Geometry: An Interactive Journey to Mastery

Introduction to Scale

Lesson 24

Topics
x The effect of scale on lengths, areas, and volumes.
x Changing units.
x The role of scale in the natural world.
)RUPXOD
In scaling by a factor k,
x all lengths change by a factor k.
x all areas change by a factor k^2.
x all volumes change by a factor k^3.
x all angles remain unchanged.
Summary
The mathematical effects of scaling are manifest in profound ways in the natural world. In this lesson, we
examine scaling in two and three dimensions, explore scaling through the viewpoint of a change of units,
and look at effects on natural phenomena.
([DPSOH
The triangle on the left side of )LJXUH has area 18 square units.
What is the area of the triangle on the right?
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7KHWZRWULDQJOHVDUHVLPLODUE\$$ͼ௘XVHYHUWLFDODQJOHVDQGFRQJUXHQWDOWHUQDWHLQWHULRUDQJOHVIRUSDUDOOHO
OLQHV௘ͽZLWKVFDOHIDFWRUk 4263 ͼ௘IURPWKHODUJHUWRWKHVPDOOHUWULDQJOH௘ͽ
Because area scales by k^2 , the area of the right triangle is 20 k^2 u 20 48998 square units.

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Figure 24.1