How to Draw Perfect Angles
If you were handed a piece of string and a pencil, could you draw perfect (or nearly perfect) 30o and 60o angles? Why might this be a useful skill? Keep on reading to find out!
Jason Marshall, PhD
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How to Draw Perfect Angles
You never know when or where a bit of math know-how will come in handy.
Case in point: A few weeks ago I was over at a friend’s house helping run a ventilation duct through the attic (what fun!), when all of a sudden we realized we needed to have a portion of the duct run at an angle, and that the angle could not be tilted more than 60o from vertical.
The problem was neither my friend or I had a tailor-made 60o angle measuring device in our metaphorical toolbelts. Sure, we probably could have somehow used a tiny protractor (which we also didn’t have) to create some sort of angle measuring tool that was large enough to work with our several feet of duct.
But that would’ve been a pain. And, once I thought about it, I realized that it was a bunch of pain that we didn’t need to suffer through. Because we could instead use some simple geometry to custom build a perfect tool for the job. And the best part was we only needed string and a pencil to do it.
What did we do? And how can you use the technique we came up with to draw not only a 60o angle but a 30o angle too? Those are exactly the questions we’ll be answering today..
How to Draw a 60o Angle
To learn how to draw “perfect” angles, you really need to draw. Which means it’s time to gather up all the tools we’re going to need—a piece of paper, a pencil, a ruler, and some string (or a compass if you want to be fancy)—and then find a cozy place to do your angle constructing. OK, ready?
The first angle we’re going to construct is a 60o angle—just like the one my buddy and I had to make to get his ducts all lined up in a row (get it?). The first step is to draw a straight line—a line that we’ll eventually be measuring our angle from. You’ll probably want to use a ruler to do this in order to keep things nice and tidy.
Next, pin the end of your string down to one end of the line you’ve just drawn (you can use an actual pin to do this if you’d like, although your finger will probably do the job just fine). Then use the string like a compass (or actually use a compass) to draw an arc that starts somewhere near the other end of the line and extends all the way up to almost directly above the pinned down end of the string.
After you’ve got that first arc drawn, it’s time to move the end of the string you’ve had pinned down over to the point where the arc intersects the line. With the end of the string now pinned down at this point, draw another arc which starts from where your string was previously pinned to and sweeps up and over to intersect the first arc.
At this point we’re almost finished. All that’s left to do is draw another straight line from the point where you initially pinned your string to up to the point where the two arcs you’ve drawn intersect.
If you stare at the picture you’ve completed, you should find a 60o angle peeking out at you between the two straight lines. If you stare a little longer, you’ll see that if you were to draw another straight line from the point where your two arcs intersect down to the place where the first arc intersects the first straight line, you’d end up with an equilateral triangle.
An equilateral triangle is “equilateral” because all three of its angles are equal. And since the interior angles of a triangle add up to 180o, each of the angles of an equilateral triangle must equal 180 / 3 = 60o. If you think about it, you’ll see that the only way all three angles can be equal is if all three sides are equal in length.
Take a look back at the steps we used to construct our 60o angle. Do you see why the process worked? If you’re having trouble figuring it out, think about what we just said about an equilateral triangle—namely, the fact that its three sides must be equal in length. Look carefully and you’ll see that the steps we carried out were designed to do nothing more than construct this type of equal-sided triangle. Pretty clever, right?
But the cleverness doesn’t stop there – we can do more……
How to Draw a 30o Angle
In particular, we can use the 60o angle we’ve drawn to create a 30o angle. Given that 30 is half of 60, it’s probably not a big surprise that we’re going to create our 30o angle by cutting our 60o angle in half. As such, we need to start by drawing a 60o angle as we just learned to do.
Now hold one end of your string at the point where the first arc intersects the first straight line, and draw an arc that extend between the straight lines and passes up near the upper line. Then pin the end of the string down to the point on the second straight line where the two arcs from the 60o angle construction meet, and draw another arc between the two lines that intersects the new arc you just drew.
If you now draw a line from the vertex of your angle to the point where these two new arcs cross, you’ll have divided your angle in half. Which means the angle formed either above or below this central line is a 30o angle.
Again, pretty clever, right? And thanks to this bit of amazing geometry—done with nothing more than a little string and a pencil—my friend is now the proud owner of several beautiful feet of shiny duct that’s positioned at a near-perfect 60o angle. Thank you, math!
Wrap Up
OK, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Compass and ruler image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.