6th Grade Math Textbook, Fundamentals

Lesson 11-11 for exercise sets. &KDSWHU 

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1.Make a conjecture: If you triple all three dimensions of a prism, how does the

surface area of the new prism compare to the surface area of the original? Use

these dimensions to verify your conjecture:2 in.; w3 in.; h1 in.

2.Discuss and Write Explain why doubling the length and halving the height

does notchange the volume of the rectangular prism. Use the formula for a

rectangular prism’s volume to help you confirm your explanation.

Key Concept

How Changes in Dimensions Affect Volume

• When one dimension of a three-dimensional
  polyhedron is tripled, the volume is multiplied by 3.


• When two dimensions of a three-dimensional
  polyhedron are tripled, the volume is multiplied by 3^2.


• When all three dimensions of a three-dimensional
  polyhedron are tripled, the volume is multiplied by 3^3.

Changes in dimensions affect changes in volume.

How would a cube’s volume change if you doubled each of its dimensions?

How would its volume change if you halved each of its dimensions?

A cube with edge length 6 mm has a volume of 6^3 or 216 mm^3.

Doubling each dimension

The new cube has edges of length 12 mm.

Its volume is 12^3 1728 mm^3.

Because 1728 8 • 216, the new volume is

8 times the original.

So multiplying each dimension by 2

multiplies the volume by 2^3 , or 8.

Triple onlythe prism’s height

So h 30 cm.

VBh( • 4 • 6)• 30

360 cm^3

The new prism’s volume is 3
times as large as the original.

1

2

Triple onlythe height of the

base of the prism

So hbase 18 cm.

VBh( • 4 • 18)• 10

360 cm^3

The new prism’s volume is 3

times as large as the original.

1

2

Triple the height of the prism

andthe height of the base

hprism 30 cm and hbase 18 cm

VBh( • 4 • 18)• 30

1080 cm^3

The new prism’s volume is

9 times, or 3^2 , as large as

the original.

1

2

Halving each dimension

The new cube has a side length of 3 mm.

Its volume is 3^3 27 mm^3.

Because 27  • 216, the new volume is

the original. So multiplying each dimension

by multiplies the volume by ()

3

(^12 12) , or. (^18)
1
8
1
8

To understand how volume changes when only oneor twodimensions

are changed, try using a volume formula and substituting different values

for each dimension.

A triangular prism has a height of 10 cm and a triangular base with a base

of 4 cm and a height of 6 cm. Its volume is Bh(^12 • 4 • 6)• 10 120 cm^3.