find the number of sides in a convex polygon if the sum of the measures of its interior angles is :540 degrees, 1080 degrees, 1800 degrees, 1620 degrees, 2340 degrees, 3600 degrees, 2880 degrees, 7560 degrees

Answer 1

To find the number of sides \(n\) of a convex polygon from the sum of its interior angles, you can use the formula:

\[ \text{Sum of interior angles} = (n - 2) \times 180 \]

We can rearrange this formula to solve for \(n\):

\[ n = \frac{\text{Sum of interior angles}}{180} + 2 \]

Now, we will apply this formula to each given sum of interior angles:

1. 540 degrees: \[ n = \frac{540}{180} + 2 = 3 + 2 = 5 \]

2. 1080 degrees: \[ n = \frac{1080}{180} + 2 = 6 + 2 = 8 \]

3. 1800 degrees: \[ n = \frac{1800}{180} + 2 = 10 + 2 = 12 \]

4. 1620 degrees: \[ n = \frac{1620}{180} + 2 = 9 + 2 = 11 \]

5. 2340 degrees: \[ n = \frac{2340}{180} + 2 = 13 + 2 = 15 \]

6. 3600 degrees: \[ n = \frac{3600}{180} + 2 = 20 + 2 = 22 \]

7. 2880 degrees: \[ n = \frac{2880}{180} + 2 = 16 + 2 = 18 \]

8. 7560 degrees: \[ n = \frac{7560}{180} + 2 = 42 + 2 = 44 \]

In summary, the number of sides \(n\) for the corresponding sums of interior angles is:

• 540 degrees: \(5\) sides (pentagon)
• 1080 degrees: \(8\) sides (octagon)
• 1800 degrees: \(12\) sides (dodecagon)
• 1620 degrees: \(11\) sides (hendecagon)
• 2340 degrees: \(15\) sides (pentadecagon)
• 3600 degrees: \(22\) sides (icosidodecagon)
• 2880 degrees: \(18\) sides (octadecagon)
• 7560 degrees: \(44\) sides (tetracontakaipentagon)

Thus, each value corresponds to a specific convex polygon with the corresponding number of sides.