https://r2.erweima.ai/mathbot/image/1726466579333ibesywna.jpg

The problem asks to calculate the number of diagonals and internal angles of various polygons, namely:

10. Heptagon (7 sides)
11. Dodecagon (12 sides)
12. Nonagon (9 sides)
13. Hendecagon (11 sides)

### Formulas:
1. **Diagonals**: The number of diagonals in an \(n\)-sided polygon is given by:
   \[
   D = \frac{n(n-3)}{2}
   \]
   where \(n\) is the number of sides.

2. **Sum of internal angles**: The sum of internal angles of an \(n\)-sided polygon is given by:
   \[
   S = 180(n-2)
   \]
   Each internal angle can then be calculated by dividing the sum by \(n\).

### Let's calculate each polygon:

---

### 10. **Heptagon (7 sides)**:
- Diagonals:
  \[
  D = \frac{7(7-3)}{2} = \frac{7 \times 4}{2} = 14
  \]
- Sum of internal angles:
  \[
  S = 180(7-2) = 180 \times 5 = 900^\circ
  \]
- Each internal angle:
  \[
  \text{Angle} = \frac{900}{7} \approx 128.57^\circ
  \]

---

### 11. **Dodecagon (12 sides)**:
- Diagonals:
  \[
  D = \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = 54
  \]
- Sum of internal angles:
  \[
  S = 180(12-2) = 180 \times 10 = 1800^\circ
  \]
- Each internal angle:
  \[
  \text{Angle} = \frac{1800}{12} = 150^\circ
  \]

---

### 12. **Nonagon (9 sides)**:
- Diagonals:
  \[
  D = \frac{9(9-3)}{2} = \frac{9 \times 6}{2} = 27
  \]
- Sum of internal angles:
  \[
  S = 180(9-2) = 180 \times 7 = 1260^\circ
  \]
- Each internal angle:
  \[
  \text{Angle} = \frac{1260}{9} = 140^\circ
  \]

---

### 13. **Hendecagon (11 sides)**:
- Diagonals:
  \[
  D = \frac{11(11-3)}{2} = \frac{11 \times 8}{2} = 44
  \]
- Sum of internal angles:
  \[
  S = 180(11-2) = 180 \times 9 = 1620^\circ
  \]
- Each internal angle:
  \[
  \text{Angle} = \frac{1620}{11} \approx 147.27^\circ
  \]

---

Let me know if you want more details or further explanations!

Here are five related questions:
1. How do you calculate the exterior angle of a regular polygon?
2. How does the number of sides of a polygon affect its diagonal count?
3. What is the relationship between diagonals and internal angles in polygons?
4. How would the internal angles change if the polygon is irregular?
5. What is the maximum possible internal angle of any polygon?

**Tip**: Remember that the sum of the external angles for any polygon always adds up to \(360^\circ\), no matter how many sides!

Instrucciones: Calcula el número de diagonales y de ángulos internos de cada polígono.
10. Heptágono
11. Dodecágono
12. Eneágono
13. Endecágono

Learn how to calculate the number of diagonals and the sum of internal angles for polygons like the Heptagon, Dodecagon, Nonagon, and Hendecagon. This geometry problem covers key concepts such as polygon properties, diagonal formulas, and internal angle calculations. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation