Three-Dimensional Drawing

It is very useful to be able to create three-dimensional drawings in Geometry. Many geometric figures are three-dimensional, especially when you are studying volume and surface area of cubes, pyramids, cones and spheres.

There are a number of different ways to draw three-dimensional objects. Examples of a cube drawn using four different types of drawings are named and illustrated below:

Of the four types of 3D drawing shown above, the best choice for drawing geometric solids is the Isometric, as it shows the solid fairly realistically and is easy to draw. You will see an Isometric Grid below, which you can print and use for your own drawing projects. Place a piece of tracing paper over the grid, and you can then experiment with drawing some 3D figures yourself.

The drawing below is an example of a brick (or box) drawn using an isometric grid. You can practice your 3D drawing-by-drawing simple solids, such as this one.

Another interesting 3D project is one I call "How Many Planes. This is a good introduction to three-dimensional visualization and drawing for students who may never have done any of this type of drawing.

Some students have trouble with 3-dimensional geometry. In this worksheet, they will gain expertise in interpreting a 3-dimensional diagram, and visualizing (as well as drawing) planes. The more that students see and draw in 3 dimensions, the more comfortable they will be with this concept.

The ability to understand, and to draw 3-dimensional diagrams will be of great benefit to the students when they are studying volume and surface area of prisms, pyramids, cylinders and cones.

The project begins with a question: How many planes are determined by the vertices of a cube? Use the drawing below to help you find the answer. Find as many planes as you can. Shade each plane in on each one of the cubes below. Group them in categories and label each category at the top of its row. There are more cubes drawn than you will need.

The answer to this question can be found at the bottom of this web page.

The drawing below shows a geometric castle. The main part of the castle is a rectangular prism, and the four towers are, starting with the front left: a square-based prism with a pyramid on top, a cylinder with a hemisphere on top, a triangular prism with a pyramid on top, and a cylinder with a cone on top. The smaller drawing below shows you the castle from above. Now, my challenge to you is to use the formulas for volume and surface area to find the total volume and surface area of this castle! I would suggest you do each geometric solid separately (of course), show all your work, and keep them in an organized fashion on your paper. You will find the answers at the bottom of this web page - but don't look until you are finished!

And you can learn more about 3D drawing at the following web address: http://mathforum.org/workshops/sum98/participants/sanders/Persp.html. You will find the answers to the castle area and volume on the next web page "(8.2 Hexatrapezoidal Solid"): 8.2hexatrapezoid.htm

Here is the answer to the question "How many planes":