The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon?

To solve this, we use the formula for the number of diagonals of a regular polygon, which is given by:

\[
D = \frac{n(n-3)}{2}
\]

Where:
- \( D \) is the number of diagonals
- \( n \) is the number of sides of the polygon

According to the problem, the number of diagonals subtracted from the number of sides results in zero:

\[
n - D = 0
\]

Substitute \( D \) from the diagonal formula into the equation:

\[
n - \frac{n(n-3)}{2} = 0
\]

Now, solve for \( n \). Start by multiplying the entire equation by 2 to eliminate the fraction:

\[
2n - n(n-3) = 0
\]

Simplify the expression:

\[
2n - n^2 + 3n = 0
\]

\[
5n - n^2 = 0
\]

Factor the equation:

\[
n(5 - n) = 0
\]

Thus, \( n = 0 \) or \( n = 5 \).

Since \( n = 0 \) is not a valid number of sides for a polygon, the only solution is \( n = 5 \).

Therefore, the polygon has **5 sides**, meaning it is a regular pentagon.

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Would you like more details, or do you have any other questions?

Here are 5 related questions for further practice:
1. What is the number of diagonals of a polygon with 7 sides?
2. Derive the formula for the number of diagonals of a polygon.
3. How many sides does a polygon have if it has 35 diagonals?
4. What is the sum of the interior angles of a regular pentagon?
5. How many diagonals does a hexagon have?

**Tip:** A polygon with \( n \) sides will always have fewer diagonals than sides, except in a triangle (where both are zero).

Solve the problem of finding the number of sides of a polygon when the number of diagonals subtracted from the number of sides equals zero. This involves understanding the formula for diagonals and basic algebraic manipulation. Suitable for students in Grades 9-10.

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