How much area in square centimeters is blue in Nacho's trademark of an isosceles triangle inscribed in a circle with a radius of 5 cm?

To determine the area of the blue section in Nacho's trademark, we first need to establish the total area of the isosceles triangle inscribed in a circle with a radius of 5 cm.

The first step is to find the area of the circle, which can be calculated using the formula for the area of a circle, ( A = \pi r^2 ). Since the radius ( r ) is 5 cm, the area of the circle becomes:

[

A = \pi (5)^2 = 25\pi \text{ cm}^2 \approx 78.5 \text{ cm}^2.

]

Next, we need to calculate the area of the isosceles triangle inscribed within this circle. For an isosceles triangle inscribed in a circle, we can use the formula:

[

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.

]

In the case of an isosceles triangle inscribed in a circle, its height can be determined based on the geometry of the situation. The height extends from the center of the circle perpendicular to the base.