Step by step
The question
Question: two similar shapes have matching lengths 5 and 10, and the small area is 12: find the large area.
lengths 5 and 10, small area = 12
1 Length scale factor
The scale factor k: every length of the large one is 2 times the small one.
k = 105 = 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 12 × 2 ✗
3 Use it
12 times 4 is 48.
large area = 12 × 4 = 48
4 Check
Let's check: the large area divided by the small one is 4, which is k squared.
So it is right!
48 ÷ 12 = 4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer
large area = 12 × 4 = 48