How Big? A look at figuring size and scale

eric

A few days ago, I needed to increase the size of an astronomical table to make it easier for my aging eyes to read. It was about half the size of a standard sheet of paper, so I decided to double its size. What scale setting should I use on the copier, 200%? Having been burned by this type of quick thinking in the past, I decided to apply some mathematical reasoning. At 200%, each side of the table doubles in length. As can be seen below, this would make the table 4 times as big!

sizescale

The size or area of a square is the product of its length and width.  If each side is doubled, or increased by a scale factor of 2, the area increases by 2 x 2 = 4. If each side is tripled (a scale factor of 3), the resulting square is 9 times larger. In general, the area increases by the square of the scale factor. While this is easiest to visualize using a square, a picture of any object on the square would scale up or down the same way, so this works with any shape.

Since area increases as the square of the scale factor, if I wish to double it, I need to use a scale factor whose square is 2. This is √2 = 1.414. So, I set the copier to 141% and doubled the size of the chart on my first try, with no wasted paper.

How about a cube? If each side of a cube is doubled, the result is a cube 8 times as big. In general, the volume of any object changes as the cube of the scale factor.

cubescale

So, how big is the Sun compared to Earth? It depends on what is meant. The Sun has a diameter 109 times larger than Earth.

So if we mean “How many Earths put side to side would be as wide as the Sun?” the answer is 109. However, if we mean “How many Earths would fit inside the Sun?” we are dealing with volume, so the answer is 109 cubed or about 1,300,000!

This relationship can be explored with all the planets by comparing the “Equatorial Diameter” and the “Volume” in the Clark Planetarium Solar System Fact Sheet. “Volume” is really telling us how many Earths would fit inside the larger planets or what fraction of Earth would fit inside the smaller planets. (Note: a planet’s volume is the cube of its average diameter, not the equatorial diameter given in the fact sheet).

Scale away!

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