An equilateral triangle is inscribed in a circle of radius r. expresss the circumference C as function of the?

length x , side of the triangle


answer is 2 pix / SR 3 .pls explain





SR: square root

An equilateral triangle is inscribed in a circle of radius r. expresss the circumference C as function of the?
a triangle with angles 30, 60,90


has sides in ratio 1: (SR 3) : 2


The equilateral triangle can be formed from


two 30,60,90 triangles.


each of those can be formed from 3 smaller 30,60,90 triangles


so the equilateral triangle can be formed from six 30,60,90 triangles.


If radius of circle is 1 unit,


each of the six small 30,60,90 triangles has


sides (1/2) ((SR 3)/2) 1


so the equilateral triangle has side SR(3) units


so if the equilateral triangle has side x, the circle has radius x /(SR(3)),


and the circle circumference is 2 pi x /((SR 3))
Reply:Best way to solve problems like this is to draw a little sketch and see how things are related. If you drop a line from a vertex to the midpoint of the opposite side you make a right triangle where you know the hypothenuse and the base. You also have a slimilar triangle with the base from the center of the circle to a side. You can get what you need from these. Draw the sketch!
Reply:draw the figure and put a radius to an angle of the triangle


draw a line segment down from the center so that it bisects a side of the triangle perpendicularly


now we have a right triangle with hypoteneuse "r" and one leg is x/2


this triangle is a 30-60-90 since the original triangle was equilateral


the other leg is then x/(2Sqrt[3])


solving for r in terms of x we get r=x/Sqrt[3]


C=2Pir= 2Pix/Sqrt[3]