Geometry Volume of Pyramids and Cones CONFIDENTIAL

Find the unknown numbers. 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. 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 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 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. Next Page: CONFIDENTIAL

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

9 m 6 m 10 m B E D C A 18 m 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. CONFIDENTIAL

Find the volume of a regular hexagonal pyramid with a 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. 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 1 acre 10 stories Next Page: CONFIDENTIAL

1 acre 10 stories CONFIDENTIAL

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

Volume of a Cones h 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

Step 1: Use the Pythagorean Theorem to find the height. 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

3) Find the volume of the cone. Now you try! 3) Find the volume of the cone. 18 m 8 m 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 24 ft Next Page: CONFIDENTIAL

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

4) The radius and height of the cone are doubled. Now you try! 4) The radius and height of the cone are doubled. Describe the effect on the volume. 9 cm 18 cm 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

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

Now some problems for you to practice ! CONFIDENTIAL

Assessment 1) Find the volume of each pyramid. Round to the nearest tenth, If necessary. A 17 in 6 in 4 in B 4 cm 4 3 cm 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 CONFIDENTIAL

3) Find the volume of each cone 3) Find the volume of each cone. Give your answer both in terms of and rounded to the nearest tenth. 14 cm 9 cm 30 in. 24 in. A B CONFIDENTIAL

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

5) Find the volume of each composite figure. Round to the nearest tenth, if necessary. 12 cm 18 cm A B 4 in. 8 in. 6 in 12 in. 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 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 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. Next Page: CONFIDENTIAL

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

9 m 6 m 10 m B E D C A 18 m 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. 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 1 acre 10 stories Next Page: CONFIDENTIAL

1 acre 10 stories CONFIDENTIAL

Volume of a Cones h 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

Step 1: Use the Pythagorean Theorem to find the height. 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 24 ft Next Page: CONFIDENTIAL

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