Comparing Perimeter, Area, and Volume of Similar Figures
Name:____________________
1. Using the grid below, create four similar squares with side lengths of 1, 2, 3, and 4 units long. Use
these squares to fill in the table below. Follow the example in the first row.
Figure
Length of one side
Perimeter of Square
Area of Square
1st
1 unit
4 units
12 = 1 unit2
2nd
2 units
3rd
3 units
4th
4 units
Ratio of Perimeters
Ratio of Areas (write as
an exponent if
possible)
nth
2. Write the ratios of each square below in simplest form.
Figure
Ratio of Side Lengths
(Scale Factor)
1st to 2nd
1st to 3rd
1st to 4th
2nd to 3rd
2nd to 4th
3rd to 4th
3. Do you notice any patterns? Is there a relationship between the scale factor and the ratio of
perimeters? What about the scale factor and the ratio of areas?
4. Using the grid below, create three similar rectangles with dimensions 1 x 2, 2 x 4 and 3 x 6. Use these
rectangles to fill in the table below.
Figure
Dimensions of Rectangle
1st
1 unit by 2 units
2nd
2 units by 4 units
3rd
3 units by 6 units
nth
n units by m units
Perimeter of Rectangle
Area of Rectangle
5. Write the ratios of the rectangles below in simplest form.
Figure
Ratio of Side Lengths
(Scale Factor)
Ratio of Perimeters
Ratio of Areas (write as
an exponent if
possible)
1st to 2nd
1st to 3rd
2nd to 3rd
nth to mth
Use the patterns you observed to answer the following questions:
Extending the tables in problem 4 and 5, the 4th rectangle would be 4 by 8 units, the 5th would be 5 by 10
units, the 6th would be 6 by 12 units, etc.
6. Following this pattern, what is the ratio of the perimeters of the 8th rectangle and the 12th rectangle?
7. What is the ratio of the areas of the 6th rectangle and 16th rectangle?
8. Based upon the results in your tables, if the ratio of the sides in two similar figures is 3/5, the ratio of
their perimeters would be ____________________ and the ratio of their areas would be ___________________.
9. Use the following cube dimensions to fill out the tables.
Dimensions
Figure of Cube
1st
2nd
3rd
nth
Volume of
Cube
1 unit by 1
units by 1
units
2 units by 2
units by 2
units
3 units by 3
units 3 units
n units by n
units by n
units
Figure
Ratio of Side
Lengths (Scale
Factor)
Ratio of Volumes
(write as an
exponent if possible)
1st to 2nd
1st to 3rd
2nd to 3rd
nth to mth
13. Do you notice any patterns? Is there a relationship between the scale factor and the ratio of volumes?
14. If the scale factor of two cubes is 4/7, find the ratio of their volumes.
Summary:
If two similar SHAPES have a scale factor of a:b, then the perimeters have a ratio of __________________.
, then the areas have a ratio of __________________.
If two similar SOLIDS have a scale factor of a:b, then the volumes have a ratio of __________________.
How do we know if shapes are similar?
How might we check if solids (cones, cylinders, etc.) are similar?
Practice!
1. There are 750 toothpicks in a regular-sized box. If a jumbo box is made by doubling all the dimensions of
the regular-sized box, how many toothpicks will the jumbo box hold?
2. Two cubes have volumes of 64 cubic feet and 200 cubic feet. What is the ratio of the surface area of the
smaller cube to the surface area of the larger cube?