Name Date PD Volume
Volume the number of cubic units needed to fill a solid. To find the volume of a prism or cylinder, multiply the base area (B) by the height h. Rectangular prisms Formula: V Bh (what is the base in a rectangular prism?) V = (length)(width)(height) Example 1: Try-It! Volume = Volume = Cylinders have two bases that are parallel, congruent circles. Formula: V Bh (what is the base in a cylinder?) V = r 2 h 2 (Since the area of the base is a circle, the area of a circle is r. We have to multiply by its height.) Example 2: Try-It! (Hint: What s the radius?) Find the volume. Volume = Volume = Find the volume to the nearest tenth. Volume Volume 1
Cone has one base that is a circle and then meets at a common vertex. Formula: V 1 1 2 Bh V r h (what is the base in a cone?) 3 3 Example 3: Try-It! (Hint: What s the radius?) Find the volume. Volume = Volume = Find the volume to the nearest tenth. Volume Volume 2
Spheres the set of all points in space that are the same distances from a center point. Formula: V 4 r 3 3 Example 4: Try-It! (Hint: What s the radius?) Find the volume. Volume = Volume = Find the volume to the nearest tenth. Volume Volume 3
Try-It! a. Bob is building a storage shed in a conical shape. The base of the shed is 4 meters in diameter and the height of the shed is 3.6 meters. What is the volume? b. A scented candle is in the shape of a cylinder with a radius of 4cm and a height of 12cm. What is the volume? c. The rectangular prism has a length of 10 inches, a width of 3 inches, and a height of 20 inches. What is the volume? d. Spaceship Earth at Epcot Center in Florida is a 180-foot geosphere. Find the volume by assuming it is a sphere with a diameter of 180 feet. 7 e. *A cylinder has a diameter and length of 1.0 10 inches and approximate volume of the solid? 5 5.0 10 inches respectively. What is the 4
Practice Find the volume of each solid. SHOW ALL WORK! 1. 2. The rectangular box has a length of 10 inches, a width of 8 inches, and a height of 20 inches. 3. Leave in terms of π. 4. Round to the nearest tenth. Volume = Volume 5
Find the volume of each solid. SHOW ALL WORK! 5. Round to the nearest tenth. 6. Leave in terms of π. A sphere with a diameter of 15 inches. Volume Volume = Find the volume of each composite solid. SHOW ALL WORK! Example 1: Example 2: Leave in terms of π. Nate uses a cube shaped bead with side lengths measuring 6mm. Each bead has a circular hole in the middle. The diameter of the circular hole is 3mm. Round to the nearest hundredth. Volume = Volume 6
Practice: Find the volume of each composite solid. SHOW ALL WORK! 1. The hemisphere has a diameter of 15 km. Round to the nearest tenth. (A hemisphere is a half of a sphere) 2. Leave in terms of π. 3. A ball is inside the cylinder. Find the volume of the empty space inside the cylinder. The height of the cylinder is 6m, the radius of the cylinder is 3m, and the ball has a radius of 3m. [not drawn to scale] 4. A hemisphere is attached to a cylinder with equal diameter. The cylinder has a diameter of 4.5 inches and a height of 2.5in. Round to the nearest hundredth. 7
5. April is filling six identical cones for her piñata. Each cone has a radius of 1.5 inches and height of 9 inches. What is the total volume of the piñata? 6. Here are the dimensions of a pill. Round to the nearest tenth.. 7. Tanya uses a cube shaped bead with side lengths measuring 12mm. Each bead has a circular hole in the middle. The diameter of the circular hole is 2mm. Find the volume of the bead. Hint: draw a picture. 8. Three tennis balls are packaged in a box. The box is 12.1cm long, 3.5cm wide and 3.5 cm tall. Each ball is 3.3cm in diameter. What is the volume of the empty space in the box? 8
Determining Missing Heights, Area of base, and Radius/Diameter Warm-Up: Find the volume of a cone with a radius of 15 ft and a height of 4 feet. Example 1: The volume of a cylinder is 405 with a diameter of 18. Find the height of the cylinder. Try-It! a. A cylindrical cake takes up 32π cubic inches. The radius of the cake is 4 inches, what is the height of the cake? b. The Roberts family uses a container shaped like a cylinder to recycle aluminum cans. It has a diameter of 1.5 feet and a volume of 2.25π ft 3. If the container is filled half way to the top, what is the height that the cans reach? 9
Example 2: The volume of a cone is 405 in 3 with a diameter of 18in. Find the height of the cone. Try-It! The volume of cone with a 30mm radius is 9420 cubic millimeters. What is the height of the cone to the nearest millimeter? Example 3: The volume of a cylinder is about 1632 in 3. The height of the cylinder is 24in. What is the area of the base? Try-It! Find the area of the base of a cone with a volume of 300 ft 3 and a height of 4ft. 10
Example 4: Find the diameter of a cone with a volume of 48 ft 3 and a height of 4ft. Try-It! Find the radius of a cylinder with a volume of 75 in 3 and a height of 3 inches. Practice Show all work. 1. A conical cake takes up 32π cubic inches. The radius of the cake is 4 inches, what is the height of the cake? 2. A funnel can hold 400π cm 3 of fluid. Its height (without the stem) is 12 cm. What is the diameter of the cone part of the funnel? 3. The Carter family uses a container shaped like a cylinder to recycle newspaper. It has a diameter of 3 feet and a volume of 13.5π ft 3. If the container is filled two-thirds of the way to the top, what is the height that the newspapers can reach? 11
4. The volume of cylinder with a 30mm radius is 9420 cubic millimeters. What is the height of the cylinder to the nearest millimeter? 5. Find the area of the base of a cylinder with a volume of 144 ft 3 and a height of 4ft. 6. Find the radius of a cone with a volume of 81 in 3 and a height of 3 inches. 12
Comparing/Analyzing Volume Example 1: a) Given the following figure, find the volume (leave in terms of π). b) Double the height, find the new volume (leave in terms of π). c) How do the two volumes compare? d) Double the radius of the original cylinder, find the volume (leave in terms of π). How does this volume compare to the one you found in part a? Why? e) What would have a greater effect on the volume of a cone: doubling its radius or doubling its height? 13
Try-It! a) Given the following figure, find the volume (leave in terms of π). b) Double the height, find the new volume (leave in terms of π). c) How do the two volumes compare? d) Double the radius of the original cone, find the volume (leave in terms of π). How does this volume compare to the one you found in part a? Why? e) What would have a greater effect on the volume of a cone: doubling its radius or doubling its height? 14
Example 2: a) Given the following figure, find the volume (leave in terms of π). b) Draw a cone with the same dimensions as the figure above. What is the cone s volume (leave in terms of π)? c) How do the two volumes compare? Try-It! a) A cylinder has a radius of 6cm and height of 12cm. Find the volume (leave in terms of π). b) Draw a sphere with the same dimensions as the figure above. What is the sphere s volume (leave in terms of π)? c) How do the two volumes compare? 15
Summary: a) Given the following figures, let s analyze their formulas. Restate their formulas under the pictures. b) How many cones fit inside a cylinder? c) How many cones fit inside a sphere? d) How many cylinders fit inside the sphere? Practice: 1. A machine uses a funnel in the shape of a cone to fill soda cans on an assembly line. The funnel has a height 10cm and a diameter of 8cm. How many times would the machine need to fill the cone to then fill a can of the same dimensions? (Show work to prove the answer.) 16
2. The diameter of the earth is approximately 7,926 miles. The diameter of the moon is approximately 2,159 miles. Approximately how many moons would fit inside the earth? Practice 1. A soda can has a diameter of 3in and a height of 5in. a) Find the volume in terms (leave in terms of π). b) Triple the radius, find the new volume (leave in terms of π). c) How do the two volumes compare? 2. A snow cone has a diameter of 3in and a height of 5in. a) Find the volume in terms of π. b) Triple the height; find the new volume (leave in terms of π). c) How do the two volumes compare? 17
3. Take a hemisphere and a cylinder with an equal base and height. Fill the hemisphere with water and then pour the water into the cylinder, how far up the cylinder will the water reach? (Hint: the hemisphere has a radius of 10mm.) Show work or explain your choice. Multiple Choice: a) b) c) 4. Marge has a cylindrical tin of popcorn that is 18 in. tall and has a radius of 4 in. She wants to use the tin for something else and needs to empty the popcorn into a box. The box is 8 in. long, 8 in. wide and 14 in. tall. Will the popcorn fit in the box? Explain. 5. If you have a cylinder and double the radius and height how does it affect the volume? 6. If the top cone was filled with sand and then emptied into the bottom cone. How much of the sand would remain in the top cone without overflowing the bottom one? 18
REVIEW 1. A cylinder has a diameter of 14 centimeters and a volume of 112π cubic centimeters. What is the height, in centimeters, of the cylinder? a. 16 b. 4 c. 16 7 d. 7 4 2. A cylinder and a cone have congruent heights and radii. What is the ratio of the volume of the cone to the volume of the cylinder? a. 1:1 b. 1:3 c. 1:6 d. 1:9 3 3. A cylinder has a radius of 3 inches and a height of 4 inches. A sphere has a radius of 3 inches. What is the 4 difference between the volumes, to the nearest tenth of a cubic inch, of the cylinder and the sphere? a. 21.2 b. 51.8 c. 68.3 d. 96.6 19
1 4. The volume of a cone is 33 cm 3 with a height of 4cm. What is the length of the diameter? 3 5. A cone has a radius of 1.2 inches and a height of 2.9 inches. What is the volume, to the nearest tenth of a cubic inch, of the cone? 6. The human eye contains rods (cylindrical in nature), primarily responsible for night vision, which have an 7 5 approximate diameter and length of 2.0 10 meters and 4.0 10 meters respectively. What is the approximate volume of the solid? 20
7. Based on the following drawing, if the top funnel was filled with water and then emptied into the bottom cone, what fraction of the bottom cone would be filled with water? Explain. 8. Find the volume. 9. Find the volume. 21
10. If r 3, what is the volume of the ball, the cylinder, and the remaining space? 11. A cylinder has a radius and height of volume of the solid? 7 2.0 10 feet and 5 4.0 10 feet respectively. What is the approximate 12. If you double the height of a cone, what does it do to its volume? 22
2 nd Review Multiple Choice: Identify the choice that best completes the statement or answers the question. (2 pts each) 1. What is the height of a cylinder with volume 6,908 cubic feet and with a radius of 10 feet? (Round to the nearest integer.) a. 33 ft b. 22 ft c. 24 ft d. 11 ft 2. What is the volume of a can of soup (cylinder) with a radius of 2 inches and a height of 5 inches? (Round to the nearest tenth.) a. 62.8 in 3 b. 31.4 in 3 c. 15.7 in 3 d. 12.6 in 3 3. A cylinder is 8 inches high. The circumference of the base is 6π inches. Find the volume. a. 288π in 3 b. 48π in 3 c. 72π in 3 d. 9π in 3 23
4. Find the volume of the cone. (Round to the nearest integer.) 9 in. 4 in. a. 1,810 in. 3 b. 151 in. 3 c. 452 in. 3 d. 276 in. 3 5. Find the missing height of the cone, if the volume is 150.72cm 3. (Round to the nearest tenth.) h 6 cm a. 2.7 cm b. 12.6 cm c. 4.0 cm d. 1.3 cm 6. Find the volume of a sphere whose diameter is 4cm to the nearest whole number. a. 34 cm 3 b. 268 cm 3 c. 50 cm 3 d. 201 cm 3 24
7. Which of the following is the volume of a hemisphere with an 11 foot radius? a. 2787.64 ft 3 b. 4181.46 ft 3 c. 5575.28 ft 3 d. 11,150.56 ft 3 8. Find the diameter of a spherical beach ball with a volume of 36π in 3. a. 6 in b. 3 in c. 9 in d. 27 in Short Answer: Make sure to show all work for each question. 9. A can contains 3 identical tennis balls. The tennis balls are packed, as shown below. (4 pts) What is the total volume, in cubic inches, of all 3 tennis balls? Round your answer to the nearest tenth of a cubic inch. 25
10. Jessica is designing a cylindrical storage container for lawn chemicals. She first designed a cylinder with radius 12 inches and height 16 inches. (2 pts each) a) What is the volume of the container? (Leave in terms of pi.) b) Jessica changes the container by doubling the radius. What is the volume of the new container? (Leave in terms of pi.) Show work or explain. 11. Jodi is making some decorations for a graduation party. A diagram of a decoration is shown. She plans to hang fifteen of these figures on wires across the room. What will be the total volume of the fifteen figures? (Note: the 7cm represents the height of the cone, not the slant height. Round to the nearest hundredth.) (3 pts) 26
12. Bridger City has a cylindrical tank for storing water used by the residents. The tank has a diameter of 30 feet and a height of 25 feet. (3 pts) a) What is the volume, in cubic feet, of the tank? b) One cubic foot of water is about 7.5 gallons of water. About how many gallons of water are in the tank, to the nearest gallon? 4 13. The diameter of the Earth is approximately 1.2 10 kilometers. What is the volume, in cubic kilometers, of the Earth? Leave your answer in scientific notation and in terms of pi. (3 pts) 27