Volumes of Pyramids and Cones Geometry 10 6

• The volume of a cube is the cube of the length of

• The volume of a solid is the sum of the volumes of

• The volume V of a prism is V = Bh, where B

• The volume V of a cylinder is V = Bh = πr

• If two solids have the same height and the same cross-sectional area

• Get your supplies – Paper – Scissors – Marker or color pencil

• Draw a net for a six sided prism on your paper •

• Using one side for a base, take your prism, and outline and

• Using an uncolored side as a base, make another right pyramid with

• What is left, that has not been colored in? • Do you

• Would this work even if the base of the pyramid was not

• The volume V of a pyramid is V = 1/3 Bh where

• When we solved for the volume of a pyramid • Did it

• What if we kept increasing the number of sides of the pyramid,

• The volume V of a cone is V = 1/3 Bh V

• Pages 554 – 557 • 6 – 18 even, 19, 22, 23,

Slides: 46

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Volumes of Pyramids and Cones Geometry 10 -6

Review

• The volume of a cube is the cube of the length of its side, or V=s 3 Volume of a cube

• The volume of a solid is the sum of the volumes of all its non-overlapping parts Volume Addition Postulate

• The volume V of a prism is V = Bh, where B is the area of a base and h is the height Volume of a Prism

• The volume V of a cylinder is V = Bh = πr 2 h where B is the area of a base, h is the height, and r is the radius of a base Volume of a Cylinder

• If two solids have the same height and the same cross-sectional area at every level, then they have the same volume Cavalieri’s Principle

New Material

• Get your supplies – Paper – Scissors – Marker or color pencil Pyramid Exploration

• Draw a net for a six sided prism on your paper • Cut it out • Fold and tape it into a prism Pyramid Exploration

• Using one side for a base, take your prism, and outline and color a section that would be a right pyramid with the same height as the prism • Color the sides that would be part of the pyramid, as shown Pyramid Exploration

• Using an uncolored side as a base, make another right pyramid with the same height as the prism • Color the sides that would be part of the pyramid, a different color Pyramid Exploration

• What is left, that has not been colored in? • Do you think this would work even if the pyramids were not right pyramids? Pyramid Exploration

• Would this work even if the base of the pyramid was not a rectangle? Pyramid Exploration

• The volume V of a pyramid is V = 1/3 Bh where B is the area of a base, h is the height Volume of a Pyramid

• When we solved for the volume of a pyramid • Did it matter how many sides the pyramid had? Cone Volume

• What if we kept increasing the number of sides of the pyramid, and the corresponding prism • what shapes do they become? Cone Volume

• The volume V of a cone is V = 1/3 Bh V = 1/3 πr 2 h where B is the area of a base, h is the height and r is the radius of the cone Volume of a Cone

Example

Sample Problems

64 units 3

70 2/3 cm 3

357. 24 in 3

64 units 3

70 2/3 cm 3

357. 24 in 3

Practice Problems

• Pages 554 – 557 • 6 – 18 even, 19, 22, 23, 43 Homework

• Pages 554 – 557 • 6 – 18 even, 19, 22, 23, 34, 43 Honors Homework