The number of diagonals in a regular polygon is 3 times its number of sides. Find the measure of one of its interior angles.

Let the number of sides of the polygon be denoted by \( n \). The number of diagonals in a polygon is given by the formula:

\[
\text{Number of diagonals} = \frac{n(n-3)}{2}
\]

According to the problem, the number of diagonals is three times the number of sides, i.e.,

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

Multiply both sides of the equation by 2 to eliminate the denominator:

\[
n(n-3) = 6n
\]

Simplifying this:

\[
n^2 - 3n = 6n
\]

\[
n^2 - 9n = 0
\]

Factor the quadratic equation:

\[
n(n - 9) = 0
\]

This gives two possible solutions: \( n = 0 \) or \( n = 9 \). Since the polygon must have sides, we take \( n = 9 \).

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

### Finding the measure of one interior angle

The measure of one interior angle of a regular polygon with \( n \) sides is given by:

\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\]

Substitute \( n = 9 \):

\[
\text{Interior angle} = \frac{(9-2) \times 180^\circ}{9} = \frac{7 \times 180^\circ}{9} = \frac{1260^\circ}{9} = 140^\circ
\]

Thus, the measure of one interior angle is \( 140^\circ \).

Would you like any details or have further questions?

Here are 5 related questions:
1. How do you find the sum of interior angles of any polygon?
2. What is the formula for the measure of one exterior angle of a regular polygon?
3. How many diagonals does a regular decagon (10-sided polygon) have?
4. Can the number of sides of a polygon be determined from its interior angle?
5. What is the relationship between interior and exterior angles in polygons?

**Tip**: The sum of the interior angles of a polygon depends only on the number of sides, not their length!

Discover how to find the interior angle of a regular polygon where the number of diagonals is three times its number of sides. This step-by-step solution uses key geometric formulas, including the diagonal formula and the interior angle theorem. Ideal for students in Grades 8-10.

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