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=s3

Volume Addition Postulate 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

Volume of a Cylinder The volume V of a cylinder is V = Bh = πr2h where B is the area of a base, h is the height, and r is the radius of a base Volume of a Cylinder

Cavalieri’s Principle 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

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

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

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

Volume of a Pyramid 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

Volume of a Pyramid 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

Cone Volume 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

Volume of a Cone The volume V of a cone is V = 1/3 Bh V = 1/3 πr2h where B is the area of a base, h is the height and r is the radius of the cone Volume of a Cone

Volume of a Cone The volume V of a cone is V = 1/3 Bh V = 1/3 πr2h 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

Sample Problems

64 units3

70 2/3 cm3

357.24 in3

64 units3

70 2/3 cm3

357.24 in3

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, 43 Homework