How many diagonals does a decagon have?

How many diagonals does a decagon have?

• A. 25
• B. 30
• C. 35
• D. 40

Answer: (C)

Counting diagonals in a polygon is about how many lines you can draw from vertices to other nonadjacent vertices. From each vertex you can connect to every other vertex except itself and its two neighbors, giving n−3 diagonals from one vertex. If you add that up for all n vertices, you’ve counted each diagonal twice (once from each endpoint), so the total number of diagonals is n(n−3)/2. For a decagon, there are 10 vertices, so the number of diagonals is 10×7/2 = 35. This matches the value that comes from the standard diagonal-count formula for a 10-sided polygon. The other numbers don’t come from this formula for a polygon with ten sides, so they don’t represent possible diagonal counts for a decagon.

Counting diagonals in a polygon is about how many lines you can draw from vertices to other nonadjacent vertices. From each vertex you can connect to every other vertex except itself and its two neighbors, giving n−3 diagonals from one vertex. If you add that up for all n vertices, you’ve counted each diagonal twice (once from each endpoint), so the total number of diagonals is n(n−3)/2.

For a decagon, there are 10 vertices, so the number of diagonals is 10×7/2 = 35. This matches the value that comes from the standard diagonal-count formula for a 10-sided polygon.

The other numbers don’t come from this formula for a polygon with ten sides, so they don’t represent possible diagonal counts for a decagon.