Worked examples
The question
Question: two similar shapes have matching lengths 3 and 6, and the small area is 10: find the large area.
lengths 3 and 6, small area = 10
1 Length scale factor
The scale factor k: every length of the large one is 2 times the small one.
k = 63 = 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 10 × 2 ✗
3 Use it
10 times 4 is 40.
large area = 10 × 4 = 40
4 Check
Let's check: the large area divided by the small one is 4, which is k squared.
So it is right!
40 ÷ 10 = 4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer large area = 10 × 4 = 40
▶ Watch it solved
The question
Question: two similar shapes have matching lengths 2 and 5, and the small area is 8: find the large area.
lengths 2 and 5, small area = 8
1 Length scale factor
The scale factor k: every length of the large one is 5/2 times the small one.
k = 52
2 Area factor
Area grows in two directions: k times k.
That is 25/4.
area factor = k² = (52)² = 254
Careful!
The area does not grow like the lengths.
It grows by k squared.
Careful: not 8 × 5/2 ✗
3 Use it
8 times 25/4 is 50.
large area = 8 × 254 = 50
4 Check
Let's check: the large area divided by the small one is 25/4, which is k squared.
So it is right!
50 ÷ 8 = 25/4 = k²
Let's look back.
First: length scale factor.
Then: area factor.
Then: use it.
Last: check.
Answer large area = 8 × 254 = 50
▶ Watch it solved
The question
Question: two similar solids have volumes 27 and 64, and the small height is 6: find the large height.
volumes 27 and 64, small height = 6
1 Volume factor
The large volume divided by the small one: 64/27.
That is k cubed.
volume factor = 6427
2 Length scale factor
To go back from k cubed to k, we take the cube root: 4/3.
k = ∛(6427) = 43
4/3 × 4/3 × 4/3 = 64/27
Careful!
64/27 is the volume factor.
Lengths grow less: by its cube root.
Careful: not k = 64/27 ✗
3 Use it
6 times 4/3 is 8.
large height = 6 × 43 = 8
4 Check
Let's check: the large height divided by the small one is 4/3, the scale factor.
So it is right!
8 ÷ 6 = 4/3 = k
Let's look back.
First: volume factor.
Then: length scale factor.
Then: use it.
Last: check.
Answer large height = 6 × 43 = 8
▶ Watch it solved
