# geometry

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Theres a circle with an equilateral triangle in the middle. The traingles edges all touch the circle. The radius of the circle is 8 meters. How do I find the area of the triangle?

Sorry The triangles edges don't touch the circle, the points do.

Did you mean the vertices of the triangle touch the circle?
If the "edges" of the triangle touch the circle, then the triangle cannot be inside the circle.

If you have an equilateral triangle inside a circle then the centre of the circle must be the centroid of the triangle.

the centroid is located 2/3 of the distance of the median from the vertex

so 2/3 of the median is 8, therefore the median must be 12.

I hope you know that the ratio of sides of a 30º,60º,90º triangle = 1:√3:2

then by the ratio
√3:2=x:12
x=6√3 which is the base of the triangle

so area = 1/2 bh = 1/2(6√3)(12)
=36√3

The perpendicular distance from the center of the circle to a side of the triangle is 8(sin30º) = 4m.

This makes the 3rd side, (x), of the (4, x, 8) triangle equal to sqrt (64 - 16) = sart 48 = 6.928m.

The area of this sixth part of the triangle area equal to 4(6.928)/2 = 13.856sm making the total area of the equilateral triangle 6(13.856) = 83.138 sq. met.