View Full Version : Need a formula

Mark Pruitt
12-02-2006, 7:17 PM
An equalateral triangle sits in a circle with all three points touching the circumference. If R is the circle's radius, how long are the triangle's sides?

...and is there a website where info like that can be quickly obtained?

David Klug
12-02-2006, 7:24 PM
It depends on the diamiter of the circle.


Mark Pruitt
12-02-2006, 7:29 PM
It depends on the diamiter of the circle.

Yeah, I figured that I should be able to calculate it from the diameter (or the radius). In this particular case the diameter is 9 1/4". (Radius 4 5/8".)

Chuck Saunders
12-02-2006, 7:34 PM
Here is what I found
Circle circumscribed around an equalateral triangle

Lee DeRaud
12-02-2006, 7:45 PM
Chuck's answer looks good.

For stuff like this I usually hike upstairs and dust off the copy of CRC's math tables book I bought in 1968 as a freshman engineering student. (I had to do this for an octagon last spring...as I recall they have a table of constants for everything up to about 16 sides and a general formula for an N-sided polygon.)

Gawd I feel old. :eek:

Mitchell Andrus
12-02-2006, 7:45 PM
Good find, I'm bookmarking this one.


Mitchell Andrus
12-02-2006, 7:47 PM
My grey cells aren't what they used to be either. I got hung up on inscribed/circumscribed. - Points touching - Sheesh!!

<edit> Actually, the question is WRONG! The points don't touch if the circle sits IN the triangle.

Trick question?


Mark Pruitt
12-02-2006, 7:54 PM
<edit> Actually, the question is WRONG! The points don't touch if the circle sits IN the triangle.

Trick question?

My question stated that the triangle sits in the circle.

Mitchell Andrus
12-02-2006, 8:01 PM
Grey cells, half gone, half useless.

Correct. Triangle in circle. No WONDER I couldn't come up with the answer!!!!


Frank Chaffee
12-02-2006, 11:47 PM
If the heavy tomes are too far away and you are out of range of WiFi, pull out your scientific calculator ($10), and sketch the problem to be solved. The booklet that comes with the calculator will give examples of triangle solutions for any combination of a known angle and one side of a right triangle.

The Center of Gravity of any triangle is 1/3 the height above any base. Graphically this can be shown as the intersection of two (and three) lines from the center of any base to the opposite corner. This point coincides with the centerpoint of the describing circle.
• Punch “60” (be sure calculator is set to “degree mode”)
• Punch “sin”
• Punch “x” (times)
• Punch “4.625”
• Punch “=” (equals)
• Result is 4.0054
• Times 2 equals the length of each side of the equilateral triangle within a circle of 4 5/8 radius.

Believe me, if I can do this, anybody can.

Frank Chaffee

Jim DeLaney
12-03-2006, 8:05 AM
Frank is spot-on.

The actual formula is: 2(sin 60 X radius)

or just: sin 60 X diameter of circle

or: 0.866025 X diameter of circle.

Mark Pruitt
12-03-2006, 9:26 AM
Thanks! That was extremely helpful. I couldn't make the other solutions work for me last night, so I just used trial and error with a divider until I finally hit the target. It was frustrating that I could not simply calculate what I wanted. Next time I'll know.

John Schreiber
12-03-2006, 11:42 AM
I got the same answer a different way. Seeing the triangle inside a circle, you could tell that the distance from the center of the triangle to one of its points is the same as the radius of the circle.

Based on that radius and one of the sides of the triangle, you can create a 30 60 90 triangle.

30 60 90 is one of the "special" triangles which have fairly easy math where you can avoid looking up trigonometric functions.

Since we know the radius is 4 5/8 or 4.625, we can substitute that for n2. Based on that, the bottom of the triangle is 4.005.

Since what we want is the bottom of the original triangle, which is twice as long as the triangle we were working with, double that and you find that the length of each side of the triangle is 8.01 inches.

Now, I've spent much too much time repeating an answer which was given before, but the nice thing about it is that it can be simplilfied to: radius of the circle * 1.732 = side of the triangle.

Chris Padilla
12-04-2006, 5:06 PM
Allow me to be the last to beat this dead horse a bit more! :D

I stole John's triangle and added a bit to it as it is indeed one of those special triangles.

Often in woodworking, dimensions can be figured out geometrically if one can "see" a right triangle hidden within the geometry. This can simply calculations quite a bit.

To extend the 30-60-90 a bit more, the picture contains the relationships between all the sides and the trig confirms it.

In this case of the inscribed equilaterial triangle within a circle, the radius is the hypontenuse of the 30-60-90 triangle or sqrt(3) side.

Incidentally, sqrt(3) = 1.732 and sqrt(3)/2 = 0.866 in case you were wondering where those numbers came from....