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

Volume

Volume 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 1

Find 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 Postulates

Volume 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 1

In 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 1

Step 1: Find the volume of each right rectangular prism. (How many cubes measuring 1 cm on an edge will fit into each solid?)

Investigation 1

Step 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 2

Use 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 Cylinders

Volume 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 3

Find the volume of the regular hexagonal prism shown.

Exercise 4

The 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 5

Find the volume of the solid of revolution formed by revolving the rectangle shown around the y-axis.

Sections

When 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 Principle

Suppose 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 Principle

The shape of the oblique rectangular prism can be approximated by a slanted stack of three reams of 8.5” x 11” paper…

Cavalieri’s Principle

The shape can be even better approximated by the individual pieces of paper in a slanted stack…

Cavalieri’s Principle

Rearranging 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 Principle

Similarly, 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 Principle

If 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 7

Name the solid shown, and then find its volume.

Exercise 8

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

Exercise 9

The volume of the cylinder is 3148 yd3. Find the length of the radius.

Investigation 2

In 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 2

Step 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 2

Step 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 Cones

Volume of a Pyramid

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

Volume of a Cone

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

1

3V Bh 21

3V r h

Exercise 10

Find the volume of the solid of revolution formed by rotating the triangle around the y-axis.

Exercise 11

Find 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 13

Find 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