Question

The perimeter of the equilateral triangle and the regular hexagon shown in the figure are equal. The circumference of the red circle inscribed in the triangle is 10 cm. What is the circumference of the blue circle inscribed in the hexagon?

Answer 1

Correct Answer: 15

We start by noting that the perimeter of the equilateral triangle and regular hexagon are equal. The inscribed circle in an equilateral triangle (inradius 'r') is related to its side 'a' by the formula \(r_{triangle} = \frac{a\sqrt{3}}{6}\). Given the circumference of the inscribed circle is 10 cm, we have \(2\pi r_{triangle} = 10\). Consequently, \(r_{triangle} = \frac{5}{\pi}\). 

The side length 'a' of the equilateral triangle connects to 'r_{triangle}' as \(r_{triangle} = \frac{a\sqrt{3}}{6}\), becoming \(\frac{5}{\pi} = \frac{a\sqrt{3}}{6}\), thus \(a = \frac{30}{\pi\sqrt{3}}\).

Now, the perimeter of the triangle is \(3a = \frac{90}{\pi\sqrt{3}}\), equal to \(6s\) where 's' is the side of the hexagon. Thus, \(6s = \frac{90}{\pi\sqrt{3}}\) leads to \(s = \frac{15}{\pi\sqrt{3}}\).

The inradius of a regular hexagon is \(\frac{\sqrt{3}}{2}s\), so \(r_{hexagon} = \frac{\sqrt{3}}{2} \times \frac{15}{\pi\sqrt{3}} = \frac{15}{2\pi}\).

The circumference of the blue circle inscribed in the hexagon is \(2\pi r_{hexagon} = 2\pi \times \frac{15}{2\pi} = 15\). The computed circumference fits perfectly within the range 15,15.

Therefore, the circumference of the blue circle is 15 cm.