Volumes of Pyramids and Cones Geometry 10 6

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Volumes of Pyramids and Cones Geometry 10 -6

Volumes of Pyramids and Cones Geometry 10 -6

Review

Review

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

• 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

• 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

• 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

• 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

• 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

New Material

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

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

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

• 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

• 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

• 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

• 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

• 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

• 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

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

• 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

• 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,

• 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

• 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

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

• 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

Example

Example

Example

Example

Example

Example

Example

Example

Example

Sample Problems

Sample Problems

64 units 3

64 units 3

70 2/3 cm 3

70 2/3 cm 3

357. 24 in 3

357. 24 in 3

64 units 3

64 units 3

70 2/3 cm 3

70 2/3 cm 3

357. 24 in 3

357. 24 in 3

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

Practice Problems

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

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

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

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