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9-3 Volume of Pyramids and Cones

Video Tutor Help Word problem: volume of a pyramid Finding the volumes of conesFinding the volumes of cones ((9-3) Finding the volumes of pyramidsFinding the volumes of pyramids (9-3) Identifying a solid from a netIdentifying a solid from a net (9-3a)

Video Tutor Help Naming a three-dimensional figure Finding the volumes of prisms Finding the volumes of cylinders Finding the volume of cones Finding the volume of pyramids Finding surface areas of spheres using a formula Finding the volumes of spheres Finding the surface area and volume of similar solid

Worksheets Daily Notetaking Guide Worksheets Version A Practice, Guided Problem Solving Lesson 9-3 Practice 9-3 Guided Problem Solving 9-3

Vocabulary Practice Vocabulary 9A: Graphic Organizer Vocabulary 9B: Reading Comprehension Vocabulary 9C: Reading/Writing Math Symbols Vocabulary 9D: Visual Vocabulary Practice Vocabulary 9E: Vocabulary C Vocabulary 9F: Vocabulary Review Puzzle Vocabulary (Electronic) Flash Cards

Additional Lesson Examples Step-by-Step Examples Lesson 9-3

Lesson Readiness Lesson Quiz Problem of the Day Lesson 9-3

Volume of a Pyramid The volume V of a pyramid is one third the area of the base B times the height h. V = 1/3Bh V = Example for a pyramid with a triangular base: V = 1/3(1/2bh)h V = 1/3(area of triangle)h Note: the height of a pyramid or cone is the distance from the vertex, perpendicular to the base. Bh 3

Cone A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex. The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related. The volume of a cone is 1/3 the volume of a cylinder with the same base area and height.

Volume of a Cone The volume V of a cone with radius r is one- third the area of the base B times the height h. V = 1/3Bh or V = 1/3  r 2 h V = ∏ r 2 h 3

Example 6-1a Find the volume of the pyramid. Volume of a pyramid Simplify. Answer: The volume is 140 cubic centimeters. Find the Volume of a Pyramid

Example 3-1a Find the volume of the pyramid. If necessary, round to the nearest tenth. Formula for volume of a pyramid The base is a square, so The height of the pyramid is 12 inches. Simplify. Answer:The volume is 900 cubic inches. Volumes of Pyramids

Example 3-1b Find the volume of the pyramid. If necessary, round to the nearest tenth. base area 19 cm 2, height 21 cm Formula for volume of a pyramid Replace B with 19 and h with 21. Answer:The volume is 133 cubic centimeters. Simplify. Volumes of Pyramids

Example 6-2a SOUVENIRS A novelty souvenir company wants to make snow “globes” shaped like a pyramid. It decides that the most cost-effective maximum volume of water for the pyramids is 12 cubic inches. If a pyramid globe measures 4 inches in height, find the area of its base. Volume of a pyramid Replace V with12 and h with 4. Use Volume to Solve a Problem

Example 6-2b Simplify. Answer: The area of the base of the snow globe is 9 square inches.

Find the volume of the pyramid to the nearest cubic foot. Volumes of Pyramids and Cones LESSON 9-3 Step 1Find the area of the base. B = s 2 area of a square formula Substitute.= 8 2 Simplify.= 64 Additional Examples

(continued) Volumes of Pyramids and Cones LESSON 9-3 Step 2Use the base area to find the volume. volume of a pyramid formulaV = Bh 1313 Substitute.= (64)(10) 1313 Multiply. = The volume of the pyramid is approximately 213 ft 3. Additional Examples

Example 6-3a Find the volume of the cone. Round to the nearest tenth. Volume of a cone Replace r with1.5 and h with 8. Simplify. Answer: The volume is about 18.8 cubic meters. Find the Volume of a Cone

Example 3-2a Find the volume of the cone. Round to the nearest tenth. Formula for volume of a cone Replace r with 5.5 and h with 8. Simplify. Answer:The volume is about cubic meters. Volume of a cone

Example 3-3a Landscaping When mulch was dumped from a truck, it formed a cone- shaped mound with a diameter of 15 feet and a height of 8 feet. What is the volume of the mulch? Formula for volume of a cone Since d = 15, replace r with 7.5. Replace h with 8. Answer:The volume of the mulch is about 471 cubic feet. Use Volume to Solve Problems

Example 3-3b Landscaping When mulch was dumped from a truck, it formed a cone- shaped mound with a diameter of 15 feet and a height of 8 feet. How many square feet can be covered with this mulch if 1 cubic foot covers 4 square feet of ground? Answer:1884 square feet can be covered with this mulch. Use Volume to Solve Problems

Find the volume of this cone to the nearest cubic centimeter. Volumes of Pyramids and Cones LESSON 9-3 Step 1Find the area of the base. area of a circle formulaB = r 2 Substitute.= (5 2 ) Simplify. = 25 Additional Examples

(continued) Volumes of Pyramids and Cones LESSON 9-3 Step 2Use the base area to find the volume. To the nearest cubic centimeter, the volume of the cone is 105 cm 3. cone volume formulaV = Bh 1313 Substitute.= (25 ) Simplify. = Additional Examples

At an amusement park, the person whose cone-shaped top spins the longest wins a prize. The volume of the top is about 14 in. 3. If the height of the top is 4 in., what is its radius to the nearest inch? Volumes of Pyramids and Cones LESSON = r ²Simplify V = r ² hvolume of a cone = r ² 4Substitute 14 for V and 4 for h Divide each side by (14) = r ² 4 = Find the positive square root of each side. Additional Examples

(continued) Volumes of Pyramids and Cones LESSON 9-3 The radius of the top is about 2 in. Use a calculator r Additional Examples