In your study of geometry so far, you have focused your attention on two-dimensional shapes. You have investigated the special properties of triangles, parallelograms, regular polygons and circles, and have developed tools to help you describe and analyze those shapes. For example, you have tools to find the interior angle of a regular hexagon, to calculate the length of the hypotenuse of a right triangle, and to measure the perimeter of a triangle or the area of a circle.

In this Chapter, you will focus on three-dimensional shapes (called solids), such as cubes and cylinders. You will learn several ways to represent three-dimensional solids and develop methods to measure their volume and surface area

In this chapter, you will learn:• how to find the surface area and volume of three-dimensional solids• investigate how prisms and cylinders relate• how to find the volume and surface area of a pyramid and a cone• how to represent a three-dimensional solid with a mat plan and a net• how to find the volume and surface area of a sphere• How the area, perimeter, surface area, and volume changes when a shape is enlarged proportionally

7.1

How Can I Name It?

Pg. 3Three-Dimensional Solids

7.1 – How Can I Name It?Three-Dimensional Solids

In the previous chapters, the shapes have been flat, two-dimensional objects. Today you are going to explore when an object is three-dimensional.

7.1 – EXPLORATIONIn the box of solids you will find 13 different shapes. Each shape has a different set of attributes.

a. Three-dimensional shapes formed with polygons have faces and edges, as well as vertices. Faces are flat surfaces of the shape, while edges are the line segments formed when two faces meet. Vertices are the points where the edges intersect. Examine the picture at right. Count how many faces, vertices, and edges it has.

6 8 12

b. Prisms have two congruent parallel faces, called bases. The above example is called a rectangular prism because it has two rectangles that are parallel and congruent. The faces connecting the bases are called the lateral faces. In prisms, the lateral faces are always rectangles. Shade the congruent bases of each prism. Determine the name of the base and then name the type of solid it is.

pentagon

rectangles

Pentagonal prism

rectangle

rectangles

rectangular prism

triangles

rectangles

triangular prism

hexagons

rectangles

hexagonal prism

c. When a solid only has one base it is called a pyramid. The lateral faces are triangles. Shade the one base of the pyramid. Then determine the name of the base, the type of lateral faces, and the name of the following solids.

rectangle

triangles

rectangular pyramid

pentagon

triangles

Pentagonal pyramid

7.2 – EULER'S THEOREMa. Count the number of faces, vertices, and edges on the shapes below. Then complete the chart for that shape. Look for a pattern in the values.

685

8125

12188

6 6 10

b. Euler found a pattern in the number of faces, vertices, and edges in a shape. He found if you add the number of faces and vertices, it will always equal the number of edges plus two. Return to part (a) and confirm that this formula works for all the shapes.

Faces + Vertices = Edges + 2

F + V = E + 2

7.3 – PRISM VS. PYRAMIDNow that you are more familiar with the solids, we are going to explore the difference between prisms and pyramids. Examine the two solids below. What is the differences between them? Describe in as much detail as possible.

• Two triangle bases

• One triangle base

• Rectangle lateral faces

• triangular lateral faces

• 5 faces, 6 vertices, 9 edges

• 4 faces, 4 vertices, 6 edges

7.4 – CIRCLES IN SOLIDSWhen a three-dimensional object has circles, the shape takes on a special name.a. What is a prism called when the bases are circles? What about pyramids with a circle base?

cylinder cone

b. What is a shape called that has no bases? What is it called when it is cut in half?

sphere hemisphere

7.5 – SORTINGExamine the 13 shapes in the box.

a. Sort the shapes by prism, pyramid, cylinder, cone, and sphere.

b. How many prisms are there?

5

c. How many pyramids are there?

4

d. How many cylinders are there?

1

e. What is a prism with square faces called?

cube

f. How many shapes will roll?

3

g. How many shapes can stack on top of each other?

6

h. What shape resembles a cereal box?

Rectangular prism

i. What shape resembles a can of soup?

cylinder