Attributes of Solids: Flat Faces, Curved Surfaces, Edges - First Grade Mathematics

Just like flat 2D shapes have sides and corners, solid 3D shapes have their own structural parts that mathematicians count and describe. When you pick up a wooden solid block, your fingers can touch flat surfaces, smooth curved surfaces, sharp straight edges, and pointy corner vertices. Knowing how to identify and count these attributes allows you to classify any solid object you encounter in the world. In this chapter, you will master the defining attributes of 3D solids: faces, curved surfaces, edges, and vertices.

What Is a Face?

A face is an individual, flat surface on a 3D solid.

The Faces of a Cube:
          +-------+   <- Top Face
         /       /|
        +-------+ |   <- Right Face
        |       | +
        | Front |/
        +-------+
       <- Bottom Face (underneath)

The 2D Shape of a 3D Face

Every flat face on a 3D solid is shaped like a 2D polygon! On a cube, every face is a 2D SQUARE (6 square faces). On a cylinder, the top and bottom faces are 2D CIRCLES (2 circular faces). On a rectangular prism, the faces are 2D RECTANGLES (6 rectangular faces).

What Is an Edge?

An edge is a straight line where two flat faces meet.

Tracing an Edge on a Cube:
              Edge (Top Front)
        +-------------------+
       /|                  /|
      +-------------------+ |
      | |                 | |
      | +-----------------|-+
      |/                  |/
      +-------------------+

Feeling the Edge

If you slide your hand across the front face of a block and reach the boundary where it turns the corner to the top face, your hand crosses an edge. A cube has 12 straight edges! A sphere and a cylinder do not have straight polygon edges.

What Is a Vertex on a Solid?

A vertex on a 3D solid is a sharp point where three or more edges come together.

A Vertex of a Cube:
                 Vertex (3 edges meet here!)
                   (o)------+
                   /|      /|
                  +-------+ |
                  | |     | |

Plural: Vertices

Just like with 2D shapes, more than one vertex is called vertices. A cube has 8 vertices: 4 on the top and 4 on the bottom!

The Complete 3D Attributes Reference Table

3D Solids Attribute Breakdown:
Solid Shape    Flat Faces   Curved Surfaces   Straight Edges   Vertices
Cube               6               0                12            8
Sphere             0               1                 0            0
Cylinder           2               1                 0            0
Cone               1               1                 0            1 (Apex)
Rectangular Prism  6               0                12            8

Practice Exercises

Exercise 1: Define what a face, an edge, and a vertex are on a 3D solid shape.

Exercise 2: How many flat faces, straight edges, and vertices does a cube have? What 2D shape is each face of a cube?

Exercise 3: How many flat faces and curved surfaces does a cylinder have? What 2D shape are its flat faces?

Exercise 4: Which 3D solid shape has zero flat faces, zero edges, and zero vertices?

Solutions and Explanations

Solution to Exercise 1: A face is a flat 2D surface of a solid. An edge is the straight line where two flat faces meet. A vertex is the sharp point where three or more edges intersect.

Solution to Exercise 2: A cube has 6 flat faces, 12 straight edges, and 8 vertices. Every face of a cube is shaped like a 2D square.

Solution to Exercise 3: A cylinder has 2 flat faces and 1 smooth curved surface. Its two flat faces are shaped like 2D circles.

Solution to Exercise 4: A sphere has zero flat faces, zero straight edges, and zero vertices; it consists entirely of one continuous curved surface.