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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.

Find the area of circle O, given triangle ABC inscribed in the circle with side lengths AB = 20, AC = 21, and BC = 13.

This problem involves finding the area of a circle where a triangle with sides AB = 20, AC = 21, and BC = 13 is inscribed. Key concepts include the circumradius formula, Heron's theorem, and circle area calculation. Suitable for students in Grades 10-12.

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