# At a Glance - The Third Dimension

Up until now, we've explored the world in two dimensions. Area, shapes, and x-y plots are all good and fine, but they're a little flat, aren't they? After all, you can't really play Minecraft in two dimensions. If we crank geometry into third gear, we'll go from kiddy cartoons to sophisticated computer animation, like Paper Mario plopped into a 3D Mushroom Kingdom.

The first step in three-dimensional space is knowing how to look at it. Yes, with your eyes, smarty-pants, but it's more than just that. We may think we see the world in three dimensions, but we really only see it in two. What a bust. That means we need to come up with ways to draw 3D space on a piece of 2D paper.

#### Example 1

 Sketch the orthogonal drawings from the front, right, left, and top.

#### Example 2

 Identify the solid and name each base, face, edge, and vertex.

#### Example 3

 Which cross-sections will result if this solid is split by a perfectly vertical and perfectly horizontal planes through the center?

#### Example 4

 Draw a net for the solid and calculate its surface area.

#### Exercise 1

Draw orthogonal drawings of the following solid from the top, front, left, and right viewpoints.

#### Exercise 2

Draw the perspective view of the following solid given its orthogonal drawings.

#### Exercise 3

Euler's formula says that the number of faces plus the number of vertices in a polyhedron equal the number of edges plus 2. Use Euler's formula and see if it's true for these two solids. State why it does or does not apply to each solid.

#### Exercise 4

Name and draw three kinds of solids that have 6 faces.

#### Exercise 5

Draw a net for the following solid.

#### Exercise 6

Identify the following solid and find the surface area using its net.

#### Exercise 7

Identify the following solid and find the surface area using its net.

#### Exercise 8

It's your little sister's birthday and you decide to buy her a cake. She still remembers how you massacred her Furby, so it'd better be a big cake. You decide to buy her this monstrous cake.

You want to cover the entire cake (even the bottom) with a layer of icing because, let's face it, icing is delicious. How much surface area do you have to cover with this icing (to the nearest square inch)?

#### Exercise 9

Your sister was really attached to that Furby, so you decide to buy her a present too. You chose a My Little Pony that fits in a cubic box that's 10 inches on each side. How many square inches of wrapping paper do you need to perfectly cover the gift box?