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Area scale and volume scale
The large cube with sides of 10 is reduced so that the sides are 1. What is the scale of the volume?
Leon is a coach for a little league soccer team. He needs a map of a soccer field that he can use when he gives instructions. The field is 70 meters times 40 meters in reality, and Leon draws it up to the scale of 1 to 200. Every meter in the field corresponds to 200dths of a meter on the map or half a centimeter. Now that's far too small.
Leon needs a map that is twice as big. So he starts again with half the scale, one to a hundred instead. Now one meter in the field is one centimeter on the map, half the scale, double the size. But hold on a moment. Shouldn't the new map be twice as big?
Now it's four times as big. How did this happen? Look closely. The larger picture is twice as high and twice as wide as the smaller picture, but it's area increased four fold. This is the difference between the linear scale and the area scale.
You remember of course that you measure the distance in meters, and the area in meters squared. And that's how it works with scales too. The area scale is equal to the linear scale squared. The smaller map is drawn on a linear scale of one to 200. If you put 200 of these pictures end to end they would stretch all the way along the soccer field.
This is the linear scale. If you take 200 squared you get 40 thousand. So the area scale is one to 40 thousand. It takes 40 thousand of the smaller pictures to cover the whole area of the soccer field. The larger picture has a linear scale of 1 to 100.
And 100 squared is 10 thousand. So the area scale is one to ten thousand. The larger picture is drawn on a half of the linear scale but a quarter of the area scale. We can take it a step further and do this in three dimensions. Here is a die.
It's edges are one centimeter long and it's volume is one cubic centimeter. Now we make a model of the dice on a linear scale of two to one, or we make the dice twice as long. The sides become two centimeters long, but what happens to the volume? It becomes one, two, three, four, five, six, seven, eight cubic centimeters. The volume scale is the linear scale to the power of three, or cubed.
When people say scale they almost always mean the linear scale. But you have to watch out, because if you increase or decrease the dimensions on a linear scale, the area increases or decreases as well, but on a linear scale squared. And the volume increases or decreases on the cube of the linear scale. The area scale is equal to the linear scale squared. The volume scale is equal to the linear scale cubed.