12.6 Dimensional Change

Predict and calculate how changing one or more dimensions of a shape affects the perimeter, area, or volume of the shape.

1. Dimensional Change
The student is able to (I can):
•Predict and calculate how changing one or more
dimensions of a shape affects the area orvolumeof that
shape.

2. What happens to the perimeter and area of a figure when we
change dimensions proportionally?
Example: What is the new perimeter and area of the
rectangle if the height is doubled?
3
7
6
7
P = 2(3)+2(7)
= 20
P = 2(6)+2(7)
= 26
A = (3)(7) = 21A = (6)(7) = 42

3. Now, what would be the effect if the rectangle’s base were
doubled?
3
7
3
14
P = 2(3)+2(7)
= 20
P = 2(3)+2(14)
= 34
A = (3)(7) = 21A = (3)(14) = 42
Notice that in both cases, doubling one
dimension doubles the area. It doesn’t
matter whether it is the base or the height.

4. What happens if we double both the base and the height?
3
7
6
14
P = 20; A = 21
P = 2(6)+2(14)
= 40
A = (6)(14)
= 84
This time, the perimeter doubled, but the
area changed by a factor of 4. Why the
difference?

5. Let’s break down the perimeter and area on the last
example:
2(3)
2(7)
A = (2)(3)(2)(7)
= (2)(2)(3)(7)
= (4)(21)
= 84
Both sides are multiplied by 2, so the
perimeter is doubled and the area is
multiplied by 2
2
.
P = 2[2(3)]+2[2(7)]
= 2[2(3)+2(7)]
= 2(20)
= 40

6. Now, let’s look at a circle. What happens to the
circumference and area if we triple the radius?
•
•
2
6
C = 2(2)=4
A=(2
2
) = 4
C = 2(6) = 12
A = (6
2
) = 36
The circumference
increased by 3, and
the area increased by
3
2
or 9.

7. Quadrilateralsandtrianglescanhaveone or both dimensions
changed. Here’s a summary of what happens when you
multiply one or two sides by some factor, f:
Forshapesthatonlyhaveonemeasurement (squares,
circles,regularpolygons),here’sasummaryofwhathappens
whenyoumultiplythatmeasurement bysomefactor,f:
Multiply Perimeter Area
1 side calculate old f
2 sides old f old f
2
Multiply Perimeter Area
whole thing old f old f
2

8. We can use these ideas to work problems going the other
way:
1.If a square’s area is quadrupled (x4), what happens to the
perimeter?
Since the area is multiplied by 4, that means that each
side was multiplied by the or 2. Thus, the perimeter
is doubled.
2.If a circle’s circumference is reduced by half, what happens
to the area?
If the circumference is multiplied by ½, then so is the
radius. Therefore, the area would be multiplied by
or ¼.4 ()
2
1
2

9. 3.An octagon hasanareaof36m
2
.Ifitisreducedtoan
areaof4m
2
,bywhatscalefactorwasitreduced?
To calculate scale factor for perimeter, we would take the
new measurement divided by the old measurement.
Since we’re dealing with area, we will take the square
root of that quotient:411
3693
==

10. Three-dimensional shapes work in a similar way, except we
are dealing with area and volume. So if the dimensions of a
rectangular prism are multiplied by some factor f:
Forothertypes of 3D shapes, we will just be multiplying
everything by some factor f:
MultiplySurface AreaVolume
1 side calculate old f
2 sides calculate old f
2
3 sides old f
2
old f
3
MultiplySurface AreaVolume
whole thing old f
2
old f
3

11. Examples
1.A rectangular prism with a volume of 54 cm
3
is cut down
by ⅓. What is the new volume?
2.A cylinderwhichhasasurfaceareaof20in
2
is expanded
by a factor of 3. What is the new surface area?
3.A sphere has a surface area of 36mm
2
. If its radius is
increased by a factor of 5, what is its new volume?3
311
54542 cm
327
V
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
 ()
22
203180 inS== 2
2
364
9
3
r
r
r
=
=
= ()
334
154500 mm
3
V==