https://file.aiquickdraw.com/mathbot/file/1726141108695t5m38eex.jpg

To solve the problem in this image, we need to calculate the sum of the interior angles for each polygon provided. The formula for finding the sum of the interior angles of a polygon is:

\[
S = 180^\circ \times (n - 2)
\]
where \( n \) is the number of sides of the polygon.

Let’s go through each one:

1. **3-gon (triangle):** 
\[
S = 180^\circ \times (3 - 2) = 180^\circ
\]

2. **5-gon (pentagon):**
\[
S = 180^\circ \times (5 - 2) = 180^\circ \times 3 = 540^\circ
\]

3. **4-gon (quadrilateral):**
\[
S = 180^\circ \times (4 - 2) = 180^\circ \times 2 = 360^\circ
\]

4. **6-gon (hexagon):**
\[
S = 180^\circ \times (6 - 2) = 180^\circ \times 4 = 720^\circ
\]

5. **7-gon (heptagon):**
\[
S = 180^\circ \times (7 - 2) = 180^\circ \times 5 = 900^\circ
\]

6. **13-gon:**
\[
S = 180^\circ \times (13 - 2) = 180^\circ \times 11 = 1980^\circ
\]

7. **21-gon:**
\[
S = 180^\circ \times (21 - 2) = 180^\circ \times 19 = 3420^\circ
\]

8. **27-gon:**
\[
S = 180^\circ \times (27 - 2) = 180^\circ \times 25 = 4500^\circ
\]

9. **35-gon:**
\[
S = 180^\circ \times (35 - 2) = 180^\circ \times 33 = 5940^\circ
\]

10. **49-gon:**
\[
S = 180^\circ \times (49 - 2) = 180^\circ \times 47 = 8460^\circ
\]

Would you like a detailed explanation of any specific step or a breakdown of the formula?

Here are 5 questions to extend this topic:
1. What is the sum of the interior angles of a 50-gon?
2. How do you calculate the individual interior angle of a regular polygon?
3. Can the formula \( S = 180^\circ \times (n - 2) \) be derived geometrically?
4. What is the sum of the exterior angles of any polygon?
5. How would the formula change for non-convex polygons?

**Tip:** To find the individual interior angle of a regular polygon, divide the sum of the interior angles by the number of sides \( n \).

Find the sum of the measures of all the interior angles of the polygons, including a 3-gon, 5-gon, 4-gon, 6-gon, 7-gon, 13-gon, 21-gon, 27-gon, 35-gon, and 49-gon.

Learn how to calculate the sum of interior angles for various polygons, including a 3-gon, 5-gon, 7-gon, and up to a 49-gon. This geometry problem involves using the formula S = 180° × (n - 2), where n is the number of sides of the polygon.

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