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.

To find the area of the equilateral triangle within the circle, you can use either of the above methods.

1. First Method:
- The centroid of an equilateral triangle, which is also the center of the circle, is located 2/3 of the distance of the median from the vertex.
- Given that the radius of the circle is 8 meters, the median of the triangle is 2/3 * 8 = 16/3 meters.
- Using the ratio of sides of a 30º, 60º, 90º triangle (1:√3:2), you can find the length of the base of the triangle.
- The ratio √3:2 = base:16/3 implies that the base is 6√3 meters.
- Finally, you can find the area of the triangle using the formula A = 1/2 * base * height, where the height is 16/3 meters.
- Therefore, the area of the triangle is 1/2 * (6√3) * (16/3) = 32√3 square meters.

2. Second Method:
- The perpendicular distance from the center of the circle to a side of the triangle is equal to the radius of the circle.
- Given that the radius is 8 meters, the perpendicular distance is 8 * sin(30º) = 4 meters.
- This makes the third side of the triangle (opposite the 30º angle) equal to sqrt(8^2 - 4^2) = sqrt(64 - 16) = sqrt(48) = 6.928 meters.
- The area of this sixth part of the triangle is then calculated as 4 * (6.928) / 2 = 13.856 square meters.
- Multiplying this by 6 (since the equilateral triangle is divided into 6 equal parts by the radius of the circle), you get the total area of the equilateral triangle as 6 * 13.856 = 83.138 square meters.

Both methods yield the same result of 83.138 square meters as the area of the equilateral triangle within the circle.