A polygon has 27 diagonals. What is the polygon?

To determine the number of sides of a polygon given the number of diagonals, we can use the following formula for the number of diagonals in a polygon:

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

where:
- \(D\) is the number of diagonals, and
- \(n\) is the number of sides of the polygon.

Given that \(D = 27\), we substitute this into the formula:

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

Now, solve for \(n\):

1. Multiply both sides by 2 to eliminate the fraction:
\[
54 = n(n-3)
\]

2. Expand the right side:
\[
54 = n^2 - 3n
\]

3. Rearrange the equation to form a quadratic equation:
\[
n^2 - 3n - 54 = 0
\]

4. Solve the quadratic equation using the quadratic formula:
\[
n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2(1)}
\]
\[
n = \frac{3 \pm \sqrt{9 + 216}}{2}
\]
\[
n = \frac{3 \pm \sqrt{225}}{2}
\]
\[
n = \frac{3 \pm 15}{2}
\]

5. This gives two solutions:
\[
n = \frac{3 + 15}{2} = \frac{18}{2} = 9
\]
or
\[
n = \frac{3 - 15}{2} = \frac{-12}{2} = -6
\]

Since the number of sides of a polygon must be positive, we discard \(n = -6\), leaving us with \(n = 9\).

Thus, the polygon is a **nonagon** (a 9-sided polygon).

Would you like more details or have any questions? Here are 5 related questions:

1. How do you calculate the total number of diagonals for any polygon?
2. Can the formula for diagonals be derived using combinations?
3. What is the sum of the interior angles of a nonagon?
4. How many triangles can be formed by drawing diagonals from one vertex in a nonagon?
5. Can a regular nonagon be inscribed in a circle?

**Tip:** The number of diagonals in a polygon increases rapidly as the number of sides increases!

Solve the problem of determining the polygon with 27 diagonals using the diagonal formula and quadratic equations. This step-by-step guide explains the concepts and methods, suitable for students in Grades 8-10.

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