# Lesson Explainer: 3D Shapes Mathematics

In this explainer, we will learn how to name 3D shapes including pyramids and prisms; describe 3D shapes by properties such as the number of edges, faces, and vertices; and build 3D shapes.

Three-dimensional (3D) figures extend, as the name suggests, in three dimensions. In contrast to two-dimensional (2D) shapes that can be drawn as such on a paper, 3D figures cannot. We can only draw them using perspective to suggest the illusion of their three dimensions.

### Useful Language

A 3D figure is delimited by its faces, edges, and vertices.

are 2D (flat) shapes that comprise the 3D figure.

are the lines of intersection of two faces.

are points where 3 or more faces end.

Note the different dimensions here: a face is 2D, an edge is 1D, and a vertex is 0D.

Let’s look at the definition of two important 3D figures: the prism and the pyramid.

### Definition: Prism

A prism is a solid object with two parallel, congruent faces called bases. Its cross section parallel to the bases is constant throughout its height (also called length).

### Definition: Pyramids

Pyramids are three-dimensional geometric shapes, or solid objects, where the base is a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles that meet at the apex or vertex.

A right pyramid is a pyramid whose apex lies above the centroid of the base.

A regular pyramid is a right pyramid whose base is a regular polygon: all the sides of the base are of equal length, and all the pyramid’s lateral edges are of equal length.

Prisms and pyramids are called after their base: a prism with a triangular base is called a triangular pyramid; a pyramid whose base is a pentagon is called a pentagonal pyramid. The following table gives the correspondence between the name of the base and the adjective used to describe the prism or pyramid.

Shape of BaseName of PrismName of Pyramid
TriangleTriangular prismTriangular pyramid
SquareCuboidSquare pyramid
RectangleCuboidRectangular pyramid
ParallelogramParallelepipedParallelogram-based pyramid
TrapezoidTrapezoidal prismTrapezoidal pyramid
PentagonPentagonal prismPentagonal pyramid
HexagonHexagonal prismHexagonal pyramid
HeptagonHeptagonal prismHeptagonal pyramid
OctagonOctagonal prismOctagonal pyramid

Let’s look at some questions to test our understanding.

### Example 1: Identifying the Vertices of a 3D Shape

How many vertices does the shape have?

The vertices are the points where 3 or more faces end. We find there are 5 of them.

### Example 2: Identifying the Faces of a 3D Shape

How many faces does the shape have?

The faces of a 3D figure are its flat parts. We count 6 of them here.

### Example 3: Identifying the Edges of a 3D Shape

How many edges does the shape have?

Edges of a 3D shape are the intersection lines between two faces. We count 12 of them.

### Example 4: Identifying a 3D Shape

For the figure below, identify the shape of the base(s). Then, classify the figure.

This 3D shape does not have a base and a vertex but has two parallel congruent faces, which are the bases. The shape of the bases is a quadrilateral that has two parallel sides: it is a trapezoid. The 3D shape is, therefore, a trapezoidal prism.

Sometimes, we are asked to answer questions which do not provide an image to help us. In these cases, it is often helpful to draw our own figure to help us address the question. In the next example, we will see one such question.

### Example 5: Properties of Three-Dimensional Shapes

Determine whether this statement is always, sometimes, or never true: A prism has 2 bases and 4 sides.

Let us consider some prisms. Starting with a triangular prism, we have two triangular bases (one at the top and one at the bottom). Then, for each edge of the triangular base, we have a side. Therefore, we have three sides, as shown in the figure.

This means that, for a triangular prism, we do have 2 bases, but we do not have 4 sides, so the statement is not true for a triangular prism.

Let us now consider a square prism, also known as a cuboid. It has two square or rectangular bases, and for each of the four edges of the base there is a side. Therefore, we have two bases and four sides.

Therefore, for a cuboid, the statement is true. Therefore, we can say that the statement “A prism has 2 bases and 4 sides” is sometimes true.

### Key Points

• 3D figure is delimited by its faces, edges, and vertices. are 2D (flat) shapes that compose the 3D figure.
are the lines of intersection of two faces.
are points where 3 or more faces end. Note the different dimensions here: a face is two-dimensional, an edge is one-dimensional, and a vertex is zero-dimensional.
• A prism is a solid object with two parallel, congruent faces called bases. Its cross section parallel to the bases is constant throughout its height (also called length).
• Pyramids are three-dimensional geometric shapes, or solid objects, where the base is a polygon (triangle, square, rectangle, pentagon, etc.) and all other sides are triangles that meet at the apex or vertex. A right pyramid is a pyramid whose apex lies above the centroid of the base.
A regular pyramid is a right pyramid whose base is a regular polygon: all the sides of the base are of equal length, and all the pyramid’s lateral edges are of equal length.