## Learning Math: Geometry

# Solids

## Explore various aspects of solid geometry. Examine platonic solids and why there are a finite number of them. Investigate nets and cross-sections for solids as a way of establishing the relationships between two–dimensional and three–dimensional geometry.

### In This Session

**Part A: **Platonic Solids

**Part B: **Nets

**Part C: **Cross Sections

Homework

In this session, you will build solids, including Platonic solids, in order to explore some of their properties. By creating and manipulating these solids, you will develop a deeper sense of some of the geometric relationships between them.

For information on required and/or optional materials for this session, see **Note 1**.

### Learning Objectives

In this session, you will do the following:

- Learn about various aspects of solid geometry
- Explore Platonic solids and why there is a finite number of them
- Examine two-dimensional properties of three-dimensional figures such as nets and cross sections

### Key Terms

**Previously Introduced**

**Congruent: **Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they “are the same size and shape,” though they may have different orientation. (One might be rotated or flipped compared to the other.)

**Regular Polygon: **A regular polygon has sides that are all the same length and angles that are all the same size.

**Vertex: **A vertex is the point where two sides of a polygon meet.

**New in This Session**

**Cross Section: **A cross section is the face you get when you make one slice through an object.

**Edge: **An edge is a line segment where two faces intersect.

**Face: **A face is a polygon by which a solid object is bound. For example, a cube has six faces. Each face is a square.

**Net: **A net is a two-dimensional representation of a three-dimensional object.

**Platonic Solid:** A Platonic solid is a solid such that all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.

**P****olyhedron: **A polyhedron is a closed three-dimensional figure. All of the faces are made up of polygons.

### Notes

**Note 1**

Materials Needed:

- Polydrons or other snap-together regular polygons
- clay
- dental floss or piano wire
- molds for a cube, tetrahedron, cylinder, cone, and sphere (optional)
- a party hat
- scissors

**Polydrons**

You can purchase Polydrons from the following source:

Polydron International Limited

Tel: 0044 (0)1285 770055

Fax: 0044 (0)1285 770171

http://www.polydron.com

An alternative to purchasing Polydrons is to make cutouts of regular polygons from stiff paper or poster board and use tape to attach them. Each individual working alone or pair of participants will need at least 32 triangles, 12 pentagons, six squares, and three hexagons, but it’s helpful if each pair has extra sets of each kind of polygon.