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**Geometry Volume of Pyramids and Cones**

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**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

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**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

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**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

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

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**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

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**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

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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

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**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

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**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

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1 acre 10 stories CONFIDENTIAL

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**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

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Volume of a Cones h r CONFIDENTIAL

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**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

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**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

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**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

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**3) Find the volume of the cone.**

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

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**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

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**Length, width, and height multiplied by ¼: Original dimensions:**

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

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**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

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**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

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**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

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**Now some problems for you to practice !**

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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 cm CONFIDENTIAL

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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

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**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

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**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

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**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

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Let’s review CONFIDENTIAL

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**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

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**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

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

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**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

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**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

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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

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**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

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1 acre 10 stories CONFIDENTIAL

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Volume of a Cones h r CONFIDENTIAL

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**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

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**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

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**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

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**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

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**Length, width, and height multiplied by ¼: Original dimensions:**

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

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**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

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**You did a great job today!**

CONFIDENTIAL

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