point B is also on the same circle. Is OA longer than OB, shorter than OB, or equal to OB?
They are equal. OA and OB are both radii of the same circle. And every radius of a circle is the same length. So OA = OB.
How to draw a circle of radius 3 cm. First, open your compass against a ruler. Put the sharp tip at the 0 mark and open the pencil leg until it sits on the 3 cm mark. Now the compass width is exactly 3 cm. Press the sharp tip down on your paper where you want the centre. Hold it still. Then gently turn the top of the compass so the pencil swings all the way around. You get a perfect circle of radius 3 cm.
Draw a circle with a radius of 4 cm. Name its centre M.
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Open the compass against your ruler. Put the sharp tip on the 0 mark. Slide the pencil leg out to the 4 cm mark. Now the gap between the tip and the pencil is exactly 4 cm.
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On your paper, mark a point and call it M. This will be the centre.
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Press the sharp tip firmly on M. Keep it fixed. Do not let it slide.
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Hold the compass by its top and swing the pencil leg all the way around, back to where it started.
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You now have a neat circle. Its centre is M, and every point on it is 4 cm from M. So its radius is 4 cm. Done!
Drawing a line segment of a given length
Drawing a straight line of an exact length is the most common construction. Let us learn it slowly. Say we want to draw a line segment AB = 5 cm (this means the segment from A to B should be exactly 5 cm long).
Figure 8.3 below shows the three simple steps.
Here are the steps written out:
- Place your ruler flat on the paper. Mark a point exactly at the 0 mark. Call it A.
- Look along the ruler to the 5 cm mark. Mark a point there. Call it B.
- Hold the ruler still. Draw a straight line from A to B along the edge of the ruler.
That line is your segment AB, exactly 5 cm long.
A student starts measuring from the metal edge of the ruler instead of the 0 mark, and stops at the 5 cm number. Will their line be exactly 5 cm?
No. On most rulers there is a small gap between the metal edge and the 0 mark. If you start from the edge, your line will be a little bit too long. You must always start counting from the 0 mark.
Copying a length with a compass
Here is a clever trick. Sometimes you want a new line to be the same length as one you already have, but you do not want to measure it with the ruler. Maybe the length is something odd, and you just want it copied exactly.
The compass can do this for you. The compass holds whatever width you open it to. So you can open it to the length of one segment, then carry that exact width somewhere else.
Say you have segment AB and you want to make a new segment CD that is exactly as long. Figure 8.4 below shows how.
The steps are:
- Open the compass so the tip is on A and the pencil is on B. Now it holds the length AB.
- Do not change the width. Draw a new straight line and mark a point C on it.
- Put the tip on C and swing a short arc that crosses the new line. Mark that crossing point D.
- Now CD = AB. You copied the length without a ruler.
This trick is very useful. We will use it soon to copy a side of a square.
Constructing a square step by step
Now let us build a full shape. We will construct a square with each side 6 cm. First, let us remember what makes a square a square.
We name the square PQRS. The four corners, going around in order, are P, Q, R, S. Figure 8.5 below shows the four steps to build it.
Here are the steps in words:
- Draw the base. Make a line segment PQ that is exactly 6 cm long.
- At corner P, draw a line going straight up, making a right angle (90°) with PQ.
- On that upright line, mark point S so that PS = 6 cm. Do the same above Q to get point R (so QR = 6 cm). The compass set to 6 cm makes this easy.
- Join S to R. Now you have square PQRS, with all sides 6 cm and all corners 90°.
Construct a rectangle ABCD with AB = 7 cm and BC = 4 cm.
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Draw the base AB exactly 7 cm long, using your ruler. Mark point A at 0 and point B at 7 cm.
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At corner B, draw a line going straight up, making a right angle (90°) with AB.
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On that upright line, mark point C so that BC = 4 cm.
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At corner A, draw another line going straight up at a right angle. Mark point D on it so that AD = 4 cm (the same as BC, because opposite sides of a rectangle are equal).
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Join C to D. You now have rectangle ABCD. Its long sides AB and DC are 7 cm, its short sides BC and AD are 4 cm, and all four corners are 90°. That is your rectangle!
Notice the difference between a square and a rectangle. The table below makes it clear.
| Square | Rectangle | |
|---|---|---|
| Sides | All four sides equal | Opposite sides equal (two long, two short) |
| Corners | All four are 90° | All four are 90° |
| Is it the other one? | A square is also a rectangle (its opposite sides are equal too) | A rectangle is a square only if all sides happen to be equal |
A perpendicular and simple artwork
You may wonder: in the square, what does “a line going straight up at a right angle” really mean? That line has a special name.
When two lines cross and make a right angle (90°), we say they are perpendicular to each other. Think of a standing pole and the flat ground. The pole is perpendicular to the ground. The “straight up” line in our square was perpendicular to the base.
You can draw a perpendicular neatly using the right-angle marks on a set square or by careful measuring with a protractor. The important thing to remember is what it means: a perfect 90° corner, like the corner of your notebook.
Now for the fun part — artwork with a compass. Remember, a compass keeps the same width. So if you draw a circle and then, without changing the width, place the tip on points along that circle and draw more arcs, the arcs cross in a pretty pattern.
Here is a lovely one: the six-petal flower. Figure 8.6 below shows it.
Try it yourself. Draw a circle. Then, keeping the same compass width, put the tip on any point of the circle and draw an arc inside. Move the tip to where that arc meets the circle and draw again. Go all the way around. You will get six petals. It feels like magic, but it is just maths.
Common Mistakes
Constructions go wrong in a few common ways. Watch out for these.
While drawing a circle, the compass width slips and changes a little as you swing it.
It is easy to think a small wobble will not matter, and a cheap compass feels loose at the hinge, so it seems fine to hold it gently.
If the width changes even a little, the curve is no longer a true circle — some points end up nearer the centre and some farther. Tighten the compass screw, hold it by the top knob, and let only the pencil leg move.
Measuring a length starting from the metal edge of the ruler instead of from the 0 mark.
The metal edge looks like the natural starting point — it is the very beginning of the ruler — so it feels right to start counting from there.
On most rulers there is a small gap between the edge and the 0 mark. Always line up the 0 mark with your starting point, then read off the length you want.
Calling a four-sided shape a square just because it looks neat, without checking the corners.
A shape with four roughly equal sides looks square at a glance, and our eyes are quick to call it a square.
A square needs both things to be true: all four sides equal AND all four corners exactly 90°. A shape can have equal sides but slanted corners (that is a rhombus, not a square). Always check the angles too.
When copying a length with the compass, opening or closing the compass before marking the new point.
After swinging the first arc, it feels natural to adjust the compass to make the next mark, since you are about to draw again.
The whole trick depends on the width staying fixed. Once you set the compass to the length, do not touch the screw until you have marked the copied point. Any change ruins the copy.
Quick Check
Let us see what stuck. Try each question, then read the explanation.
What is the radius of a circle?
Why does a compass always draw a perfect circle?
A four-sided shape has all four sides equal to 5 cm, but its corners are not 90°. Is it a square?
You want segment CD to be exactly as long as segment AB without using a ruler. What do you do?
Practice Problems
Try each one yourself before opening the solution. Drawing it on paper is the best way to learn.
Easy
Draw a circle with a radius of 2 cm. Mark its centre and call it O.
- Open the compass against your ruler. Put the sharp tip on the 0 mark and the pencil on the 2 cm mark. Now the width is 2 cm.
- Mark a point on your paper and call it O. This is the centre.
- Press the sharp tip on O and hold it still.
- Swing the pencil all the way around.
- You get a circle with centre O and radius 2 cm. Every point on it is 2 cm from O.
Draw a line segment XY that is exactly 6 cm long.
- Place your ruler on the paper.
- Mark point X right at the 0 mark.
- Mark point Y at the 6 cm mark.
- Join X to Y with a straight line along the ruler.
- That line is segment XY = 6 cm. Remember: start at 0, not at the metal edge.
Medium
Construct a square with each side 5 cm. Name it ABCD.
- Draw the base AB exactly 5 cm long. Mark A at 0 and B at 5 cm.
- At corner B, draw a line straight up, making a right angle (90°) with AB.
- On that upright line, mark point C so that BC = 5 cm.
- At corner A, draw another line straight up at a right angle. Mark point D so that AD = 5 cm.
- Join C to D.
- Check it: all four sides should be 5 cm, and all four corners should be 90°. Now ABCD is a square.
You are given a line segment PQ. Without using a ruler, copy its length onto a new line so that the new segment RS is exactly as long as PQ.
- Open the compass so its sharp tip is on P and its pencil is on Q. Now the compass holds the length PQ.
- Do not change the width.
- Draw a new straight line. Mark a point R on it.
- Put the tip on R and swing a small arc that crosses the new line.
- Mark the crossing point S.
- Now RS = PQ, because the compass width never changed.
Challenge
Construct a rectangle that can be divided into two identical squares. (Hint: think about how the long side compares to the short side.)
First, plan with a rough sketch. If the rectangle is made of two equal squares placed side by side, and each square has side 4 cm, then:
- the short side of the rectangle = 4 cm (the side of one square), and
- the long side of the rectangle = 4 cm + 4 cm = 8 cm (two squares in a row).
So we need a rectangle that is 8 cm by 4 cm. Here is how to build it:
- Draw the base AB exactly 8 cm long.
- At A, draw a line straight up at a right angle. Mark D so that AD = 4 cm.
- At B, draw a line straight up at a right angle. Mark C so that BC = 4 cm.
- Join D to C. Now ABCD is an 8 cm by 4 cm rectangle.
- To show the two squares, mark the middle of AB (at 4 cm from A) — call it M — and the middle of DC — call it N. Join M to N.
- The line MN splits the rectangle into two squares, each 4 cm by 4 cm.
Draw a six-petal flower using only a compass.
- Open the compass to any width, say 3 cm. Draw a circle. Keep this width for the whole problem — do not change it.
- Mark any point on the circle. Call it the starting point.
- Put the compass tip on that starting point. Draw an arc inside the circle. This arc will cross the circle at two new points.
- Move the tip to one of those crossing points. Draw another arc.
- Keep moving the tip to each new crossing point and drawing an arc. Go all the way around.
- You will make exactly six arcs, and they cross to form six petals meeting at the centre. It works because the radius and the spacing are the same — six steps fit perfectly around a circle.
Summary
- A ruler draws straight lines and measures exact lengths. A compass draws circles and arcs, and can copy a length from one place to another.
- A circle is all the points that are the same distance from one centre point. That centre is the centre, and the fixed distance is the radius.
- A compass draws a perfect circle because its width stays the same, so the pencil is always the same distance (the radius) from the fixed tip.
- To draw a line segment of a given length, line up the ruler, mark the start at the 0 mark, mark the end at the length you want, and join them.
- To copy a length, open the compass to that length, keep the width fixed, and swing it onto a new line.
- A square has all four sides equal AND all four corners 90°. A rectangle has opposite sides equal AND all four corners 90°.
- You build a square or rectangle step by step: draw the base, make right angles at the corners, mark the other corners using the correct lengths, and join them up.
- Two lines that cross at a right angle (90°) are perpendicular. Using one compass width, you can also draw fun artwork like a six-petal flower.
What’s Next
You can now draw perfect circles, squares and rectangles. Next, in Chapter 9 — Symmetry, you will look at shapes that match themselves when you fold or turn them. The flower you just drew with your compass is full of symmetry — so you have already met the idea. Get ready to spot it everywhere.
Frequently Asked Questions
What tools do you use for constructions in class 6 maths and why?
You use a ruler (to draw and measure straight lines) and a compass (to draw circles and copy lengths). These two tools together let you make perfectly accurate shapes. A compass keeps one end fixed and draws a perfect circle because every point on the circle is the same distance from the centre.
What is the radius of a circle and how is it related to the diameter?
The radius is the distance from the centre of the circle to any point on its edge. The diameter is a straight line through the centre touching the circle on both sides -- it is exactly twice the radius. If the radius is 5 cm, the diameter is 10 cm.
How do you draw a line segment of a given length using a ruler and compass?
First draw a straight line with your ruler and mark one endpoint. Set the compass to the required length by opening it to that measurement on the ruler. Place the compass point on your endpoint and draw a small arc crossing the line. Where the arc cuts the line is the second endpoint. Join the two points.
How do you construct a square using a ruler and compass step by step?
Draw one side of the required length. At each end of that side, draw a perpendicular (right-angle) line using your compass. Mark off the same length along each perpendicular. Join the two new endpoints. All four sides will be equal and all angles will be 90°.
What is a perpendicular line and how do you know two lines are perpendicular?
Two lines are perpendicular if they meet at exactly 90° (a right angle). You can check this with a protractor or the corner of a set square. In constructions, you create perpendiculars using a compass by drawing arcs from two points to find a crossing point directly above.