Question

Find the perimeter of each regular polygon with the given side lengths. triangle, \(16 \mathrm{~km}\)

Short answer

The perimeter of the triangle is 48 km.

Step-by-step solution

Identify the Polygon

We are given a shape that is a regular triangle (also known as an equilateral triangle). This means that all three sides of the triangle are of equal length. Therefore, the number of sides is 3.

Understand the Side Length

Each side of the triangle is given to be 16 km. Since the shape is regular, all sides of the triangle are equal.

Calculate the Perimeter using Formula

The perimeter of a regular polygon is calculated by multiplying the number of sides by the length of one side. For a triangle, this formula is: \( P = n imes s \), where \( n = 3 \) (number of sides) and \( s = 16 \text{ km} \) (length of a side).

Substitute Values and Compute

Substitute the values into the formula: \( P = 3 imes 16 \text{ km} \). Calculate the product: \( P = 48 \text{ km} \).

Key concepts

Regular Polygon

A regular polygon is a geometric shape where all sides are of equal length, and all interior angles are the same. This symmetry gives regular polygons some distinct properties:

• Every vertex angle in a regular polygon is identical, making structures stable and aesthetically pleasing.
• The sides and angles help in calculating areas and perimeters easily.
• Examples of regular polygons include equilateral triangles, squares, and regular pentagons.

For any regular polygon, if you know the number of sides and the length of one side, you can calculate the perimeter easily. This is done with the formula: \[ \text{Perimeter} = \text{Number of sides} \times \text{Length of one side} \]
This formula highlights the fundamental property of regular polygons: uniformity. Knowing this makes understanding perimeters intuitive and straightforward.

Equilateral Triangle

An equilateral triangle is a type of regular polygon specifically with three sides. In this triangle, all sides are equal, and each interior angle measures 60 degrees. This leads to several interesting characteristics:

• All interior angles add up to 180 degrees, consistent with all triangles.
• Being regular, the equilateral triangle offers elegance through its symmetry.
• Due to equal side lengths, determining the perimeter is simplified.

To find the perimeter of an equilateral triangle, use the formula:
\[ \text{Perimeter} = 3 \times \text{side length} \]
This formula reinforces the simplicity of equilateral triangles. With knowing just one side, you can deduce several properties instantly.

Side Length

Side length is a crucial measurement in geometry, especially when dealing with regular polygons and triangles. It determines several attributes of geometric shapes:

• The side length in regular polygons allows for easy calculation of the perimeter.
• In an equilateral triangle, knowing any side lets you compute the entire perimeter.
• A uniform side length is essential for confirming a polygon's regularity.

When engaging with problems involving regular polygons, always check for equal side lengths. This ensures the shape is regular and calculations proceed accordingly. Knowing this aids in efficient problem-solving and a deeper understanding of geometric principles.