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HALLAR LA SUPERFECIE APLANADA DE LA CIRCUNFERENCIA CON DIAMETRO DE 60 MM

To find the "superficie aplanada" (flattened area) of the circular segment in the provided diagram, we need to consider the circular segment area formula. The circular segment is the area bounded by a chord (the straight line in the diagram) and the arc of the circle.

### Given:
- The diameter of the circle: 60 mm
- The chord length \( s \) (which can be deduced from the diagram).

### Steps to Find the Flattened Area:
1. **Calculate the Radius**:
   \[
   r = \frac{D}{2} = \frac{60}{2} = 30 \text{ mm}
   \]
2. **Determine the Angle (Central Angle)**:
   The central angle provided is 90°. 

3. **Calculate the Height \( h \) of the Segment**:
   \[
   h = r - \sqrt{r^2 - \left(\frac{s}{2}\right)^2}
   \]
   However, since we are given the central angle, we can also use the segment area formula directly.

4. **Area of the Circular Segment**:
   The area of a segment of a circle can be calculated using the formula:
   \[
   A_{\text{segment}} = \frac{r^2}{2} \left( \theta - \sin(\theta) \right)
   \]
   where \( \theta \) is in radians. For \( 90^\circ \), \( \theta = \frac{\pi}{2} \) radians.

   Substituting the values:
   \[
   A_{\text{segment}} = \frac{30^2}{2} \left( \frac{\pi}{2} - \sin\left(\frac{\pi}{2}\right) \right)
   \]

5. **Simplify**:
   \[
   A_{\text{segment}} = \frac{900}{2} \left( \frac{\pi}{2} - 1 \right)
   \]
   \[
   A_{\text{segment}} = 450 \times \left( 1.5708 - 1 \right)
   \]
   \[
   A_{\text{segment}} \approx 450 \times 0.5708 \approx 256.86 \text{ mm}^2
   \]

Thus, the flattened area of the circular segment is approximately **256.86 mm²**.

Would you like further clarification or have any questions?

Here are 8 related questions you might consider:
1. How can you derive the formula for the area of a circular segment?
2. What is the chord length \( s \) in this problem, and how is it calculated?
3. How would the area change if the central angle were different?
4. What is the area of the entire circle with the given diameter?
5. How does this segment area relate to the sector area?
6. How would you calculate the arc length of the segment's arc?
7. What is the significance of the central angle in calculating the segment area?
8. How can the height of the segment \( h \) be found directly from the chord length?

**Tip:** When dealing with segments and sectors of a circle, remember to distinguish between the area formulas for each and always convert angles to radians when using trigonometric functions in geometry problems.

Learn how to find the flattened area of a circular segment with a diameter of 60 mm using the area formula. This problem involves calculating the area bounded by a chord and the arc of a circle. Suitable for high school students studying circle geometry and segment area calculations.

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