Question

Find the perimeter of each regular polygon with the given side lengths. nonagon, \(3.8 \mathrm{~mm}\)

Short answer

The perimeter of the nonagon is 34.2 mm.

Step-by-step solution

Identify the Number of Sides in a Nonagon

A nonagon is a polygon with 9 sides. Therefore, the number of sides, \( n \), for a nonagon is 9.

Understand the Formula for Perimeter of a Regular Polygon

The formula for the perimeter \( P \) of a regular polygon is given by \( P = n \times s \), where \( n \) is the number of sides and \( s \) is the length of each side.

Substitute the Given Values into the Formula

We know from the problem that the side length \( s \) is given as \( 3.8 \) mm. Using the perimeter formula: \( P = 9 \times 3.8 \).

Calculate the Perimeter

Multiply the number of sides by the length of each side:\[P = 9 \times 3.8 = 34.2 \text{ mm}\]. Thus, the perimeter of the nonagon is 34.2 mm.

Key concepts

Regular Polygon

A regular polygon is a shape where all sides and all angles are equal. This means, regardless of the number of sides, each one is the same length, and each angle is the same measure. Regular polygons are an important category of shapes, simplifying perimeter and area calculations by allowing us to use uniform measurements.

When you hear the term 'regular polygon,' think of uniformity:

• Equal side lengths
• Equal angles

This uniformity allows us to apply simple mathematical formulas to calculate properties like perimeter and area. The beauty of regular polygons lies in this symmetry, making many geometric calculations straightforward.

Nonagon

A nonagon is a specific type of polygon that has 9 sides. Each side is equal in length if it is a regular nonagon, which also means all the interior angles are also equal.

The term "nonagon" comes from the Latin and Greek words for nine—which is why it's sometimes called an enneagon as well.

Here are some quick facts about nonagons:

• Number of sides: 9
• Sum of interior angles: \[ (9 - 2) imes 180 = 1260^\circ \]
• Measure of each interior angle in a regular nonagon: \[\frac{1260}{9} = 140^\circ\]

These characteristics are unique to nonagons and help to distinguish them from other polygons.

Polygon Formula

To find the perimeter of a regular polygon, we use the formula:\[P = n \times s\]where:

• \(n\) is the number of sides
• \(s\) is the side length.

The calculation is simple: multiply the number of sides by the length of one side. This works because in a regular polygon, all side lengths are equal, making the perimeter easily calculable.

In the exercise above, the nonagon's perimeter comes from:

• Number of sides (\(n\)) is 9
• Side length (\(s\)) is 3.8 mm
• Perimeter: \[P = 9 \times 3.8 = 34.2 \text{ mm}\]

The formula applies universally to all regular polygons, making perimeter calculations both easy and efficient.