How to Calculate the Radius of a Circumscribed Circle of an Equilateral Triangle with Side Length (x) and Why You Should Care.

So, do I win the award the award for the most objectionable blog post title ever?

What? There’s no award? What a waste.

Well, anyway, one of my dear readers (and you are dear, trust me, I can count you on my fingers and toes) asked me about how I located the peg holes on the underside of the top of my sassafras kitchen table (from this post). The intuitive way would be to flip the table upside-down and set the base on top, then push it around until the distance from the point atop the legs to the nearest edge of the tabletop is equivalent for each leg.

equilateral
The tops of the legs form an equilateral triangle. Your mission, should you choose to accept it, is to mark the corners of a congruent triangle that is precisely centered on the underside of the tabletop.

That would work, I suppose, but it would be finicky and prone to error. As my reader surmised, there is indeed an easier, more precise, more elegant way. First, we just need to envision that the top corners of the three legs form an equilateral triangle. The length of the sides is easy enough to measure. In this case, it was 28.75″ precisely.

Let’s call the length of the side (x). If you recall from high school geometry, there is a fixed relationship between the length of the sides of an equilateral triangle and the radius of a circle – we’ll refer to the radius as (y) – that passes through all three corners (i.e., a circumscribed circle).

That relationship is: y = x (√3) ÷ 3

circle 001
Mmm…numbers. For the record: No, I didn’t remember this formula. I had to look it up.

So, we just plug in 28.75 for (x) and we get 16.598820…Let’s call it 16.6″. I don’t have a ruler that’s marked in tenths of an inch, so I just used 16 38/64 (16.594″). I adjusted my trammel to the radius and scribed a circle on the on the underside of the table (you can see it on the picture above).

Okay, now we have a circle of the proper radius that is perfectly concentric to the edge of the table, but we still need to accurately locate the three corners of the triangle. Well, that’s dead-simple now that we have a trammel set to the radius. Just pick a point on the perimeter and start “walking” the trammel around the perimeter, making a mark at each intersection. You should end up at the same point that you started at (or very close to it) with six equally spaced marks around the circumference of the circle, forming a perfect hexagon. Drill a hole in every other point, and there you have your perfect equilateral triangle.

hexagon
Mathematical constants are fun and useful!

So there you have it. Hopefully I made the process tolerably clear – it took far longer to write about it than it did to actually complete it.

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2 thoughts on “How to Calculate the Radius of a Circumscribed Circle of an Equilateral Triangle with Side Length (x) and Why You Should Care.

  1. I would just consider each side a cord of the circle and do a perpendicular bisect of 2 sides to find the circle center and radius. Where the bisectors intersect is the circles center. Then just set your compas from the center/intersect to one of your triangles cornors and swing your circle…no math needed!!! Simple geometry.

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    1. That would certainly do the trick, the only problem would have been finding a surface big enough to lay out the geometry! I generally try to avoid numbers while woodworking, but I don’t mind them when it’s expedient to use them. I found it easy just to do the calculation while sitting at my bench. Thanks for the alternative viewpoint, though!

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