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Geometry Volume of Pyramids and Cones

Geometry Volume of Pyramids and Cones. Warm Up. Find the unknown numbers. The difference of two numbers is 24. the large number is 4 less than 3 times the smaller number.

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Geometry Volume of Pyramids and Cones

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  1. GeometryVolume of Pyramids and Cones CONFIDENTIAL

  2. Warm Up Find the unknown numbers. • The difference of two numbers is 24. the large • number is 4 less than 3 times the smaller number. • 2) Three times the first number plus the second number is 88. The first number times 10 is equal to 4 times the second. • 3) The sum of two numbers is 197. The first number is 20 more than ½ of the second number. 1)14 and 38 2) 16 and 40 3) 79 and 118 CONFIDENTIAL

  3. Volume of Pyramids and Cones The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. Next Page: CONFIDENTIAL

  4. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. CONFIDENTIAL

  5. Volume of a Pyramid The volume of a pyramid with base area B and height h is V = 1/3 Bh. h h B B CONFIDENTIAL

  6. Finding Volumes of Pyramids • Find the volume of each pyramid. • A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft. • b) The square pyramid • the base is a square with a side • length of 4 in., and the height is 6 in. 6 in. 4 in. 4 in. Next Page: CONFIDENTIAL

  7. Find the volume of each pyramid. c) The trapezoidal pyramid with base ABCD, where AB || CD and AE plane ABC. E 9 m 10 m B Step 1 Find the area of the base. A 6 m C D 18 m Next Page: CONFIDENTIAL

  8. E 9 m 10 m B A Step 2 Use the base area and the height to find the volume. Because AE plane ABC, AE is the altitude, so the height is equal to AE. 6 m C D 18 m CONFIDENTIAL

  9. Now you try! • Find the volume of a regular hexagonal pyramid with a • base edge length of 2 cm and a height equal to the area of the base. 1) 108 cm3 CONFIDENTIAL

  10. Architecture Application The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft) 2 10 stories 1 acre Next Page: CONFIDENTIAL

  11. 10 stories 1 acre CONFIDENTIAL

  12. Now you try! 2) What would be the volume of the Rainforest Pyramid if the height were doubled? 10 stories 2) The volume will be doubled. 1 acre CONFIDENTIAL

  13. Volume of a Cones h h r r CONFIDENTIAL

  14. Finding Volumes of a Cones Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth. A) A cone with radius 5 cm and height 12 cm Next Page: CONFIDENTIAL

  15. B) A cone with a base circumference of 21 cm and a height 3 cm less than twice the radius Step 1: Use the circumference to find the radius. Step 2: Use the radius to find the height. 2(10.5) – 3 = 18 cm The height is 3 cm less than twice the radius. Step 3: Use the radius and height to find the volume. Next Page: CONFIDENTIAL

  16. 25 ft 7 ft Step 1: Use the Pythagorean Theorem to find the height. Step 2: Use the radius and height to find the volume. CONFIDENTIAL

  17. Now you try! 3) Find the volume of the cone. 18 m 8 m 3) 648∏ m3 CONFIDENTIAL

  18. Exploring Effects of Changing Dimensions The length, width, and height of the rectangular pyramid are multiplied by ¼ . Describe the effect on the volume. 20 ft 20 ft 24 ft Next Page: CONFIDENTIAL

  19. 20 ft Length, width, and height multiplied by ¼: Original dimensions: 20 ft 24 ft CONFIDENTIAL

  20. Now you try! 4) The radius and height of the cone are doubled. Describe the effect on the volume. 18 cm 9 cm 4) The volume will increase by 8 times. CONFIDENTIAL

  21. Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 2 in 4 in 5 in CONFIDENTIAL

  22. Now you try! 5) Find the volume of the composite figure. 15 ft 12 ft 25 ft 5) 3000 ft3. CONFIDENTIAL

  23. Now some problems for you to practice ! CONFIDENTIAL

  24. Assessment 1) Find the volume of each pyramid. Round to the nearest tenth, If necessary. 4 3 cm B A 17 in 6 in 4 in 4 cm 1a) 136 in3. 1b) 96 cm3 CONFIDENTIAL

  25. 2) A crystal is cut into the shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. what is the volume to the nearest cubic millimeter of the crystal? 3 mm 5.7 mm 2) 64.98 mm3. CONFIDENTIAL

  26. 3) Find the volume of each cone. Give your answer both in terms of and rounded to the nearest tenth. B A 14 cm 30 in. 24 in. 9 cm 3a) 378∏ cm3 3b) 1440∏ in3. CONFIDENTIAL

  27. 4) Describe the effect of the each change on the volume of the given figure. b) The dimensions are multiplied by ½. a) The dimensions are tripled 3 cm 15 ft 9 ft 5 cm 9 ft 4a) volume increases by 27 times. 4b) volume decreases by 8 times. CONFIDENTIAL

  28. 5) Find the volume of each composite figure. Round to the nearest tenth, if necessary. A B 18 cm 8 in. 4 in. 6 in 12 cm 12 in. 12 cm 12 cm 5a) 2592 cm3. 5b) 1920∏ in3. CONFIDENTIAL

  29. Let’s review CONFIDENTIAL

  30. Volume of Pyramids and Cones The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. Next Page: CONFIDENTIAL

  31. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. CONFIDENTIAL

  32. Volume of a Pyramid The volume of a pyramid with base area B and height h is V = 1/3 Bh. h h B B CONFIDENTIAL

  33. Finding Volumes of Pyramids • Find the volume of each pyramid. • A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft. • b) The square pyramid • the base is a square with a side • length of 4 in., and the height is 6 in. 6 in. 4 in. 4 in. Next Page: CONFIDENTIAL

  34. Find the volume of each pyramid. c) The trapezoidal pyramid with base ABCD, where AB || CD and AE plane ABC. E 9 m 10 m B Step 1 Find the area of the base. A 6 m C D 18 m Next Page: CONFIDENTIAL

  35. E 9 m 10 m B A Step 2 Use the base area and the height to find the volume. Because AE plane ABC, AE is the altitude, so the height is equal to AE. 6 m C D 18 m CONFIDENTIAL

  36. Architecture Application The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft) 2 10 stories 1 acre Next Page: CONFIDENTIAL

  37. 10 stories 1 acre CONFIDENTIAL

  38. Volume of a Cones h h r r CONFIDENTIAL

  39. Finding Volumes of a Cones Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth. A) A cone with radius 5 cm and height 12 cm Next Page: CONFIDENTIAL

  40. B) A cone with a base circumference of 21 cm and a height 3 cm less than twice the radius Step 1: Use the circumference to find the radius. Step 2: Use the radius to find the height. 2(10.5) – 3 = 18 cm The height is 3 cm less than twice the radius. Step 3: Use the radius and height to find the volume. Next Page: CONFIDENTIAL

  41. 25 ft 7 ft Step 1: Use the Pythagorean Theorem to find the height. Step 2: Use the radius and height to find the volume. CONFIDENTIAL

  42. Exploring Effects of Changing Dimensions The length, width, and height of the rectangular pyramid are multiplied by ¼ . Describe the effect on the volume. 20 ft 20 ft 24 ft Next Page: CONFIDENTIAL

  43. 20 ft Length, width, and height multiplied by ¼: Original dimensions: 20 ft 24 ft CONFIDENTIAL

  44. Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 2 in 4 in 5 in CONFIDENTIAL

  45. You did a great job today! CONFIDENTIAL

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