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Math Help - Diameter of circle--equilateral triangle

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    Diameter of circle--equilateral triangle

    If I have an equilateral triangle with sides of 4", how do I figure out what the diameter of a circle would be that touches the three points?
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    Re: Diameter of circle--equilateral triangle

    Draw it out; make sure the diagram isn't too small. Write EVERYTHING you know about the circle and the triangles (notice I said triangles--there's a hint).
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    Re: Diameter of circle--equilateral triangle

    I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:

    Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to find.

    Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.
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    Re: Diameter of circle--equilateral triangle

    If we deconstruct the equilateral triangle into 3 congruent isosceles 30-30-120 triangles, we find the two equal sides of these isosceles triangles is the radius of the circle.

    Then, we may find the radius r in inches is:

    r=2\sec(30^{\circ})=\frac{4}{\sqrt{3}}

    And so the diameter d is:

    d=2r=\frac{8}{\sqrt{3}}\approx4.62\text{ in}.
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    Re: Diameter of circle--equilateral triangle

    Quote Originally Posted by Jambe View Post
    I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:
    Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to find.

    Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.

    This page answers your question.

    You want the radius of the inscribed circle. It is right there.
    Thanks from Jambe
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    Re: Diameter of circle--equilateral triangle

    Quote Originally Posted by Plato View Post
    This page answers your question.

    You want the radius of the inscribed circle. It is right there.
    Your link seems to provide the simplest solution.

    (I wanted the circumscribed circle, not inscribed.)

    The square root of 3, divided by 3, times 4"; times 2 for diameter.

    My answer: 4.6188".

    Thanks everyone.
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    Re: Diameter of circle--equilateral triangle

    Quote Originally Posted by Jambe View Post
    Your link seems to provide the simplest solution.
    (I wanted the circumscribed circle, not inscribed.)
    The square root of 3, divided by 3, times 4"; times 2 for diameter.
    My answer: 4.6188".

    When most mathematicians see "circle would be that touches", we think tangent. That would be an incircle.
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