Step by step
The question
Question: two similar shapes have matching lengths 2 and 6, and the small area is 5: find the large area.
lengths 2 and 6, small area = 5
1 Length scale factor
The scale factor k: every length of the large one is 3 times the small one.
k = 62 = 3
2 Area factor
Area grows in two directions: k times k.
That is 9.
area factor = k² = 3² = 9
A square 1 by 1, made 3 times longer both ways, holds 9 little squares.
Careful!
The area does not grow like the lengths.
It grows by k squared.
Careful: not 5 × 3 ✗
3 Use it
5 times 9 is 45.
large area = 5 × 9 = 45
4 Check
Let's check: the large area divided by the small one is 9, which is k squared.
So it is right!
45 ÷ 5 = 9 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
large area = 5 × 9 = 45