Diagonal scale

We have seen that the plain scales only give two dimensions, such as a unit and it's subunit or it's a fraction.

The diagonal scales give us three successive dimensions that is a unit, a subdivision of subunits.

The principle of construction of a diagonal scale is as follows.

• Let the XY in figure be a suitable height.


• From Y draw a perpendicular YZ to a suitable height.


• joint XZ, Divide YZ into 10 equal parts.


• Draw parallel lines to XY from all these divisions and number them as shown.


• From geometry, we know that similar triangles have their like sides proportional.


• Consider two similar triangles XYZ and 6'6Z,

we have      [katex]\frac{6Z}{YZ}\ =\ \frac{6'6}{XY} [/katex]

means         [katex]6^\prime6\ =\ \frac{6}{10}\ \times\ XY[/katex]

                            [katex]=\ 0.6\ XY[/katex]

similarly

1'1 = 0.1 XY
2'2 = 0.2 XY

• Thus, It is very clear that, the sides of small triangles, which are parallel to divided lines, become progressively shorter in length 0.1 XY.

Problems on diagonal scale

1 ] The distance between Delhi and Agra is 200 km. In a railway map is represented by a line 5 cm long. Find it's R.F Draw a diagonal scale to show single km. And a maximum of 600 km. indicate on it following distances.

1 ) 122 km

2 ) 336 km

3 )569 km

Solution Steps:

R.F = 5 cm /40,00,000

length of scale = R.F × 600 × 105 
                        = 1/ 40,00,000 × 600 × 105
                        =15 cm

1. Draw a line 15 cm long. it will represent 600 km.


2. divide it into it six equal parts. ( each will represent 100 km )


3. Drive the first division in ten equal parts. Each will represent 10 km. Draw a line upward from the left end and mark 10 part on it of any distance.


4. Name those parts 0 to 10 as shown.join 9^th sub-division of horizontal scale with 10^th division of the vertical division.


5. Then draw parallel lines to this line from the remaining subdivision and complete diagonal scale.

2 ] A rectangular plot of land measuring 1.28 hector is represented on a map by a similar rectangle of 8 sq. cm. Calculate RF of the scale. Draw a diagonal scale to read a single meter. show a distance of 438m on it.

Solution Steps

1 hector         = 10,000 sq. meters
1.28 hectors  = 1.28 × 10,000 sq. meters
                      =1.28 × 104 × 104 sq. cm

8 sq. cm area on map represents = 1.28 × 104 × 104 sq. cm on land
1 cm sq. on map represents         = 1.28 × 104 × 104 /8 sq. cm on land

1 cm on map represent =[katex] \sqrt{\frac{1.28\ \times{10}^4\times{10}^4}{8}} [/katex] cm
                                      = 4,000 cm

1 cm on drawing represent 4,000 cm, means R.F = 1/4000

Assuming the length of the scale 15 cm, it will represent 600 m.

1. Draw a line 15 cm long. it will represent 600 m. Divide it into six equal parts. ( each will represent 100 m. )


2. Divide first division in ten equal parts. Each will represent 10 m.


3. Draw a line upward from the left end and mark 10 parts on it of any distance.


4. Name those parts 0 to 10 as shown. join 9^th sub-divisions of horizontal scale with 10^th division of the vertical divisions.


5. Then draw parallel lines to this line from remaining subdivisions and complete diagonal scale.

3 ] Draw a diagonal scale of R.F 1: 2.5, showing centimeters and millimeters and long enough to measure up to 20 centimeters.

Solution Steps:

R.F =1/2.5
length of scale = 1/2.5  ×  20 cm
                        = 8 cm

1. Draw a line 8 cm long and divide it into 4 equal parts. ( Each part will represent a length of 5 cm.)


2. Divide the first part into 5 equal divisions. ( Each will show 1 cm.)


3. At the left-hand end of the line, draw a vertical line and on step off 10 equal divisions of any length.


4. Complete the scale as explained in previous problems.


5. Show the distance 13.4 cm on it.

Please write comments if you find anything incorrect and if you have any queries about this topic or you want to share more information about the topic discussed above.