7.4

How Much Does It Hold?

Pg. 11

Volume of Prisms and Cylinders

7.4 – How Much Does It Hold?___________Volume of Prisms and Cylinders

Today you are going to explore a different measurement found with three-dimensional shapes, called volume.

7.16 – VOLUMEUsing blocks provided by your teacher, work with your team to build the three-dimensional solid at right. Assume that blocks cannot hover in mid-air. That is, if a block is on the second level, assume that it has a block below it to prop it up.

a. Is there more than one arrangement of blocks that could look like the solid drawn at right? Why or why not?

Yes, there could be a hidden one in the back left

b. To avoid confusion, a mat plan can be used to show how the blocks are arranged in the solid. The number in each square represents the number of the blocks stacked in that location if you are looking from above. For example, in the lower right-hand corner, the solid is only 1 block tall, so there is a "1" in the corresponding corner of its mat plan.

Verify that the solid your team built matches the solid represented in the mat plan above.

c. What is the volume of the solid? That is, if each block represents a "cubic unit," how many blocks (cubic units) make up this solid?

13 cubic units 13 un3

7.17 – VOLUME TO MAT PLANSFor each of the solids below, build a mat plan for the solid. Then find the volume. Don't forget units.

0 or 1

3

0

2

2

1

0

1

0

V = 9 u3

orV = 10 u3

3

1

1

0

1

0

2

1

0

V = 9 u3

7.18 – MAT PLANS TO VOLUMEFor each of the mat plans below, find the volume of the solid. Don't forget units.

V = 12 u3V = 11 u3

7.19 – PRISMSPaul built a tower by stacking six identical layers of the shape at right, one on top of each other.

a. What is the number of cubes on each layer?

7

b. Make a mat plan of the shape.

6

6

6

0

6

6

6

6

0

c. What is the volume of his tower?

6

6

6

0

6

6

6

6

0

V = 42 u3

d. Paul's tower is an example of a prism because it is a solid and two of its faces (called bases) are congruent and parallel. A prism also has sides that connect the bases (called lateral faces).

For each of the prisms below, find the volume. Be ready to share any shortcuts you have developed. Don't forget any units.

4

5

20 un3

6

4

24 un3

12

5

60 un3

7.20 – VOLUME OF ANY PRISMCome up with a formula that will work to find the volume of any prism.

V = BH

7.21 – SPECIAL PRISMSThe prism at right is called a triangular prism because the two congruent bases are triangles.

a. Shade in the bases of the prism. Find the area of the base of the prism.

A = ½(3)(4)

A = 6 cm2

b. Find the volume of the triangular prism.

V = BH

V = (6)(9)

V = 54 cm3

7.22 – HEXAGONAL PRISMSWhat if the bases are hexagonal, like the one shown at right?

a. Why are the bases the hexagons and not the rectangles? Shade in one of the bases.

Hexagons are congruent parallel bases

b. Find the area of the hexagonal base. Leave answers in square root form.

30

444 3

18 4 3 6

2A

296 3A cm

c. Find the volume of the hexagonal prism. Leave answers in square root form.

30

444 3

V BH

2480 3V cm

96 3 5V

7.23 – CYLINDERSCarter wonders, "What if the bases are circular?"

a. Shade in one base of the circular prism. Change the volume formula to include circles.

V BH2V r H

b. Find the volume. Don't forget units. Leave answers in terms of .

2V r H

25 8V

3200V un

2SA r r

27SA 7 1549SA 105

2154SA in