What percent of the area of a circle (radius 6 cm) is covered by an inscribed equilateral triangle?

To determine what percent of the area of a circle is covered by an inscribed equilateral triangle, we need to calculate the areas of both the circle and the triangle.

First, let's find the area of the circle. The formula for the area of a circle is given by:

[

\text{Area of the circle} = \pi r^2

]

where ( r ) is the radius of the circle. In this case, the radius is 6 cm:

[

\text{Area of the circle} = \pi \times (6^2) = \pi \times 36 \approx 113.1 \text{ cm}^2

]

Next, we need to calculate the area of the inscribed equilateral triangle. For an equilateral triangle inscribed in a circle (circumcircle), the relationship between the radius ( r ) of the circumcircle and the side length ( s ) of the triangle is given by:

[

s = r \sqrt{3}

]

Therefore, for our circle with a radius of 6 cm, the side length ( s ) of the inscribed triangle is:

[

s = 6 \sqrt{3}