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Volume of Cones and Pyramids Geometry Unit 5, Lesson 8 Mrs. King

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Reminder: What is a Pyramid? Definition: A shape formed by connecting triangles to a polygon. Examples:

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Reminder: What is a Cone? Definition: A shape formed from a circle and a vertex point. Examples: s

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Volume Of A Cone. Consider the cylinder and cone shown below: The diameter (D) of the top of the cone and the cylinder are equal. D D The height (H) of the cone and the cylinder are equal. H H If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ? 3 times. This shows that the cylinder has three times the volume of a cone with the same height and radius. www.ltscotland.org.uk/Images/volumesofsolids_tcm4-123355.ppt

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Formulas Volume of a Cone: V= 1/3 r 2 h Volume of a Cylinder: V = r 2 h

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Example #1 Calculate the volume of: V= 1/3 r 2 h V= 1/3 ( )(7) 2 (9) V = 147 m 3

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Example #2 Calculate the volume of: V= 1/3 r 2 h V= 1/3 ( )(5) 2 (12) V = 100 cm 3

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An ice cream cone is 7 cm tall and 4 cm in diameter. About how much ice cream can fit entirely inside the cone? Find the volume to the nearest whole number. r = = 2 d2d2 V = πr 2 h 1313 V = π(2 2 )(7) 1313 V ≈ 29.321531 About 29 cm 3 of ice cream can fit entirely inside the cone. Example #3:

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Compare Compare a Prism to a Pyramid. Make a conjecture to what the formula might be for Volume of a Pyramid.

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Formulas Volume of a Prism: Volume of a Pyramid: V = 1/3 Bh

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Example #4 Calculate the volume of: 10” 15” V = 1/3 Bh V = 1/3 (10 2 )(15) V = 500in 3

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Find the volume of a square pyramid with base edges 15 cm and height 22 cm. Because the base is a square, B = 15 15 = 225. V = Bh 1313 = (225)(22) 1313 = 1650 Example #5

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Find the volume of a square pyramid with base edges 16 m and slant height 17 m. The altitude of a right square pyramid intersects the base at the center of the square. Example #6 Because each side of the square base is 16 m, the leg of the right triangle along the base is 8 m, as shown below.

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Step 1: Find the height of the pyramid. 17 2 = 8 2 + h 2 Use the Pythagorean Theorem. 289 = 64 + h 2 225 = h 2 h = 15 Example #6, continued Step 2: Find the volume of the pyramid. = 1280 V = Bh 1313 = (16 x 16)15 1313

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