Question

Find the perimeter of each regular polygon with the given side lengths. pentagon, \(20 \mathrm{in} .\)

Short answer

The perimeter of the pentagon is 100 in.

Step-by-step solution

Understand the problem

The problem asks for the perimeter of a regular pentagon, given the side length. A regular polygon is one where all sides are of equal length.

Identify the characteristics of a pentagon

A pentagon is a polygon with 5 sides. Since it is a regular pentagon, each side is of the same length.

Use the perimeter formula for a regular polygon

The formula for the perimeter ( P ) of a regular polygon is given by P = n imes ext{side length} , where ... is the number of sides. For a pentagon, ... is 5.

Substitute the given values

Substitute the given side length of 20 ext{ in}. and the number of sides, 5 , into the formula. This gives P = 5 imes 20 .

Calculate the perimeter

Perform the multiplication: 5 imes 20 = 100 . Thus, the perimeter of the pentagon is 100 ext{ in}.

Key concepts

Regular Pentagon

A regular pentagon is a specific type of polygon. It is called "regular" because all of its sides are of equal length, and all its internal angles are equal, too. Each internal angle in a regular pentagon measures 108 degrees.

This characteristic symmetry of regular polygons simplifies many calculations, such as computing the perimeter or area. Understanding the concept of a regular pentagon is important in geometry because it serves as a basic building block for more complex shapes.

Regular polygons like the pentagon are often used in architecture and design due to their pleasing aesthetic and structural benefits.

Polygon Characteristics

Polygons are multi-sided shapes that can have any number of sides. The word "polygon" comes from Greek and means 'many angles.' Each polygon's name is derived from the number of its sides, such as triangle (3 sides), quadrilateral (4 sides), and pentagon (5 sides).

Key characteristics of polygons include:

• Sides: The line segments that form the perimeter of the polygon.
• Vertices: The points where two sides meet.
• Angles: The spaces between two intersecting sides, inside the polygon.

Understanding these features helps in identifying and categorizing polygons, which is fundamental in fields like mathematics and engineering.

Perimeter Formula

To compute the perimeter of a regular polygon, you can use a straightforward formula. The perimeter is the total distance around the shape.

The formula for the perimeter (P) of a regular polygon is:
\[ P = n \times \text{side length} \]
where:

• \( n \)is the number of sides of the polygon.
• \( \text{side length} \)is the length of one side.

For a regular pentagon, \( n \) is always 5 since it has five equal sides.

This formula makes it easy to find the perimeter by just multiplying the number of sides by the length of one side. In practical applications, knowing how to calculate the perimeter can be useful in a variety of tasks, from calculating materials needed in construction projects to fabricating objects in manufacturing.