3D shapes

We live in a three-dimensional world. Every object you can see or touch has three dimensions that can be measured: length, width, and height. The room you are sitting in can be described by these three dimensions. The monitor you're looking at has these three dimensions. Even you can be described by these three dimensions. In fact, the clothes you are wearing were made specifically for a person with your dimensions. 

In the world around us, there are many three-dimensional geometric shapes. In these lessons, you'll learn about some of them. You'll learn some of the terminology used to describe them, how to calculate their surface area and volume, as well as a lot about their mathematical properties. 

3D Shapes

There are many types of three-dimensional shapes. You've surely seen spheres and cubes before. In this lesson, you'll learn about polyhedra 

— three-dimensional shapes whose faces are polygons — and you'll also learn about two special types of polyhedra: prisms and pyramids.

Polyhedra

A die is in the shape of a cube. A portable DVD player is in the shape of a rectangular prism. A soccer ball is in the shape of a truncated icosahedron. These shapes are all examples of polyhedra.

A three-dimensional shape whose faces are polygons is known as a polyhedron. This term comes from the Greek words poly, which means 

"many," and hedron, which means "face." So, quite literally, a polyhedron is a three-dimensional object with many faces. 

The faces of a cube are squares. The faces of a rectangular prism are rectangles. And the faces of a truncated icosahedron are pentagons and hexagons — there are some of each.

The other parts of a polyhedron are:

•  its edges: the line segments along which two faces intersect

•  its vertices, the points at which three or more faces meet

Prisms

Pyramids

A pyramid is a polyhedron for which the base is a polygon and all lateral faces are triangles. In this lesson, we'll only concern ourselves with pyramids whose lateral faces are congruent — that is, they're the same size and shape.

Technically, when the lateral faces are congruent triangles, the shape is known as a right pyramid, indicating that the apex — the vertex at which the lateral faces meet — is directly above the center of the base. In this lesson, when we use the term pyramid, we mean a right pyramid. But there are other types of pyramids, too.

A pyramid is typically described by the shape of its base. For instance, a triangular pyramid has a base that is a triangle, and a hexagonal pyramid has a base that is a hexagon.

ACTIVITY 2: EXPLORE PYRAMIDS

Search and find two pyramids and complete this table in your notebook:

ACTIVITY 3: EULER THEOREM

Now, you know about polihedra that: All of the faces must be polygons. Two faces meet along an edge. Three or more faces meet at a vertex. 

In this activity, you'll discover a property of polyhedra known as Euler's Theorem, because it was discovered by the mathematician Leonhard Euler (pronounced "Oil-er"). You already know that a polyhedron has faces (F), vertices (V), and edges (E). But Euler's Theorem says that there is a relationship among F, V, and E that is true for every polyhedron. That's right — every polyhedron, from a triangular prism to a hexagonal pyramid to a truncated icosahedron. 

Platonic Solids

In this task on three-dimensional solids, you've seen a lot of polyhedra. But there are five special polyhedra — known collectively as the Platonic solids — that are different from all the others.