Question

Find the perimeter of each regular polygon with the given side lengths. decagon

Short answer

The perimeter is 50 units (assuming sides are 5 units long).

Step-by-step solution

Understanding the Problem

A decagon is a polygon with 10 sides. To find the perimeter of a regular polygon, we multiply the length of one side by the total number of sides.

Identify the Given Value

Since the problem statement does not specify a side length, let's assume a side length, say 5 units, for illustration. You will need to ensure you use the actual side length if specified.

Calculate the Perimeter

The formula for the perimeter of a regular decagon is \( P = 10 \times \text{side length} \). Assuming a side length of 5 units, the perimeter is: \( P = 10 \times 5 = 50 \) units.

Key concepts

Decagon

A decagon is a type of polygon that has exactly ten straight sides. It is classified under polygons which are the most basic shapes in geometry. When you think of a decagon, it fits into a category called **"regular polygons"** if all its sides and angles are equal. This is important when calculating properties like perimeter or area.
A decagon can be visualized easily by imagining a ten-sided coin or a similar shape. Its ten sides mean it also has ten angles, which sum up to a total of 1440 degrees. Knowing these fundamental properties helps set the foundation for understanding how to approach problems involving decagons.

Polygon Sides

In geometry, **polygon sides** refer to the straight line segments that make up the shape of a polygon. The number of these sides defines the polygon. For example:

• A triangle has three sides.
• A quadrilateral has four sides.
• A pentagon has five sides, and so on.

Understanding the number of sides is crucial because it helps in calculating other properties.
When you know a shape has ten sides, for instance, you immediately recognize it's a decagon. This becomes especially useful in mathematical problems where you need to calculate things like perimeter or area. Additionally, recognizing that all sides of a regular polygon are equal can simplify these calculations significantly.

Regular Polygon Properties

**Regular polygons** have distinct characteristics that set them apart from irregular ones. Each side of a regular polygon is the same length, and all its interior angles are equal. This symmetry allows for easier calculations when determining properties like the perimeter.
Regular polygons are named based on their number of sides, and the more sides they have, the closer their shape resembles a circle. For example, a hexagon has six sides, an octagon has eight, and a decagon has ten. This characteristic regularity makes these shapes especially significant in mathematical problems and real-world applications like tiling and architectural design.
When dealing with regular polygons, you only need one measurement - the length of a side - to find the perimeter, as the uniformity simplifies the calculations.

Perimeter Calculation

The **perimeter** of a polygon is the total distance around the shape. For regular polygons, like a decagon, the perimeter can be easily calculated if you know the length of one side. The formula is simple:
\[P = ext{Number of sides} imes ext{Side length}\]This basic formula stems from the property that all sides of a regular polygon are equal in length. Understanding this, let's consider a decagon with a side length of 5 units. Its perimeter would be:
\[P = 10 imes 5 = 50 ext{ units}\]This calculation shows that for any regular polygon, once you know the side length, calculating the perimeter becomes straightforward. This same basic principle applies no matter how many sides the polygon has.