Assignment• P. 822-825: 1, 2, 3-

21 odd, 24, 26, 32, 33, 35, 36

• P. 832-836: 1, 2-24 even, 28, 30-36, 40, 41

• Challenge Problems

Units, Units2, and Units3

Recall that length is measured in units:

And area is measured in square units:

1

Length: 1 unit

1

1

Area: 1 square unit

Units, Units2, and Units3

The volume of something (a polyhedron, a room, a bottle) is measured in cubic units: in3, ft3, cm3, m3, etc. It’s a three-dimensional measurement.

1

1

1

Volume: 1 cubic unit

VolumeVolume is the measure

of the amount of space contained in a solid, measured in cubic units.– This is simply the

number of unit cubes that can be arranged to completely fill the space within a figure.

Exercise 1Find the volume of

the given figure in cubic units.

12.4-12.5: Volume of Prisms, Cylinders, Pyramids, and Cones

Objectives:1. To derive and use the formulas for the

volume of prisms, cylinders, pyramids, and cones

Volume PostulatesVolume of a Cube

– The volume of a cube is V = s3.Volume Congruence

– If two polyhedra are congruent, then their volumes are equal.

Volume Addition– The volume of a solid is the sum of the

volumes of all of its nonoverlapping parts.

Investigation 1In this Investigation,

you will begin by examining the volumes of simple rectangular solids. You will then generalize your observations to apply to other kinds of solids.

Investigation 1Step 1: Find the volume of each right

rectangular prism. (How many cubes measuring 1 cm on an edge will fit into each solid?)

Investigation 1Step 2: To get the volume of the prism, you

could use a principle of multiplication to find the number of cubes:

Number of cubes in the base = (2)(4) = 8 cubes

Area of the base, B

Since the prism is 3 layers high, V = (8)(3) = 24 cubes

Height of prism, h

Exercise 2Use the formula for the volume of a prism to

help derive a formula for the volume of a cylinder with radius r and a height h.

Volume of Prisms and CylindersVolume of a Right

Prism

• B = area of the base• h = height of prism

Volume of a Right Cylinder

• r = radius of cylinder• h = height of cylinder

V Bh 2V r h

Exercise 3Find the volume of the

regular hexagonal prism shown.

Exercise 4The rectangle shown can be rotated around the y-

axis or the x-axis to make two different solids of revolution. Which solid would have the greater volume?

Exercise 5Find the volume of the solid of revolution formed by

revolving the rectangle shown around the y-axis.

SectionsWhen a solid is cut by a plane, the resulting

plane figure is called a section. A section that is parallel to the base is a cross-section.

Exercise 6

Exercise 6

Cavalieri’s PrincipleSuppose you wanted to find the volume of

an oblique rectangular prism with a base 8.5 inches by 11 inches and a height of 6 inches…

Cavalieri’s PrincipleThe shape of the oblique rectangular prism

can be approximated by a slanted stack of three reams of 8.5” x 11” paper…

Cavalieri’s PrincipleThe shape can be even better approximated

by the individual pieces of paper in a slanted stack…

Cavalieri’s PrincipleRearranging the paper formed into an

oblique rectangular solid back into a right rectangular prism changes the shape, but does it change the volume?

Cavalieri’s PrincipleSimilarly, you could

use a stack of coins to show that an oblique cylinder has the same volume as a right cylinder with the same base and height.

Cavalieri’s PrincipleIf two solids have the same height and the

same cross-sectional area at every level, then they have the same volume.

All 3 of these shapes have the same volume.

Exercise 7Name the solid

shown, and then find its volume.

Exercise 8Given the dimensions shown

in the diagram, how much concrete would be used to make 20 cinderblocks?

Exercise 9The volume of the cylinder is

3148 yd3. Find the length of the radius.

Investigation 2In this Investigation you will

discover the relationship between the volumes of prisms and pyramids with congruent bases and the same height and between cylinders and cones with congruent bases and the same height.

Investigation 2Step 1: Choose a prism and a

pyramid that have congruent bases and the same height.

Step 2: Fill the pyramid, then pour the contents into the prism. About what fraction of the prism is filled by the volume of the volume of one pyramid?

Step 3: Check your answer by repeating Step 2 until the prism is filled.

Investigation 2Step 4: Choose a cone and a

cylinder that have congruent bases and the same height and repeat Steps 2 and 3.

Step 5: Did you get similar results with both your pyramid-prism pair and the cone-cylinder pair?

Volume of Pyramids and ConesVolume of a Pyramid

• B = area of the base• h = height of pyramid

Volume of a Cone

• r = radius of cone• h = height of cone

13

V Bh 213

V r h

Exercise 10Find the volume of the solid of revolution

formed by rotating the triangle around the y-axis.

Exercise 11Find the volume of the solid of revolution

formed by rotating the triangle around the y-axis.

Exercise 12 You are using the funnel shown to

measure the coarseness of a substance. It takes 2.8 seconds for the substance to empty out of the funnel. Find the flow rate of the substance in mL per second (1 mL = 1 cm3).

Exercise 13Find the volume of the composite figure.

Assignment• P. 822-825: 1, 2, 3-

21 odd, 24, 26, 32, 33, 35, 36

• P. 832-836: 1, 2-24 even, 28, 30-36, 40, 41

• Challenge Problems