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FORMULAS Y CALCULOS

a.Cálculo de ángulos a = 2 sen^-1 ( C/ 2r )

b.Calculo de la sumatoria de ángulos internos
Σa1+a2…an   donde a es alfa
c.Cálculo de la sumatoria teórica de ángulos internos Σt=(n-2) * 180, donde n es el número de vértices

d.Cálculo de error de cierre angular
E=Σt-Σo

e. Corrección por vértice
C = E/n

f. Cálculo del área por triángulos
A = b * h/ 2 = b * c/ 2 * sena = √p (p -a) * (p -b) *(p-c)*(p-n)

Donde p es el semiperímetro es decir la suma de los lados de cada triángulo dividido en dos.

g. Cálculo del polígono
Apoligono = Al1+A2 + …+ An

Para el cálculo de los detalles, se halla el área de las figuras geométricas que se encuentren. Área total del terreno se calcula con la suma del área total del polígono con el área de los detalles que se encuentran por fuera de este, al resultado de la operación anterior se le resta el área de los detalles que se encuentran dentro del polígono

Let's analyze the diagram and apply the relevant formulas to calculate the angles, internal angle sums, area, and other requested values. Here's a breakdown of the steps based on the formulas provided:

### 1. **Calculation of Angles**:
   - Formula: \( a = 2 \sin^{-1} \left(\frac{C}{2r}\right) \)
   - In this case, you'll need the lengths of sides and the radius \( r \) if applicable. 

### 2. **Sum of Internal Angles**:
   - Formula: \( \Sigma a_1 + a_2 + \dots + a_n \)
   - This requires summing up the angles at each vertex.

### 3. **Theoretical Sum of Internal Angles**:
   - Formula: \( \Sigma t = (n-2) \times 180^\circ \)
   - For a quadrilateral (\( n = 4 \)): \( \Sigma t = (4-2) \times 180^\circ = 360^\circ \)

### 4. **Calculation of Angular Closure Error**:
   - Formula: \( E = \Sigma t - \Sigma o \)
   - This finds the discrepancy between the theoretical and observed angle sum.

### 5. **Correction per Vertex**:
   - Formula: \( C = \frac{E}{n} \)
   - This divides the error among the vertices.

### 6. **Area Calculation by Triangles**:
   - Formula: \( A = \frac{1}{2} \times b \times h = \frac{1}{2} \times b \times c \times \sin a = \sqrt{p(p-a)(p-b)(p-c)} \)
   - \( p \) is the semi-perimeter: \( p = \frac{a+b+c}{2} \)

### 7. **Total Polygon Area**:
   - Formula: \( A_{\text{polygon}} = A_1 + A_2 + \dots + A_n \)
   - Sum up the areas of the triangles within the polygon.

### 8. **Adjustment for External/Included Areas**:
   - Add or subtract areas outside or inside the main polygon as needed.

#### **Next Steps**:
1. **Calculation of Angles (Step 1)**:
   - We need the circle's radius \( r \) to use the formula directly. If it's not provided, the formula may be more complex depending on other factors.

2. **Sum of Internal Angles (Step 2)**:
   - Sum the internal angles provided or measured.

3. **Area Calculation (Steps 6 & 7)**:
   - Identify the triangles within the shape, calculate their areas, and sum them up.

Would you like to proceed with a detailed step-by-step calculation based on the image, or do you want to focus on specific aspects? Let me know your preference!

Explore how to calculate angles using trigonometric formulas, sum of internal angles, error of angular closure, correction per vertex, and area by triangles. This problem involves advanced high school-level trigonometry and geometry.

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