The number of diagonals in a polygon can be calculated using the formula: D = n(n – 3) / 2, where D is the number of diagonals and n is the number of sides in the polygon.
Let’s break down the calculations for each polygon:
- Dodecagon (12 sides):
Using the formula:
D = 12(12 – 3) / 2 = 12(9) / 2 = 108 / 2 = 54 diagonals. - Pentadecagon (15 sides):
Using the formula:
D = 15(15 – 3) / 2 = 15(12) / 2 = 180 / 2 = 90 diagonals. - Icosahedron (20 sides):
Using the formula:
D = 20(20 – 3) / 2 = 20(17) / 2 = 340 / 2 = 170 diagonals. - Hectagon (100 sides):
Using the formula:
D = 100(100 – 3) / 2 = 100(97) / 2 = 9700 / 2 = 4850 diagonals.
In summary:
- Dodecagon: 54 diagonals
- Pentadecagon: 90 diagonals
- Icosahedron: 170 diagonals
- Hectagon: 4850 diagonals
This approach helps us understand how the number of diagonals increases with the number of sides in a polygon, highlighting the geometric complexity as polygons grow in size.