Volume of Prisms and Cylinders Section 9.4

Objectives: Find the volume of prisms and cylinders.

Key Vocabulary Prism Cylinder Volume

The volume of a solid is the number of cubic units contained in its interior.

Rectangular Prism A prism is a polyhedron with two congruent faces, called bases, that lie in parallel planes. The other faces called lateral faces, are parallelograms formed by connecting the corresponding vertices of the bases. The segments connecting these vertices are lateral edges.

Finding the Volume of a Prism The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box?

Finding the Volume of a Prism The base of the box is 5 units by 3 units. This means 5 3, or 15 unit cubes, will cover the base. Three more layers of 15 cubes each can be placed on top of the lower layer to fill the box. Because the box contains 4 layers with 15 cubes in each layer, the box contains a total of 4 15 cubes, or 60 unit cubes.

Conclusion - Volume of a Prism Because the box is completely filled by the 60 cubes, and each cube has a volume of 1 cubic unit, it follows that the volume of the box is 60 1, or 60 cubic units. The area of the base, 15 square units, multiplied by the height of 4 units, yields the volume of the box, 60 cubic units. So, the volume of the prism can be found by multiplying the area of the base by the height.

Area of base = B Height of Prism = h Volume of Prism = Area of base x height = Bh w h w Volume of Prism Base

Volume of a Prism

Find the volume of the prism. V = lwh l = 8 in w = 4 in h = 1 ft = 12 in V = (8)(4)(12) V = 384 in 3 1 ft 4 in 8 in

Example 1 Find the Volume of a Rectangular Prism SOLUTION The base is 5 units by 3 units. So, 3 · 5, or 15 unit cubes are needed to cover the base layer. There are 4 layers. Each layer has 15 cubes. So, the total number of cubes is 4 · 15, or 60. Find the volume of the box by determining how many unit cubes fit in the box. ANSWER The volume of the box is 60 cubic units.

Example 2 Find the Volume of a Prism ANSWER The volume is 140 cubic inches. Find the volume of the prism. a. b. SOLUTION a. V = Bh Write the formula for volume of a prism. = (7 · 4) · 5 Area of rectangular base = l · w = 7 · 4. = 140 Simplify.

Area of triangular base = 1 2 – · 8 · 6. = 1 2 – · 8 · 6 · 3 Example 2 Find the Volume of a Prism ANSWER The volume is 72 cubic feet. = 72 Simplify. b. V = Bh Write the formula for volume of a prism.

Your Turn: Find the volume of the prism ANSWER 216 ft 3 ANSWER 125 cm 3 ANSWER 245 in. 3

Cylinder A cylinder is a solid with congruent circular bases that lie in parallel planes. The altitude, or height of a cylinder is the perpendicular distance between its bases. The radius of the base is also called the radius of the cylinder. A cylinder is called a right cylinder if the segment joining the centers of the bases is perpendicular to the bases.

r h Volume of Cylinder Formation of Cylinder by adding up bases Area Base = Area of Circle = π r 2 r r = radius h = height Or V = Bh B = Area Base h = height

Volume of a Cylinder

Find the volume of the cylinder. V = π r 2 h π = 3.14 r = 6 h = 10 V = (3.14)(6 2 )(10) V = in 3 10 in 6 in

Example 3 Compare Volumes of Cylinders How do the radius and height of the mug compare to the radius and height of the dog bowl? a. How many times greater is the volume of the bowl than the volume of the mug? b. SOLUTION The radius of the mug is 2 inches and the radius of the dog bowl is 6 inches. The radius of the bowl is three times the radius of the mug. The height of the mug is the same as the height of the bowl. a.

Example 3 Compare Volumes of Cylinders ANSWER The volume of the bowl is nine times the volume of the mug. b. Volume of mugVolume of dog bowl V = πr 2 h Write the formula for volume. V = πr 2 h = π(2 2 )(4)= π(6 2 )(4) Substitute for r and for h. = 16π Simplify. = 144π To compare the volume of the bowl to the volume of the mug, divide the volume of the bowl by the volume of the mug. –––––––––––––– Volume of bowl Volume of mug 144π 16π ––––– == 9

Your Turn: ANSWER 38 ft 3 ANSWER 16 in. 3 ANSWER 126 m 3 Find the volume of the cylinder. Round your answer to the nearest whole number

Assignment Pg , #1 – 57 odd