Three inscribed objects

Okay, so there's a square, and inside it, there's a circle that fits perfectly (inscribed). Then, inside that circle, there's an equilateral triangle with each side measuring 4 √3 . I need to find the diagonal of the square.

First, let's visualize the shapes:

1. Square: Let's say the square has side length s .
2. Circle: Inscribed in the square, so its diameter is equal to the side of the square. Therefore, the radius r of the circle is s/2 .
3. Equilateral Triangle: Inscribed in the circle, with each side 4 √3 .

I need to find the diagonal of the square, which is s √2 (since the diagonal of a square with side s is s √2 ).

So, my goal is to find s , and then calculate s √2 .

Step 1: Relate the Equilateral Triangle to the Circle

An equilateral triangle inscribed in a circle has a specific relationship between its side length and the radius of the circle.

For an equilateral triangle with side length a , the radius R of the circumscribed circle (circumradius) is given by:


In this problem, the side length a = 4 √3 , so:


Wait, that simplifies to:


So, the radius of the circle is 4.

Step 2: Relate the Circle to the Square

The circle is inscribed in the square, so the diameter of the circle is equal to the side length of the square.

Diameter of the circle = 2 × radius = 2 × 4 = 8

Therefore, the side length of the square s = 8

Step 3: Find the Diagonal of the Square

The diagonal d of a square with side length s is:


So, plugging in s = 8 :


Final Answer: