To find the area of the largest isosceles triangle that can be inscribed in a circle of radius 6, we start by using the formula for the area of a triangle: Area = 0.5 × base × height.
When an isosceles triangle is inscribed in a circle, the apex of the triangle will touch the circle’s circumference, while the base will lie along a chord of the circle. For maximum area, we want the base to be as long as possible.
The maximum base of the isosceles triangle occurs when the triangle is actually a right triangle, where the apex creates a right angle with the base. In this scenario, the base will be the diameter of the circle, which measures 2 × radius = 2 × 6 = 12.
Next, we can find the height of the triangle. The height extends from the apex (which lies on the circumference) down to the midpoint of the base (the diameter). In a right triangle inscribed in a circle, this height will also be the radius of the circle, which equals 6.
Now, we can substitute these values into the area formula:
Area = 0.5 × base × height
Area = 0.5 × 12 × 6
Area = 0.5 × 72 = 36
Thus, the area of the largest isosceles triangle that can be inscribed in a circle of radius 6 is 36 square units.