If you care about angles, you're going to love this. If you don't care that much, watch it once so that someday when you want to learn it, you'll know where to find it. Also, I'm going to tell a joke at the end that you won't get unless you sit through the whole thing and pay attention.

Using Triangles for Common Angles

The 90-Degree Angle

The zero to 180-degree line is straightforward. For the 90-degree line, you can use either triangle.

The 45-Degree Angle

With the 45-degree triangle, you get 45 degrees right away. Here's a little special something, if you flip it the other way, it gives you 135 degrees. We added 90 degrees, and came up with 135. You don't have to do this math, I'm doing it for you.

The 60-Degree and 30-Degree Angles

Let's use the 60-degree angle. It gives us 60 degrees. It also gives us 120 degrees, double what it was. And this 60-degree triangle doubles as a 30-degree triangle, which gives us 30 degrees. If we flip it, we get 150 degrees, an extra 120, which was the result of that double 60.

Combining Triangles for Additional Angles

We're mostly done, but there are four angles that take two triangles. Sometimes it takes a couple. There's an old proverb in Ecclesiastes about this, but I won't quote it because this is a perspective class. You can look it up, something about two are better than one.

Getting 15 Degrees

How can we get a 15-degree measurement with a 45 and 30 triangle? The 45-degree triangle gives us 45 degrees. What if we slant it back and remove 30 degrees from 45? It worked, we got 15 degrees. I bet if we flipped it, we'd get the minus 15 from 180 over here, which gives us 165 degrees.

Getting 75 Degrees

We have one key position left: 75 degrees. What if we add 30 and 45? It adds up to 75. We can flip it the other way and get its counterpart of 105 degrees. Fifteen degrees below 90 over here at 75, fifteen degrees above 90 over here at 105. Success!

The Pizza Wedge Analogy

Now, you may wonder, do you have to memorize all these numbers? You don't. This is absolutely voluntary. You can be good at freehand perspective without knowing these, but they help you see, and it's not that much work to get familiar with the 15-degree increments around the circle.

• The 90-degree wedge is easy.
• The 45-degree wedge is a good-sized piece of pizza.
• The 30-degree wedge is when you're trying to cut back.
• The 15-degree wedge is for when you're in trouble or guilt over eating.
• The 60-degree pizza is indulgent.
• The 75 degrees is kind of hogging it.

What about the rest? I don't have images for them, but your generation should do really well in developing vivid images for these higher numbers that previous generations like mine couldn't handle.

Do You Need to Memorize All These Angles?

As I said, you don't need to know all this. But if you get stranded on a desert island and need to build things for your own survival and you can't do it because you didn't take this seriously... Well, I'm not saying you deserve to die. But think about it.

A (Failed) Joke

Now for a joke.

An artist, an art teacher, and an engineer walk into a bar. The artist orders wine. The art teacher orders coffee. The engineer pulls out, oh wait, that was the wrong joke. Sorry, I'll start again.

An artist, an art teacher, and an engineer on a desert island are trying to build a shelter, so they can survive. I better not tell that one either, it's too violent.

Here, this will be the one.

A math-challenged art student tried to memorize the 12 line angles of the half circle in 15-degree increments. They got confused about the 45 to 15-degree angle and why the 30 and 45 add up to it, and how the difference between 75 and 105 hardly seemed analogous to the 15 and the 165. But then after watching this video 14, 13 times, taking notes, and doing repetitive memorization exercises, it stuck in their brains in a constant image that made them unable to carry on normal conversations. But they did find a way to make a living, aligning broadcast patterns on antique television sets.

Oh, I suck at jokes.

Anyway, thank you for watching.

Join the premium course for additional lessons, demos, critiques, and more!