1. Area Topic 13 Lesson 5 Effects of Changing Dimensions Proportionally Holt Geometry Texas ©2007

2. Objectives and Student Expectations
TEKS Focus:
(10)(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional changes.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(11)(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure.

3. Find the perimeter and area of the rectangle.
Example: 1
Find the perimeter and area of the rectangle.
Length
Width
Perimeter
P = 2(4) +2(10)
P = 28 cm
Area
A = 10(4)
A = 40 cm2

4. Find the perimeter and area of the rectangle if the length is tripled.
Example: 2
Find the perimeter and area of the rectangle if the length is tripled.
Perimeter
P = 2(4) +2(30)
P = 68 cm
Area
A = 30(4)
A = 120 cm2

5. Example: 3 Perimeter P = 2(16) +2(40) P = 112 cm Area A = 40(16)
Find the perimeter and area of the rectangle if the length and width is multiplied by 4.
Perimeter
P = 2(16) +2(40)
P = 112 cm
Area
A = 40(16)
A = 640 cm2

6. Data from Examples 1-3.
Rectangle
Original Dimensions
Only Length Tripled
Length & Width Quadrupled
Perimeter
28 cm
68 cm
112 cm
Area
40 cm2
120 cm2
640 cm2
How does the area change if only the length is tripled?
Area is multiplied by 3
If both length and width are quadrupled, how does the perimeter change?
How does the area change?
Perimeter is multiplied by 4
Area is multiplied by 16 or 42

7. Find the perimeter and area of the triangle.
Example: 4
Find the perimeter and area of the triangle.
Perimeter
P =
P = 42 in
Area
A =
A = 84 in2

8. Example: 5
Find the perimeter and area of the triangle if all dimensions are multiplied by 5.
Perimeter
P =
P = 210 in
Area
A =
A = 2100 in2

9. Data from Examples 4-5.
Triangle
Original Dimensions
All Dimensions Multiplied by 5
Perimeter
42 in
210 in
Area
84 in2
2100 in2
If all dimensions are multiplied by 5, is the height also multiplied by 5?
Yes
If all dimensions are multiplied by 5, how does the perimeter change?
How does the area change?
Perimeter is multiplied by 5
Area is multiplied by 25 or 52

10. Find the circumference and area of the circle.
Example: 6
Find the circumference and area of the circle.
Circumference
C = 2(8)
C = 16 yd
Area
A = (8) 2
A = 64  yd2

11. Example: 7
Find the circumference and area of the circle, if the radius is multiplied by 6. 6
Circumference
C = 2(48)
C = 96 yd
Area
A = (48) 2
A = 2304 yd2

12. Data from Examples 6-7.
Circle
Original Dimensions
Radius multiplied by 6
Perimeter
16 yd
96 yd
Area
64 yd2
2304 yd2
If the radius is multiplied by 6, how does the circumference change?
Circumference multiplied by 6
If the radius is multiplied by 6, how does the area change?
Area is multiplied by 36 or 62

13. Conjectures:
If only one dimension of a two dimensional figure is changed, then the two figures (original and new figure) are NOT _________.
If all dimensions of a two dimensional figure are changed, then the two figures (original and new figure) are __________.
If all dimensions of a two dimensional figure are changed by a factor of a, then the perimeter changes by a factor of ____ and the area changes by a factor of ____.
similar
similar a a2

14. Describe the effect of each change on the area of the given figure.
Example: 8 & 9
Describe the effect of each change on the area of the given figure.
8. If the height of a triangle is doubled, then the area is ________.
9. If only the height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by 1/3, then the area is multiplied by ________________.
doubled
1/3

15. Example: 10 & 11
Describe the effect of each change on the perimeter or circumference and the area of the given figure.
10. If the base and height of a triangle with base 12 in and height 6 in are both tripled, then the perimeter is ________ and the area is multiplied by ________.
11. If the base and height of a rectangle with length 18 ft and width 6 ft are both multiplied by ½, then the perimeter is multiplied by _______ and the area is multiplied by _________.
tripled
9 or 32
1/2
¼ or (1/2) 2

16. Example: 12
Describe the effect on the area for changing the following.
The diagonal SU of the kite with vertices R(2, 2), S(4, 0), T(2, –2), and U(–5,0) is multiplied by .
original dimensions:

17. Example: 13
The base and height of the triangle with vertices P(2, 5), Q(2, 1), and R(7, 1) are tripled. Describe the effect on its area and perimeter.
original dimensions:
The perimeter is tripled, and the area is multiplied by 9.
dimensions tripled:

18. Example: 14
A circle has a circumference of 32 in. If the area is multiplied by 4, what happens to the radius?
and the area is A = r2 = 256 in2. If the area is multiplied by 4, the new area is 1024 in2.
The original radius is
r2 = 1024
Set the new area equal to r2.
r2 = 1024
Divide both sides by .
Take the square root of both sides and simplify.
r = √1024 = 32
Notice that 32 = 2(16). The radius is multiplied by 2,
which is the square root of 4.