The diagram shows a circle with center $O$, and a triangle inscribed in the circle, with vertices labeled $A$, $B$, and $C$. The sides of the triangle are labeled as follows:

• $AB = 20$
• $AC = 21$
• $BC = 13$

The task is to find the area of the circle.

Approach

To find the area of the circle, we first need to find the radius $r$ of the circle. One efficient method to determine the radius is to use the formula for the circumradius $R$ of a triangle when its side lengths are known:

$R = \frac{abc}{4A}$

where:

• $a = 13$, $b = 21$, and $c = 20$ are the sides of the triangle,
• $A$ is the area of the triangle.

We can find the area of the triangle using Heron's formula:

1. Heron's formula for the area $A$ of a triangle is: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is the semi-perimeter: $s = \frac{a + b + c}{2}$

Once we have the area $A$, we can use the circumradius formula to find $R$, which is the radius of the circle. The area of the circle is then:

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

Let's begin by calculating the necessary values.The area of the circle is approximately $368.70 \, \text{square units}$.

Would you like more details on any part of the solution?

Here are some related questions to explore:

1. How does Heron's formula work for finding the area of a triangle?
2. What is the significance of the circumradius in a triangle?
3. Can the circumradius be calculated using other methods?
4. How would this problem change if the triangle were not inscribed in a circle?
5. What other properties can be derived from a triangle inscribed in a circle?

Tip: Remember that the circumradius $R$ can also provide insight into other properties of the triangle, such as its angles and symmetry.