SIMILARITY OF LENGTH AND AREA
#28
Once you know two figures are similar with a SCALE FACTOR or
a
RATIO OF SIMILARITY b , the following proportions for the SMALL (sm)
and LARGE (lg) figures (which are enlargements or reductions of each other) are
true:
a
sidesm
=
side lg
b
a
Psm
=
P lg
b
a2
Asm
= 2
A lg
b
where P = perimeter, A = area of a face or the total surface area.
a
b
In each proportion above, the data from the smaller figure is written on top
(in the numerator) to help be consistent with correspondences. When working
with area, the scale factor (ratio of similarity) is squared. NEVER square the
actual areas themselves, only the scale factor.
Example
The two rectangular prisms above are similar. Suppose the ratio of their vertical edges is 37 .
a)
Find the ratio of their surface areas.
b)
Find the perimeter of the front face of the small prism if the perimeter of the front face of the
large prism is 20 units.
c)
Find the area of the front face of the small prism if the area of the front face of the large prism is
24 square units.
Solutions
a)
The ratio of the surface areas is
parts.
b)
3
7
=
P
20
7P = 60
P =
60
7
≈ 8.57 units
c)
A sm 32
9
=
=
A l g 7 2 49
2
()=
A
24
9
49
A
24
3
7
=
. Use proportions to solve for the rest of the
49A = 216
A = 216
49 ≈ 4.41 units
Note that the ratio of similarity
must be squared when using it in
area calculations.
Problems
1. Two rectangular prisms are similar. The smaller, P, has a height of
four units while the larger, Q, has a height of five units.
a) What is the scale factor from prism P to prism Q?
b) What is the ratio, small to large, of their surface areas?
2.
P
4
x
Q
5
y
figures are for problems 1 and 2
What would be the ratio of the lengths of the edges labeled x and y in the prisms above?
For problems 3–16, you may want to draw two rectangles, label them A and B, and write in the specific data
from each problem.
3.
If rectangle A and rectangle B have a ratio of similarity of 53 , what is the area of rectangle B if
the area of rectangle A is 30 square units?
4.
If rectangle A and rectangle B have a ratio of similarity of 53 , what is the area of rectangle A if
the area of rectangle B is 30 square units?
5.
If rectangle A and rectangle B have a ratio of similarity of 25 , what is the area of rectangle B if
the area of rectangle A is 46 square units?
6.
If rectangle A and rectangle B have a ratio of similarity of 25 , what is the area of rectangle A if
the area of rectangle B is 46 square units?
7.
If rectangle A and rectangle B have a ratio of similarity of 37 , what is the area of rectangle B if
the area of rectangle A is 82 square units?
8.
If rectangle A and rectangle B have a ratio of similarity of 37 , what is the area of rectangle A if
the area of rectangle B is 82 square units?
9.
If rectangle A and rectangle B have a ratio of similarity of 19 , what is the area of rectangle B if
the area of rectangle A is 24 square units?
10.
If rectangle A and rectangle B have a ratio of similarity of 19 , what is the area of rectangle A if
the area of rectangle B is 24 square units?
11.
Rectangle A is similar to rectangle B. The area of rectangle A is 121 square units while the area
of rectangle B is 49 square units. What is the ratio of similarity between the two rectangles?
12.
Rectangle A is similar to rectangle B. The area of rectangle A is 36 square units while the area
of rectangle B is 64 square units. What is the ratio of similarity between the two rectangles?
13.
Rectangle A is similar to rectangle B. The area of rectangle B is 81 square units while the area
of rectangle A is 25 square units. What is the ratio of similarity between the two rectangles?
14.
Rectangle A is similar to rectangle B. The area of rectangle B is 289 square units while the area
of rectangle A is 121 square units. What is the ratio of similarity between the two rectangles?
15.
Rectangle A is similar to rectangle B. The area of rectangle A is 324 square units while the
area of rectangle B is 121 square units. If the perimeter of rectangle A is 12 units, what is the
perimeter of rectangle B?
16.
Rectangle A is similar to rectangle B. The area of rectangle A is 324 square units while the
area of rectangle B is 121 square units. If the perimeter of rectangle B is 12 units, what is the
perimeter of rectangle A?
17.
The corresponding diagonals of two similar trapezoids are in the ratio of 11
23 . What is the ratio
of their areas?
18.
The corresponding diagonals of two similar trapezoids are in the ratio of 17
19 . What is the ratio
of their areas?
19.
5 . What is the ratio of their areas?
The ratio of the perimeters of two similar parallelograms is 13
20.
11 . What is the ratio of their areas?
The ratio of the perimeters of two similar parallelograms is 15
21.
4 . What is the ratio of their heights?
The ratio of the areas of two similar trapezoids is 81
22.
The ratio of the areas of two similar trapezoids is 59 . What is the ratio of their heights?
23.
The areas of two circles are in the ratio of 36
1 . What is the ratio of their radii?
24.
The areas of two circles are in the ratio of 36
49 . What is the ratio of their radii?
Answers
1.
a) 45
5.
9.
b) 16
25
2.
x
y
= 45
287.5 units2
6.
7.36 units2
1944 units2
10.
≈ 0.30 units2
11.
11
7
12.
6
8
13.
5
9
14.
11
17
15.
7.3 units
16.
19.6 3 units
17.
121
529
18.
289
361
19.
25
169
20.
121
225
21.
2
9
22.
5
3
23.
6
1
24.
6
7
3.
10.8 units2
4.
83. 3 units2
7.
446.4 units2
8.
≈ 15.06 units2
or 34