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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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GeometryVolume of Pyramids and Cones CONFIDENTIAL
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
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
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
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
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
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
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
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
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
10 stories 1 acre CONFIDENTIAL
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
Volume of a Cones h h r r CONFIDENTIAL
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
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
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
Now you try! 3) Find the volume of the cone. 18 m 8 m 3) 648∏ m3 CONFIDENTIAL
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
20 ft Length, width, and height multiplied by ¼: Original dimensions: 20 ft 24 ft CONFIDENTIAL
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
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
Now you try! 5) Find the volume of the composite figure. 15 ft 12 ft 25 ft 5) 3000 ft3. CONFIDENTIAL
Now some problems for you to practice ! CONFIDENTIAL
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
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
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
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
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
Let’s review CONFIDENTIAL
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
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
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
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
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
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
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
10 stories 1 acre CONFIDENTIAL
Volume of a Cones h h r r CONFIDENTIAL
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
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
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
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
20 ft Length, width, and height multiplied by ¼: Original dimensions: 20 ft 24 ft CONFIDENTIAL
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
You did a great job today! CONFIDENTIAL