Volume Polyhedra A

• “poly” means “many” • “hedron” means “base” or “seat”

• A polyhedron is a solid in 3 dimensions with: – flat polygonal faces – straight edges Regular Tetrahedron Small Stellated Dodecahedron Icosidodecahedron Great Cubicuboctahedron Rhombic Triacontahedron Octagonal Prisms

• A prism is a polyhedron with two congruent n-sided polygonal bases and n faces joining the corresponding sides Uniform Prisms Uniform Prisms Tetragonal Prism () Uniform Prisms Uniform Prisms Dodecagonal Prism What is “Volume”?

• In your table groups, agree on a single sentenc e or phrase that explains what “volume” is. Volume of a Prism

• Starting small… – What is the volume of a cube (tetragonal prism) with side lengths of 1 foot?

1 foot

1 foot 1 foot Volume of a Cube

• Now, generalize… – What is the volume of a cube (tetragonal prism) with side lengths of s feet?

s feet

s feet s feet Volume of a Rectangular Prism (also called a cuboid) • Now, generalize even further… – What is the volume of a rectangular prism with sides lengths l (for length), w (for width) and h (for height)? Volume of a Prism • More generalization… – How can we find the volume of any regular prism? Beyond Polyhedra

• What is the volume of a ? Beyond Polyhedra

• What is the volume of a generalized cylinder?

h Back to Polyhedra

• What is the volume of this oblique rectangular prism:

h

w l Beyond Polyhedra

• What is the volume of an oblique generalized cylinder?

h Pyramids

• A Pyramid is a polyhedron with a polygonal base (triangle, square, pentagon, …) that connects to a single point (called the apex) Pyramid Examples Parts of a Pyramid

The height of a pyramid is the (shortest) distance from the apex to the base. Volume of a Pyramid

To find the volume of a pyramid, we can use a Habit of a Mathematician and… Volume of a Pyramid

… take apart a cube

a

a a

This creates 3 congruent pyramids (that are known as yangmas) Volume of a Pyramid

The volume of one yangma must be 1/3 the volume of the cube!

1 a � = � a 3 a Scaling 3-D Shapes • If a I have a 1 x 1 x 1 cube, what happens to the volume when I – Double the length? – Triple the width? – Halve the height?

• If I have a cube with side lengths a, by what factor does the volume change if I scale the height to h?

h

a a a a a Volume of a Pyramid Of course, we can scale the height to any size, and the volume also scales. What is the volume of this pyramid?

h

a a Volume of a Pyramid

What is the volume of this pyramid?

1 � = �ℎ h 3

a a Volume of a Pyramid If we move the apex parallel to the base (so the height does not change), what is the volume of this new pyramid?

h

a a a a Volume of a Pyramid If we move the apex parallel to the base (so the height does not change), the volume does not change.

1 � = �ℎ h 3

a a a a The Calvalieri Principle

If two solids have the same cross- sectional area at every level of their height, then they have the same volume. Pyramid Volume How can we use the Calvalieri Principle to find the volume of any pyramid? Pyramid Volume

1 � = � ℎ 3 Volume of Cone

A cone can be thought of as a pyramid with a circular base. From the Cavalieri Priniciple, what is it’s volume? Volume of Cone

1 � = � ℎ 3 1 = ��ℎ 3