Question

Grid In Find the perimeter in centimeters of a regular nonagon whose sides are \(3.9\) centimeters long. (Lesson 10-1)

Short answer

The perimeter of the regular nonagon is 35.1 cm.

Step-by-step solution

Understand the Shape

A nonagon is a polygon with nine equal sides and nine angles. Since the nonagon is regular, all sides are of equal length.

Determine Formula for Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides. For a regular nonagon, the perimeter can be calculated using the formula: \[ P = n \times s \] where \( n \) is the number of sides and \( s \) is the length of each side.

Substitute Values

Substitute the given values into the formula. Here, \( n = 9 \) (because it is a nonagon) and \( s = 3.9 \) cm. The formula becomes: \[ P = 9 \times 3.9 \]

Calculate the Perimeter

Perform the multiplication to find the perimeter of the nonagon. \[ P = 9 \times 3.9 = 35.1 \] cm

State the Result

The perimeter of the regular nonagon is \( 35.1 \) centimeters.

Key concepts

Perimeter Calculation

The perimeter of a shape is essentially the total distance around it. To calculate the perimeter of a regular polygon, like a nonagon, it's vital to know the number of sides and the length of each side. A regular nonagon has sides of equal length, which simplifies the calculation.

To find the perimeter, the fundamental formula is:

• Perimeter (\[ P \]) = Number of sides (\[ n \]) × Length of one side (\[ s \])

By multiplying the number and length of its sides, you determine what is effectively the boundary length of the entire shape.This formula is applicable not only to nonagons but to all regular polygons.For the given nonagon with side length of\(3.9\)centimeters, simply plug in values to get an exact perimeter calculation of\(35.1\)centimeters.

Polygon Properties

Polygons are geometric shapes with straight sides. The term "regular polygon" indicates that all sides and angles are equal. Understanding these properties provides the basis for many geometric calculations, such as perimeter and area.

A nonagon is a specific type of polygon with:

• Nine sides
• Nine angles, which sum up to \((n-2) \times 180\)° where \(n = 9\)
• Equal side lengths if it is a regular nonagon

Knowing that every side is identical in a regular polygon helps us use simple formulas for calculations. These properties not only assist in forming accurate geometric models but also simplify complex geometric tasks. Consider these properties as the backbone for understanding and manipulating many types of shapes.

Geometric Shapes

Geometric shapes form the basis of geometry, each having specific attributes and rules. Regular polygons like the nonagon have defined characteristics that set them apart. By being regular, all sides and angles are equal, which ensures symmetry and simplifies calculations.

Understanding geometric shapes at a deeper level involves knowing:

• How they are constructed
• Their symmetry and balance
• How they relate to other types of shapes

In essence, a nonagon, with its nine sides, stands as an interesting geometric figure used in various applications from design to architecture. Grasping these foundational aspects is critical for further exploration in geometry, making learning about shapes not only about memorizing properties but also about recognizing their place in the universe of geometry.