Step by step
The question
Question: two similar shapes have matching lengths 4 and 8, and the large area is 60: find the small area.
lengths 4 and 8, large area = 60
1 Length scale factor
The scale factor k: every length of the large one is 2 times the small one.
k = 84 = 2
2 Area factor
Area grows in two directions: k times k.
That is 4.
area factor = k² = 2² = 4
A square 1 by 1, made 2 times longer both ways, holds 4 little squares.
Careful!
The area does not grow like the lengths.
It grows by k squared.
Careful: not 60 × 2 ✗
3 Use it
Going to the small one, we divide: 60 divided by 4 is 15.
small area = 60 ÷ 4 = 15
4 Check
Let's check: the large area divided by the small one is 4, which is k squared.
So it is right!
60 ÷ 15 = 4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
small area = 60 ÷ 4 = 15