Transcript Geometry

Geometry
Similar Solids
Definition:
Similar Solids
• Two solids of the same type with eaqual ratios of corresponding
linear measures, such as heights or radii, are called similar solids.
• The common ratio is called the scale factor of one solid to the
other solid
• Any two cubes are similar
• Any two spheres are similar
Similar Solids Theorem
• If two similar solids have a scale factor of
𝑎: 𝑏, then corresponding areas have a ratio of
𝑎2 : 𝑏 2 , and corresponding volumes have a ratio
3 3
of 𝑎 : 𝑏 .
Example 1: Are they similar?
Check the ratios:
25 50 50
=
=
45 90 80
25 50 50
=
≠
45 90 80
Therefore, these two figures are not
similar
Example 2: Are they similar?
Check the ratios:
7
3
=
21 9
Yes they are in fact equal,
therefore these two figures
are similar and have a scale
factor of 1:3.
Example 3: Given the information, find the
scale factor
Set up the ratio of their volumes:
8000
1728
We know that the volumes have a relationship of
𝑎3 : 𝑏 3 . So we cube root the ratio.
3
8000 20 5
=
=
3
1728 12 3
Therefore, the scale factor is 5:3
Example 4: The figures are similar, find the scale
factor, the ratio of their surface areas, and the ratio of
their volumes.
Their scale can be found by finding the ratio of
their linear measures:
144
8
= → 8: 3 is the scale factor
54
3
Therefore the ratio of the surface areas then is:
82 64
=
2
3
9
And, the ratio of their volumes is:
83 512
=
3
3
27
Example 5: Find the missing information about
the similar figures described.
128
32
If the surface areas have a ratio of:
Then the solids have a scale factor of:
2
2
128
32
=
2
8 2
2
4 2
2
1
= → 2: 1
Thus the volumes are going to be in a
ratio of 8:1 so we can find the missing
volume by dividing the volume we know
by 8 → 12288 ÷ 8 = 1536𝑚3
Homework
Assignment 10-5