In the previous section we discussed the direct proportional changes in 'angular measurements'. In this section we will see 'scales' in Maps and Drawings.
Consider an iron rod of length 300 cm. We want to draw a picture of this rod on a sheet of paper. The object that we have is an iron rod. So it's picture will be a straight line. But we cannot draw a 300 cm long line on a sheet of paper. The maximum size possible is 118 cm on an A0 size paper. (See here for details of various sizes of paper sheets). So we need a method to represent such large objects on paper. The method that is used is 'scaling':
In this method we assume that 1 cm on the paper represent a certain distance on the real object. Let us see the different possibilities:
Assume that 1 cm on paper represents 100 cm of the real object. The real object (which is the iron rod in our case) is 300 cm long. It has three '100 centimetres' in it. Each of these '100 centimetres' will become a '1cm' in the drawing. This is shown in the fig.3.4 below:
From the above fig., we can see that each 100 cm in the real iron rod 'shrinks' to 1 cm on the paper. So the total 300 cm shrinks to 3 cm. To represent the iron rod fully, we need to draw a line 3 cm long on the sheet of paper. But in the drawing, the length of the line will be marked as '300 cm' not '3 cm'. Also it is compulsory to write the scale in the drawing. Here the scale used is 1 cm = 100 cm. It can also be written as 1:100. This is shown in the fig.3.5 below:
If we measure the 'line drawn on the sheet', it will be exactly 3 cm.
But this 3 cm looks very small on an ordinary A4 size paper. (Because A4 paper has a length of 29 cm, and a width of 21 cm) Especially if there is nothing else drawn on the paper. This 3 cm line will look appropriate if it is drawn on a page of a small 'pocket book'. On an A4 paper, there is plenty of more space available. So let us try to make the drawing bigger:
For this, we change the scale. Assume that 1 cm represents 50 cm of the actual iron rod. There are six '50 centimetres' in the actual iron rod. Each of these '50 centimetres' will become a '1cm' in the drawing. This is shown in the fig.3.6 below:
From the above fig., we can see that each 50 cm in the real iron rod 'shrinks' to 1 cm on the paper. So the total 300 cm shrinks to 6 cm. To represent the iron rod fully, we need to draw a line 6 cm long on the sheet of paper. But in the drawing, the length of the line will be marked as '300 cm' not '6 cm'. As it is compulsory to write the scale in the drawing, we write 1 cm = 50 cm. It can also be written as 1:50. This is shown in the fig.3.7 below:
If we measure the 'line drawn on the sheet', it will be exactly 6 cm.
If there is more space available on the paper, we can make the drawing even bigger. This is by changing the scale to 1:25 or 1:10.
• If it is 1:25, the line will be 12 cm long because, there are twelve '25 centimetres' in the real rod. (∵ 12 × 25 = 300).
• If it is 1:10, the line will be 30 cm long because, there are thirty '10 centimetres' in the real rod. (∵ 30 × 10 = 300).
So we see that, as the right side of the scale decreases, the size of the drawing increases. The drawing becomes more and more like the real object. If the right side decrease even more and become equal to '1', then it is a 1:1 scale. Which means the drawing is of the same size as the real object. Small objects can be drawn to a scale of 1:1.
We can write the converse also: When the right side increases, the drawing becomes smaller. In any case we must remember that the left side is for the drawing and right side is for the actual object. This is shown in the fig.3.8 below:
We divide the length of the real object by the right side of the scale. The result is the length to be drawn on the paper. Thus we can write:
If • Actual length of the object = L
• Scale = 1:S
• Length on paper which represents L = l
Then l = L/S
So now we know how to represent a given length in the real world onto a sheet of paper. In the next section we will see more applications of this method.
Consider an iron rod of length 300 cm. We want to draw a picture of this rod on a sheet of paper. The object that we have is an iron rod. So it's picture will be a straight line. But we cannot draw a 300 cm long line on a sheet of paper. The maximum size possible is 118 cm on an A0 size paper. (See here for details of various sizes of paper sheets). So we need a method to represent such large objects on paper. The method that is used is 'scaling':
In this method we assume that 1 cm on the paper represent a certain distance on the real object. Let us see the different possibilities:
Assume that 1 cm on paper represents 100 cm of the real object. The real object (which is the iron rod in our case) is 300 cm long. It has three '100 centimetres' in it. Each of these '100 centimetres' will become a '1cm' in the drawing. This is shown in the fig.3.4 below:
 |
| Fig.3.4 |
 |
| Fig.3.5 |
But this 3 cm looks very small on an ordinary A4 size paper. (Because A4 paper has a length of 29 cm, and a width of 21 cm) Especially if there is nothing else drawn on the paper. This 3 cm line will look appropriate if it is drawn on a page of a small 'pocket book'. On an A4 paper, there is plenty of more space available. So let us try to make the drawing bigger:
For this, we change the scale. Assume that 1 cm represents 50 cm of the actual iron rod. There are six '50 centimetres' in the actual iron rod. Each of these '50 centimetres' will become a '1cm' in the drawing. This is shown in the fig.3.6 below:
 |
| Fig.3.6 |
 |
| Fig.3.7 |
If there is more space available on the paper, we can make the drawing even bigger. This is by changing the scale to 1:25 or 1:10.
• If it is 1:25, the line will be 12 cm long because, there are twelve '25 centimetres' in the real rod. (∵ 12 × 25 = 300).
• If it is 1:10, the line will be 30 cm long because, there are thirty '10 centimetres' in the real rod. (∵ 30 × 10 = 300).
So we see that, as the right side of the scale decreases, the size of the drawing increases. The drawing becomes more and more like the real object. If the right side decrease even more and become equal to '1', then it is a 1:1 scale. Which means the drawing is of the same size as the real object. Small objects can be drawn to a scale of 1:1.
We can write the converse also: When the right side increases, the drawing becomes smaller. In any case we must remember that the left side is for the drawing and right side is for the actual object. This is shown in the fig.3.8 below:
 |
| Fig.3.8 |
If • Actual length of the object = L
• Scale = 1:S
• Length on paper which represents L = l
Then l = L/S
So now we know how to represent a given length in the real world onto a sheet of paper. In the next section we will see more applications of this method.
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