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Calcule el área de la corona circular si O es centro, A es punto de tangencia y OB = 6 u.
A. 27π u² B. 36π u² C. 25π u²
D. 28π u² E. 30π u²

To calculate the area of the circular annulus (corona circular), we need to determine the difference in areas between the larger circle and the smaller circle.

### Information given:
- \( O \) is the center of both circles.
- \( A \) is the point of tangency of the inner circle.
- \( OB = 6 \, \text{units} \).
- The angle \( \angle AOB = 30^\circ \).

The point \( A \) is the point of tangency between the line and the inner circle, which means \( OA \) is the radius of the smaller circle. We can use this information to find \( OA \) by applying trigonometry.

### Step 1: Using the trigonometric relation
In the triangle \( \triangle OAB \), we can use the cosine of \( 30^\circ \) to relate \( OB \) and \( OA \):
\[
\cos(30^\circ) = \frac{OA}{OB}
\]
\[
OA = OB \times \cos(30^\circ)
\]
\[
OA = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}
\]

### Step 2: Calculating the areas
- The area of the smaller circle is given by:
\[
A_{\text{small}} = \pi \times (OA)^2 = \pi \times (3\sqrt{3})^2 = 27\pi
\]
- The area of the larger circle is:
\[
A_{\text{large}} = \pi \times (OB)^2 = \pi \times 6^2 = 36\pi
\]

### Step 3: Calculating the area of the annulus (corona circular)
The area of the annulus is the difference between the areas of the larger and smaller circles:
\[
A_{\text{annulus}} = A_{\text{large}} - A_{\text{small}} = 36\pi - 27\pi = 9\pi
\]

Thus, the area of the circular annulus is \( 9\pi \, \text{units}^2 \).

### Conclusion
There seems to be a mistake in the answer options, as the correct area should be \( 9\pi \, \text{units}^2 \), but this does not match any of the given choices. Please confirm if there are any further details to check.

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Do you have any questions, or would you like more details on any step?

Here are 5 related questions to deepen understanding:
1. How can we use trigonometric functions in right triangles for similar problems?
2. What is the general formula for calculating the area of an annulus?
3. Why is the cosine function appropriate for finding the radius of the smaller circle here?
4. How does the tangent point affect the geometry of the inner circle?
5. How would this problem change if the given angle were different?

**Tip**: When calculating areas involving angles and circles, trigonometry is often useful for relating sides to the radii of circles.

Calcule el área de la corona circular si O es centro, A es punto de tangencia y OB = 6 u.

Learn how to calculate the area of a circular annulus given the center O, tangent point A, and OB = 6 units, with an angle of 30°. The solution involves trigonometry and geometry concepts suitable for Grades 9-11.

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