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Try this beautiful problem from Geometry based on Circumscribed Circle

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

- \(\frac{5\sqrt3}{\pi}\)
- \(\frac{3\sqrt3}{\pi}\)
- \(\frac{3\sqrt3}{2\pi}\)

Geometry

Triangle

Circle

But try the problem first...

Answer: \(\frac{3\sqrt3}{\pi}\)

Source

Suggested Reading

AMC-10A (2003) Problem 17

Pre College Mathematics

First hint

Let ABC is a equilateral triangle which is inscribed in a circle. with center \(O\). and also given that perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle.so for find out the peremeter of Triangle we assume that the side length of the triangle be \(x\) and the radius of the circle be \(r\). then the side of an inscribed equilateral triangle is \(r\sqrt{3}\)=\(x\)

Can you now finish the problem ..........

Second Hint

The perimeter of the triangle is=\(3x\)=\(3r\sqrt{3}\) and Area of the circle=\(\pi r^2\)

Now The perimeter of the triangle=The Area of the circle

Therefore , \(3x\)=\(3r\sqrt{3}\)=\(\pi r^2\)

can you finish the problem........

Final Step

Now \(3x\)=\(3r\sqrt{3}\)=\(\pi r^2\) \(\Rightarrow {\pi r}=3\sqrt 3\) \(\Rightarrow r=\frac{3\sqrt3}{\pi}\)

- https://www.cheenta.com/geometry-based-on-triangle-prmo-2018-problem-6
- https://www.youtube.com/watch?v=7AlfBAPWEMg

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