3-D Shapes Topic 14: Lesson 6 3-D Shapes Topic 14: Lesson 6 Volume of Cylinders Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS: G2B, G3B, G6B, G8D, G11D The student will make conjectures about 3-D figures and determine the validity using a variety of approaches. The student will construct and justify statements about geometric figures and their properties. The student will use nets to represent and construct 3-D figures. The student will find surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite figures. The student will describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed.

Example: 1 Find the volume of the cylinder. Give your answer in terms of π and rounded to the nearest tenth. V = r 2h V = (9)2(14) V = 1134 in3  3562.6 in3

Example: 2 Find the volume of a cylinder with a diameter of 16 in. and a height of 17 in. Give your answer both in terms of π and rounded to the nearest tenth. V = r 2h Volume of a cylinder V = (8)2(17) Substitute 8 for r and 17 for h. V = 1088 in3 V  3418.1 in3

Example: 3 Find the volume of a cylinder with base area 121 cm2 and a height equal to twice the radius. Give your answer in terms of  and rounded to the nearest tenth. Step 1 Use the base area to find the radius. r 2 = 121 Substitute 121 for the base area. r = 11 Solve for r. Step 2 Use the radius to find the height. The height is equal to twice the radius. h = 2(r) h = 2(11) = 22 cm Step 3 Use the radius and height to find the volume. V = r 2h Volume of a cylinder. V = (11)2(22) Substitute 11 for radius and 22 for height. V = 2662 cm3  8362.9 cm3

Example: 4 The radius and height of the cylinder are multiplied by . Describe the effect on the volume. original dimensions: radius and height multiplied by :

Example: 4 continued The radius and height of the cylinder are multiplied by . Describe the effect on the volume. Notice that . If the radius and height are multiplied by , the volume is multiplied by , or .

Example: 5 Vcylinder — Vsquare prism = 45 — 90  51.4 cm3 Find the volume of the composite figure. Round to the nearest tenth. Find the side length s of the base: The volume of the square prism is: The volume of the cylinder is: The volume of the composite is the cylinder minus the rectangular prism. Vcylinder — Vsquare prism = 45 — 90  51.4 cm3

Example: 6 Find the dimensions and volume of the largest cylinder that can be packed inside of a box that has dimensions 14 inches by 7 inches by 2 inches. How much extra space would there be in the box? Hint: draw the box! 7 in 2 in 7 in 2 in 7 in 2 in 14 in 14 in 14 in

Solution: There are three ways that the cylinder could fix in the box: It could fit “standing up” with dimensions of diameter of 7 inches and height of 2 inches. It could fit “laying down left to right” with dimensions of diameter of 2 inches and height of 14 inches. It could fit “laying down front to back” with a diameter of 2 inches and height of 7 inches. This would be smaller than the second choice because the diameter is the same but the height is only half as much.

The cylinder on the left is the largest V = r 2h V =  (3.5)2(2) V = 24.5  in3 V = r 2h V =  (1)2(14) V = 14  in3 The cylinder on the left is the largest cylinder that will fit in the box.

Now you also need to know the volume of the box. V = Bh V = (14)(7)(2) 7 in 2 in Now you also need to know the volume of the box. V = Bh V = (14)(7)(2) V = 196 in3 14 in V = r 2h V =  (3.5)2(2) V = 24.5  in3 The extra space is the volume of the box minus the volume of the largest cylinder: V = Bh - r 2h V = 196 – 24.5  V ≈ 119.03 in3