The Basics of 3-Dimensional Shapes

The shapes that we deal with in plane geometry (geometry contained within a plane) is 2-dimensional.  Shapes have length (y-axis) and width (x-axis).

But, of course, there is a whole world of 3-dimensional figures out there as well. Three dimensional shapes have one more dimension than 2-dimensional shapes: height (typically graphed on the z-axis, but that's beyond the scope of this lesson).

Before you work with three-dimensional shapes, you should learn the vocabulary we use to talk about them:

• Three dimensional (3d) figure: A figure with three dimensions (usually width, length, and height).
• Faces: All of the surfaces of a 3d figure.
• Edges: Lines where the surfaces (planes) of 3d figures intersect.
• Vertices (Vertex): Points where lines of a 3d figure intersect.
• Surface area: The sum of the areas of each face of a 3d figure.
• Lateral surface area: The surface area of just the sides, not the bases, of a 3d figure.
• Bases: The shapes that make up the top and bottom of a prism or cylinder, or the base of a pyramid or cone.
• Volume: The amount that a 3d figure can hold (usually area of the base, times the height).

There are several different types of 3-dimensional shapes.

Here are some common ones:

 Rectangular Prism Cube Cylinder Triangular Prism
• Rectangular prism: 3d rectangle. Bases have the same area.  Pairs of lateral faces have the same area. Volume = length x wide x height.
• Cube: 3d square.  All edges are equal; all faces have the same area. Because all sides are equal, volume=side x side x side. Surface area=side x side x 6 (because there are six sides).
• Cylinder: Bases are circles.  Lateral face is a rectangle (curved around). The two circles have the same area. There are only three faces here (two bases and wrapped rectangle).
• Triangular prism: Bases (which can be the top and bottom or the two ends of the prism) are triangles. Lateral sides are rectangles. Bases have the same area.

It's important to know some overall facts about how shapes are formed and made:

Prisms: What makes a prism is that there are two shapes (bases) that are the same size.  Those two bases are connected by faces (lateral faces).  Prisms are named after their bases.  So when two triangles are connected, it's called a triangular prism. When two rectangles are connected, it's called a rectangular prism. The distance between the bases is called the "height" even if the connection between the bases (as in the prisms below, is horizontal and not vertical). The trick to finding the bases is to find two shapes that are identical and connected (the squares at the left and right ends of the rectangular prism and the equilateral triangles at the ends of the triangular prism). Note that the triangular prism has rectangular faces, but triangular bases. There can also be octagonal prisms (octagons as bases -- sides are still rectangles), pentagonal prisms, etc.

Cylinders: These are essentially circular prisms. The bases are circles. The height of the cylinder is the distance between the circles (bases).  Note: the lateral side can be unwrapped into a rectangle.

Pyramids: Pyramids have a base. From the corner of each base a line goes upward and slants inward.  The lines all meet at a point.  If the base is a rectangle, it's a rectangular pyramid.  If the base is a triangle, it's a triangular pyramid.  The faces of a pyramid are always triangular. The height of a pyramid is the length of a line, that is perpendicular to the base, from the base to the point.

Cones: These are essentially circular pyramids.  The base is a circle. Height goes up from all around the base to a point.  The lateral face of a cone, if unwrapped, is a almost a triangle (it's a triangle with a curved base). The height of a cone is the length of a line, that is perpendicular to the base, from the base to the point.