To find the area of an inscribed equilateral triangle in a circle with a radius of 6 inches, we start by understanding the relationship between the circle and the triangle. When a triangle is inscribed in a circle, the vertices of the triangle touch the circumference of the circle.
For an equilateral triangle inscribed in a circle, the radius (R) of the circle is related to the side length (s) of the triangle by the formula:
s = R × √3
Substituting the radius of our circle:
s = 6 × √3
This simplifies to:
s ≈ 10.39 inches
Next, we can calculate the area (A) of the equilateral triangle using the formula:
A = (√3 / 4) × s²
Substituting the side length we found:
A = (√3 / 4) × (10.39)²
A little math gives us:
A = (√3 / 4) × 107.99 ≈ 46.8
Thus, the area of the inscribed equilateral triangle is approximately 46.8 square inches.