1. Similar Shapes

2. Similar Shapes
We sometimes come across figures that have similar shapes.

3. Similar Plane Figures
For two similar plane figures, there is a relationship between the ratio of their areas and the ratio of their corresponding sides.
Let’s consider a square with side a.
After enlarging the square by a scale factor k, then the length of its side = ka ka a
scale factor = k

4. ka a
scale factor = k
Then
and ∴

5. In fact, we have:
For any two similar plane figures, the ratio of their areas is equal to the square of the ratio of any pair of their corresponding sides (or line segments).
In other words,
where l1 and l2 represent the corresponding sides (or line segments) of two similar plane figures; A1 and A2 represent their corresponding areas of the figures.

6. Try to find the value of x.
Refer to the following pair of similar figures.
area = 12 cm2
area = x cm2
8 cm
16 cm
Try to find the value of x.

7. Follow-up question Find the value of x in the pair of similar figures.
5 cm
x cm
area = 9 cm2
area = 16 cm2
Solution

8. Similar Solids
For two similar solids, there is a relationship among the ratio of their volumes, the ratio of the areas of their corresponding surfaces and the ratio of their corresponding sides.
Let’s consider a cube with side a.
After enlarging the cube by a scale factor k, then the length of its side = ka ka a
scale factor = k

9. ka a
scale factor = k
Then
and ∴

10. ka a
scale factor = k
Similarly, ∴

11. In fact, we have:
For any two similar solids,
the ratio of the areas of their corresponding surfaces is equal to the square of the ratio of any pair of corresponding sides (or line segments),
the ratio of their volumes is equal to the cube of the ratio of any pair of corresponding sides (or line segments).
In other words,
where l1 and l2 represent the corresponding sides (or line segments) of two similar solids; A1 and A2 represent the areas of the corresponding surfaces; V1 and V2 represent the corresponding volumes of the solids.
and

12. Refer to the following pair of similar solids.
12 cm
volume = 24 cm3
volume = 81 cm3
x cm
Can you find the value of x?

13. Follow-up question The figure shows a pair of similar solids.
6 cm
8 cm
volume = x cm3
volume = 96 cm3
solid A
solid B
The figure shows a pair of similar solids.
(a) Find the value of x.
Solution
(a)

14. Follow-up question (cont’d)
6 cm
8 cm
volume = x cm3
volume = 96 cm3
solid A
solid B
The figure shows a pair of similar solids.
(b) Find the ratio of the total surface areas of A to B.
Solution
(b)
∴ The ratio of the total surface areas of A to B is