Venn Riddle

Other than remembering that the area of a circle = ¶r squared (the r is squared, not the ¶), and that the circumference is 2¶r, I'm stumped (just substitute r for x, they mean the same thing in this equation).

¶ is made by holding down the alt key and hitting 0182 on the keypad.

Could this be a variation of the "goat in the circular field" problem where the goat is tethered to a point on the circumference and you have to determine the ratio of radii for the goat to eat half the field.

If so it requires some tricky calculus.

It doesn't require calculus, and you only need to know basic trig and really basic geometry.

The centres of all three circles form an equilateral triangle. The radii of the circles are all the same, as the circles are equal, and the edges of the circles pass through the centres of the adjacent circles. Therefore there's an equal distance between each centre, therefore it's an equilateral triangle.

Therefore the area of the segment of one of the circles that includes that triangle is one sixth of the area of the circle, as the segment has an angle of 60 degrees whilst the complete circle is 360.

So one of the segments is pxx/6.

(Where p = pi, and x is the radius of the circle. xx is x squared)

Now, you need to find the area of the equilateral triangle. That's just a matter of using Pythagoras' theorem to find the height from the base of the triangle, and then multiplying it by the length of the base (x), and 1/2 (the area of a triange is half the base times the height).

Now you have to take the area of the triangle away from the area of the segment, and you'll get the area of the dicky little curvy bit that's in the segment but not the triangle.

Then you times the area of the curvy bit by three (one for each circle) and add on the area of the equilateral triangle in the middle.

This isn't really a riddle, more like a maths problem. So while I don't usually do other people's homework for them, I'm making an exception here, to make a point that you shouldn't always jump to the conclusion that you need to use calculus. Furthermore, saying that "this requires calculus", doesn't make you smart. Sorry if that sounds harsh.

i never made it to calculus, is it hard?

lol, i hate math so much. bleh, ugh.

It ranges from pretty easy to very hard.

It depends on what level you're talking about.

Basic calculus is pretty easy once you get the concept behind it.

But if you don't like maths then I wouldn't suggest trying to learn it.

You're better off sticking with things you enjoy, I reckon. I enjoy maths - crazy I know but solving math problems gives me a thrill and a sense of accomplishment, it's a very similar feeling to solving riddles.

I agree with milhouseandhagg's solution which does not require calculus. The "goat in the field" problem is another matter.

Seems to be a weakness in the logic in that the " equilateral triangle " isn't a triangle: it has curved sides> however this area can be determined as a compound shape: involving an equilateral triangle and 3 chords. In fact this is a classical maths problem that is harder than it looks. The conclusive answer is attributed to Gauss in "disquitones arithmaticae " (1801)

Sorry I'm 5 years late, I was playing with my Reuleaux Triangle until a half x squared times (pi - root3).

Fascinating, except it's

Disquisitiones Arithmeticae,

and you didn't answer the question.

read the question and my response again.

I'm correct.

The equilateral triangle is formed by the radii, not the partial circumferences.

Hence they are straight lines.

Then I go on to encorporate the segments later. AS A COMPOUND SHAPE.

How can you not see that? And then go on to talk crap about how it's actually more difficult than it looks.

It doesn't prove anything. Solving the problem (as I did) proves I know how to.

yeah, what glooper said