13.1 VOLUMES OF PRISMS AND

CYLINDERS

By Nathan Critchfield and Ben Tidwell

Objectives Find volumes of prisms. Find volumes of cylinders.

Volumes of Prisms

The volume of a figure is the measure of the amount of space that a figure encloses. Volume is measured in cubic units. You can create a rectangular prism from different views of the figure to investigate its volume.

Volumes of Prisms If a prism has a

volume of V cubic units, a height of h units, and each base has an area of B square units, then V = Bh

Area of base = B

h

Find the volume of the right triangular prism

Example 1: Volume of a Triangular Prism

24

15

a

20

Use the Pythagorean Theorem to find the length of the base of the prism.

*Note: Remember, you can only use P.T. on Right Triangles!

Use Pythagorean Theorem to find the length of the base of the prism.

Example 1: Volume of a Triangular Prism

24

20

15

a

a² + b² = c² Pythagorean Theorem

a² + 15² = 24²

a² + 225 = 576

a² = 351

a = √351

a ≈ 18.7

Find the volume of the prism.

Example 1: Volume of a Triangular Prism

24

20

15

18.7

V = Bh Volume of a Prism

BUT WAIT!.. Since it is a triangle, not a rectangle, it is…V =½(18.7)(15)(20)

B = 18.7(15) h = 20

V = 2,805 cubic centimeters

Find the volume in feet of the rectangular prism

Example 2: Volume of a Rectangular Prism

Convert feet to inches.

12 in. 25 ft.

10 ft

25 feet = 25 x 12 or 300 inches

10 feet = 10 x 12 or 120 inches

Find the volume in feet of a rectangular prism

Example 2: Volume of a Rectangular Prism

12 in. 300 in.

120 in.

300 in. x 120 in. = 36,000 in.

36,000 in. x 12 in. = 432,000 cubic inches.

432,000 / 123 = 250 cubic feet.

Volumes of Cylinders

If a cylinder has a volume of V cubic units, a height of h units, and the bases have radii of r units, then V = Bh or V = πr²h

Area of base = πr²

h

r

Find the volume of each cylinder

Example 3: Volume of a Cylinder

9.4m

1.6m

a. The height h is 9.4 meters, and the radius r is 1.6 meters.

V = πr²h

= π(1.6²)(9.4)

≈ 75.6 meters

Find the volume of each cylinder

Example 3: Volume of a Cylinder

b.

7 in.15 in.

The diameter of the base, the diagonal, and the lateral edge of the cylinder form a right triangle. Use the Pythagorean Theorem to find the height.

a² + b² = c² Pythagorean Theoremh² + 7² = 15²

h² + 49 = 225

h² = 176

h ≈ 13.3

Find the volume of each cylinder

Example 3: Volume of a cylinder

b.

7 in.13.3 in.

V = π(3.5²)(13.3)

V = 511.8

The volume is approximately 511.8 cubic inches.

Cavalieri’s PrincipleKey Concept!

If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

Which basically means, that whether it is right or oblique, it’s volume is V=Bh

Find the volume of the oblique cylinder

Example 4: Volume of an Oblique Cylinder

8 yd

13 yd

To find the volume, use the formula for a right cylinder.

V = πr²h

= π(8²)(13)

= 2,613.8

The volume is approximately 2,613.8 cubic yards.

AssignmenT

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