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# Drawing Solids and Their Nets

In this guided construction, we will draw a solid and its corresponding net. A net is a two-dimensional figure that can be cut out and folded up to make a three-dimensional solid. The figure on the left is the net of the solid on the right.

The picture we'll draw has isometric perspective, which means that all the edges are the same length as they are on the box, instead of changing them as in a regular drawing. The segments of the net are colored differently to indicate which ones will be shared when the net is foldedÑif you wanted to find the perimeter of the solid figure, you would count only the thick edges (why?).

### Drawing the Net

We're going to draw the net for a triangular prism. We'll draw a triangle and attach rectangles to the edges that will serve as the sides of the prism. These all need to be the same length. Then we'll attach another triangle which will serve as the bottom.
Draw a triangle. On one of the edges, draw a rectangle.

Construct rectangles on the other two sides which have the same height as the first rectangle.
Your picture should look something like this:

Now add the triangle on one of the rectangles that will form the top of our solid. The triangle needs to have edges that are equal to the edges of the original triangle. To do this, select one of the points on the end of a rectangle and shift-select one of the other edges of the original triangle. From the construct menu, choose "Construct by Center and Radius". This will give you a circle centered at one corner of the rectangle with radius equal to an edge of the original triangle. Do the same thing for the other side of the original triangle. Pay attention the order (the triangle has to be oriented properly for this whole thing to work).

Hide the circles. Now we have the whole net, which should look like the picture below.

We want to draw a picture of the three-dimensional solid that will be made from this net. Let's start by drawing a copy of the triangle. Do this by using the "Circle by center and radius" in the Construct menu.

Now we need to represent the rectangles in our solid. However, they will turn out as parallelograms! Choose the lead point of your new triangle and one of the sides of the corresponding rectangle. Construct a circle with that center and radius. Draw any segment from the center of the triangle to the edge of the circle and parallels through the other two points on the triangle (see first picture next page). Draw the bottoms of the rectangles in, making them parallel to the edge of the triangle. Add the other edge of the triangle in behind, making it dashed to show that it's hidden. See the pictures. Now move the vertices of your original triangle and see if the whole thing moves. Vague, but it's a start!

### Adding Bits on Area, Perimeter, and Volume

You could add some parts to this sketch that would show the area, perimeter, volume, and their ratios. Try it!