SIMILAR SOLIDS

Taking it to the 3rd Dimension

When we study similar figures in 2 dimensions, we learn that the ratio of their areas equals the square of the ratio of their sides. When we add the third dimension and study similar solids, we find that the ratio of their volumes is the cube of the ratio of their "sides" (or dimensions)

In similar solids:

The level one ratio -- that of the sides or dimensions -- determines the other two.
Note:
We usually say "area" instead of surface area -- but we mean surface area.

Example

We know the volume of both cans, so we know the level 3 ratio of 125 : 64.
It is the cube of the level 1 ratio (that of the radii), so we take the cube root.

This means the level 1 ratio of BIG RADIUS to small radius is 5 to 4.
We know r = 2 cm. so

and now we find the base area of the big can

We could find the area of the small base then multiply it by 25/16 -- the square of 5/4.

If we know level 3 and need level 2 -- we take the cube root then square the result.

Caution: Watch for statements like "the area of the larger is 4 times the area of the smaller". It tells you the level 2 ratio of 4 : 1.

Note: we can tell the ratio level by the exponent or power of the units.
Level 1: length, radius, etc. units to the 1st power -- like centimeters, feet or yards.
Level 2: area -- the units are squared, raised to the 2nd power
Level 3: volume -- units are cubed -- raised to the 3rd power.

In this example, we knew the level 3 ratio and the radius of the small can, which is a level 1 value. Once we take the cube root of the level 3 ratio, we use it to get to the level 2 ratio or area.

Example: The Listenup Company makes audio speakers in 2 sizes. The speakers are similar rectangular prisms. The small one is 40 cm high, its base area is 250 cm². The area of the larger speaker's base is 490 cm². Find the height and volume of the larger speaker?

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s),then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice

Make a diagram for questions 1, 2 and 3.

1) A company makes 2 similar cylindrical vases.
The smaller one has a diameter of 10 cm. and a height of 20 cm.
The total area of the larger vase is 4 times that of the smaller.
Find the dimensions and volume of the larger vase.

2) Two right prisms with rectangular bases are similar.
The base area of the smaller is 3.14 cm² and its height is 1.5 cm.
The base area of the larger is 78.5 cm².
What is the volume of the larger prism rounded to the nearest unit?

3) Two cylindrical food cans are similar.
The diameter of the smaller can equals the radius of the larger can.
How many of the smaller cans could we fill with the full contents of the larger one?

4)

.

Solutions

1) h = 40 cm, r = 10 cm and volume = 12566 cm³.

2)

3) We could fill 8 small cans with the contents of the larger one.

4) The level 1 ratio is 5 : 20 or ¼, so the level 3 ratio = (¼)³.
The volume of each small piece will be 1/64 th the volume of the mother piece.
4800 ÷ 64 = 75 cm³.

(all content © MathRoom Learning Service; 2004 - ).