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Perimeter, area, and volume: changes in scale

# Perimeter, area, and volume: changes in scale

If you increase the length of side by scale factor N
• the perimeter increases by scale factor N.
• the area increases by scale factor N2
• the volume increases by scale factor N3

Look at this square:

If the side lengths are doubled, then which of the following statements about its perimeter will be true?

a) The new perimeter will be 2 times the old perimeter.
b) The new perimeter will be 4 times the old perimeter.
c) The new perimeter will be 3 times the old perimeter.
d) The new perimeter will be 12 of the old perimeter.

You can solve this problem without using the measurements given in the diagram.

The original square had this perimeter:

P = 4s

The new square will have sides that are 2 times as long. Since the original side lengths were s, the new side lengths will be 2s. Calculate the perimeter:

P = 4(2s) = 8s

Divide the new perimeter by the original perimeter and simplify.

$\frac{new\;perimeter}{original\;perimeter}=\frac{8s}{4s}=2$

The new perimeter will be 2 times the old perimeter.

Look at this rectangular prism

 5 yd 4 yd 5 yd

If all three dimensions are doubled, then which of the following statements about its surface area will be true?

A. The new surface area will be $\frac{1}{4}$ of the old surface area.
B. The new surface area will be 4 times the old surface area.
C. The new surface area will be 2 times the old surface area.
D. The new surface area will be 9 times the old surface area.