Geometry of Solids

How much deeper would oceans be if sponges didn’t live there?

Steven Write

Objectives

• Learn the vocabulary of polyhedrons• Learn the vocabulary of spheres, cylinders,

and cones. • Discover formulas for finding the volumes of

prisms and cylinders.

What is volume?

• Volume is the measure of the amount of ________ contained in a __________. space solid

Vocabulary

Term

• ______ Polyhedron• ______ Faces• ______ Edges• ______ Vertex• ______ Tetrahedron• ______ Regular Polyhedron

Definition• A) Each face is congruent to the

other faces. Faces meet at each vertex in exactly the same way.

• B) Solid formed by polygons that enclose a single region of space.

• C) Polygonal surfaces of a polyhedron.

• D) A polyhedron with four faces.• E) A segment where two faces

intersect.• F) Point of intersection of three of

more edges.

B

C

E

F

D

A

Vocabulary

Term• ______ Prism• ______ Altitude• ______ Pyramid• ______ Lateral Edges• ______ Right Prism• ______ Lateral Faces

Definition• A) A Polyhedron with one base.• B) Parallelograms that connect the

corresponding sides of the bases. • C) Segments where the lateral faces

meet. • D) A type of polyhedron with two

bases that are congruent, parallel polygons.

• E) Any perpendicular segment from one base to the plane of the other base.

• F) A prism whose lateral faces are rectangles.

D

E

A

C

F

B

Vocabulary

Term• _____ Sphere• _____ Cylinder• _____ Hemisphere• _____ Oblique Cylinder• _____ Great Circle • _____ Right Cone

Definition• A) The circle that encloses the base of

a hemisphere.• B) A type of solid with a curved

surface where the line segment connecting the vertex to the center of the base is perpendicular to the base.

• C) Cylinder that is not a right cylinder. • D) A solid with a curved surface that

has 2 bases that are parallel and congruent.

• E) The set of all points in space at a given distance from a given point.

• F) Half a sphere and has a circular base.

E

D

F

C

A

B

Label the Shape

• Bases• Lateral Faces• Lateral Edges• Vertex• What is this shape? Bases

Lateral Face

Lateral Edge

Hexagonal Prism

Vertex

Label the Shape

• Vertex• Altitude• Base• Radius• What is this shape?

Vertex

Altitude

RadiusBaseOblique Cone

Volume of Prisms and Cylinders• Investigation: The Volume Formula for Prisms and Cylinder

P. 514At each table there is a different right rectangular prism. You have 3 minutes to find the volume of the shape on your table, then we will switch tables. We will do this until each group has gone to each table.

Volume Chart

Shape Length Width Height Total Volume

Table 2

Table 3

Table 4

Table 5

Table 6

Use this table to organize the information you collect as you move from table to table.

Conjecture:If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula for the volume is V = ______.

Volume = (Length * Width) * Height Volume = (Area of Base) * HeightVolume = BH

BH

Volume of Prisms and Cylinders• Now continue the investigation with your groups to discover the volume

formula for a right prism or cylinder.

Conjecture:If B is the area of the base of a right prism (or cylinder) and H is the height of the solid, then the formula for the volume is V = _____.

Conjecture:The volume of an oblique prism (or cylinder) is the same as the volume of a right prism (or cylinder) that has the same __________ and the same ______.

BH

base area height

Prism-Cylinder Volume Conjecture

The volume of a prism or a cylinder is the __________ multiplied by the __________. base area height

Cylinder:

Right Trapezoidal prism:

Cube:

V = r2(H)

V = (½ * (b1+b2)h)(H)

V = (l * w) H

Exit ActivityTable 2 Table 3 Table 4 Table 5 Table 6 Class

Triangular Prism

Rectangular Prism

Pentagonal Prism

Hexagonal Prism

Octagonal Prism

N-gonal Prism

Lateral Faces

Total Faces

Edges

Vertices

Volume Formula

3

5

9

6

(½ * b * h)H

4

6

12

8

(l*w)H

5

7

15

10

(½*a*s*5 )H

6

8

18

12

(½*a*s*6 )H

8

10

24

16

(½*a*s*8 )H

n

n+2

3n

2n

(½*P*a )H