# Surface Area and Volume

### Topics

One way to represent 3D figures is with an **orthogonal drawing**, which is a fancy word for drawing all the different sides of the three-dimensional shape. That means a cube would look like this:

It's useful, because we can figure out the actual 3D shape if we have the orthogonal drawings. But if you ask us, it's a little repetitive and not very interesting. We signed up for 3D, and by golly, we're going to get 3D.

Our eyes see shapes with perspective (3D glasses optional), so what better way to draw these shapes than through **corner** or **perspective drawings**?

Telling you that the picture above you is a cube would be unnecessary. You know it's a cube. You've seen it a million times in your freezer's ice tray. But that's exactly the point: to draw shapes the way we see them.

So what if we had orthogonal drawings and we wanted to find out the perspective drawing of the shape? Let's say we have this orthogonal drawing:

The thin lines on the front and top indicate some kind of change in the surface when we look at the image from those viewpoints. The side view doesn't have any change. If we look at the images and piece them together like Legos, we should get a perspective drawing that looks like this:

With a little practice, you'll be able to translate orthogonal drawings to perspective drawings faster than you can say, "translate orthogonal drawings to perspective drawings."

So what kinds of things are three-dimensional shapes? Well, lots! Anything that exists in the real world is a three-dimensional shape. Like your computer. And your eyeball. And your little sister's Furby that you're convinced is going to kill you in your sleep.

Terrifying, isn't it?

But before we go and classify Furbies as anything at all, we should start with the basics of the third dimension.

If we have a solid with all flat surfaces that completely encloses a region of space, we call it a **polyhedron**. "Poly-" means many and "-hedron" means face.

The line segments of intersection are called **edges** and the points are called **vertices**.

A **regular polyhedron** has all regular congruent polygons for faces, and the edges are all congruent. There are only five types of regular polyhedrons, called the **Platonic solids**, named after the Ancient Greek philosopher, Plato, who also invented the plate (except not really).

A **prism** is a special type of polyhedron with two parallel congruent faces called **bases**. The other sides are parallelograms. If a prism is **regular**, that means its bases are regular polygons.

A **pyramid** has one base with all the other faces intersecting at one vertex. Some pyramids are named after a city or civilization (the Cairo Pyramids, the Mayan Pyramids, etc.), but we'll name our pyramids for their bases.

Since the base for this pyramid is a square, it's a square pyramid.

Some solids aren't polyhedrons. These are shapes that have faces that aren't flat like polygons. They're round or curved.

A **cylinder** is a pair of congruent circular bases on two different parallel planes. A **cone** is what you eat your ice cream in, a circular base with a single vertex. A **sphere** is a ball, but its official definition is a set of points equidistant from a given point in 3D space. They're all drawn below.

These perspective drawings of the shapes are their ID badges. If we see one, we should be able to identify it without any problems.

### Sample Problem

What is the name of the shape? What are the faces, vertices, and edges?

The faces are the sides: ∆*ABC*, ∆*ACD*, ∆*ADB*, and ∆*BCD*.

The vertices are just *A*, *B*, *C*, and *D*.

The edges are the line segments *AB*, *AC*, *AD*, BC, *BD*, and *CD*.

Since all the faces are congruent equilateral triangles, this shape would be a regular polyhedron (one of the five Platonic solids). Since it has four faces total, it's called a tetrahedron. ("Tetra" means 4. And don't you tetra-get it.)

If you are still confused on 3D shapes, check out our video right here to help you out: