Areas and Volumes in Map Scale Problems

When the scale in the map is 1 : n, the area scale is { 1 }^{ 2 }:{ n }^{ 2 } and the volume scale is { 1 }^{ 3 }{ n }^{ 3 }.
Also 1m=100cm, squaring both sides gives
{ 1m }^{ 2 }={ 100 }^{ 2 }{ cm }^{ 2 }
{ 1m }^{ 2 }={ 100 }00{ cm }^{ 2 }
Cubing both sides gives
1m = 100cm
{ 1m }^{ 3 }={ 100 }^{ 3 }{ cm }^{ 3 }
{ 1m }^{ 3 }={ 100 }0000{ cm }^{ 3 }

More examples



A map is drawn to a scale of 1500.
Calculate the actual length, in metres, of a canal which is 5cm long on the map
The actual area of a region is 7.5{ km }^{ 2 }. What is its area in { cm }^{ 2 } on the map?


Solution:
map/actual =\frac { 1 }{ 500 }
Map given is 5cm long, substituting in the above gives
5cm/actual =\frac { 1 }{ 500 }
Actual =5cm\times 500
=2500cm
=25cm since 1m=100cm.
Map/Actual =\frac { 1 }{ 500 }Squaring gives \frac { Map }{ Actual } ={ (\frac { 1 }{ 500 } ) }^{ 2 }=\frac { 1 }{ 250000 }
Given actual area =\frac { 7.5{ km }^{ 2 } }{ 250000 }
Now 1km = 100,000cm, so { 1km }^{ 2 }={ 100,000 }^{ 2 }{ cm }^{ 2 }=1,000,000,000{ cm }^{ 2 }
Map =\frac { 7.5 }{ 250,000 } \times 10,000,000,000=\frac { 7.5 }{ 2.5 } \times { 10 }^{ 5 }\times { 10 }^{ 10 }{ cm }^{ 2 }
Map =\frac { 7.5 }{ 2.5 } \times { 10 }^{ 5 }
=3\times { 10 }^{ 5 }{ cm }^{ 2 }
=300,000{ cm }^{ 2 }
If we wish to give this answer in { m }^{ 2 } instead, we know
1m = 100cm
So, { 1m }^{ 2 }={ 100 }^{ 2 }{ cm }^{ 2 }
And, therefore area on the map =300,000{ cm }^{ 2 }
=\frac { 300,000 }{ 10,000{ m }^{ 2 } }
=30{ cm }^{ 2 }



A map is drawn to a scale of 12500
The actual volume of a dam is 3,000,000{ m }^{ 3 }. What is its volume in { cm }^{ 3 } on the map.
Solution:
\frac { Map }{ Actual } =1/2500
Since we are dealing with volume, we cube both sides
\frac { Map }{ Actual } =\frac { 1 }{ { 2500 }^{ 3 } }
\frac { Map }{ Actual } = 1/625000000
Given the actual volume of the dam =3,000,000{ m }^{ 3 }, substituting
\frac { Map }{ 3,000,000{ m }^{ 3 } } =\frac { 1 }{ 625,000,000 }
Map =\frac { 3,000,000{ m }^{ 3 } }{ 625,000,000 }
={ 0.0048m }^{ 3 }

Now 1m = 100cm
1{ m }^{ 3 }={ 100 }^{ 3 }{ cm }^{ 3 }
1{ m }^{ 3 }={ 1,000,000 }{ cm }^{ 3 }
Map =0.0048\times { 1,000,000 }{ cm }^{ 3 }
=4800{ cm }^{ 3 }


An actual area of 16{ km }^{ 2 } is represented by an area of 1{ cm }^{ 2 } on a map. Find the scale to which this map is drawn.

Solution:
\frac { Map }{ Actual } =\frac { 1 }{ 160,000,000,000 } since 16{ km }^{ 2 }=16\times { 100,000 }^{ 2 }{ cm }^{ 2 }
Because this is an area scale, (squared) we take the square root of 160,000,000,000=16\times { 10 }^{ 10 }=\sqrt { 16\times { 10 }^{ 10 } }
4\times { 10 }^{ 5 }
=400,000
The scale used is 1400000


A map is drawn to a scale of 1100000.
Find the area in { km }^{ 2 } of a village which is represented on the map by an area of 600{ cm }^{ 2 }.


Solution:
\frac { Map }{ Actual } =\frac { 1 }{ 100,000 }
Squaring, \frac { Map }{ Actual } ={ (\frac { 1 }{ 100,000 } ) }^{ 2 }={ (\frac { 1 }{ 10 } })^{ 10 }
We are given an area on the map =600{ cm }^{ 2 }. We substitute in the above:
\frac { 600{ cm }^{ 2 } }{ Actual } =\frac { 1 }{ { 10 }^{ 10 } }
Actual =600\times { 10 }^{ 10 }{ cm }^{ 2 }
=6\times { 10 }^{ 12 }{ cm }^{ 2 }
Now 1{ km }=100,000cm={ 10 }^{ 5 }cm
So, 1{ km }^{ 2 }=({ { 10 }^{ 5 }) }^{ 2 }={ 10 }^{ 10 }{ cm }^{ 2 }.
Therefore, the actual area of village, in { km }^{ 2 }, is
\frac { 6\times { 10 }^{ 12 }{ km }^{ 2 } }{ { 10 }^{ 10 } }.



On a plan, a piece of land is represented by an area of dimension 4cm\times 5cm. Given that a scale of 2cm to 5cm is used, find the actual area of the piece of land, giving an answer in { m }^{ 2 }.


Solution:
Area on the map =4cm\times 5cm
={ 20cm }^{ 2 }
Scale: \frac { Map }{ Actual } =\frac { 2cm }{ 5m } =\frac { 2cm }{ 500cm } =\frac { 1 }{ 250 }
Squaring gives \frac { Map }{ Actual } =({ =\frac { 1 }{ 250 } ) }^{ 2 }=\frac { 1 }{ 62,500 }
Substituting Map =20{ cm }^{ 2 } gives
\frac { 20{ cm }^{ 2 } }{ Actual } =\frac { 1 }{ 62,500 }
Actual =20\times { 62,500cm }^{ 2 }
Now, 1m=100cm
So, 1{ m }^{ 2 }={ 100 }^{ 2 }{ cm }^{ 2 }=10,000{ cm }^{ 2 }.
Therefore, the actual area of land =\frac { 20\times 62,500 }{ 10,000{ m }^{ 2 } }
=125{ m }^{ 2 }