Understanding the Coordinates of the Vertices

Oct 13, 2025 by miodeya.com 115 views

Introduction

A hexagon is a six-sided polygon with all sides of equal length. In this article, we will explore the concept of a hexagon and how to calculate the length of each side using the coordinates of its vertices. We will use a scale drawing of a fountain in the shape of a hexagon to demonstrate this concept.

Understanding the Coordinates of the Vertices

The coordinates of the vertices of the fountain are given as (10.5,8), (145,5), (10.5,2), (5.5,2), (1.5,5), and (5.5,8). These coordinates represent the x and y values of each vertex, which can be used to calculate the length of each side.

Calculating the Distance Between Two Points

To calculate the length of each side, we need to calculate the distance between two points. The distance between two points (x1, y1) and (x2, y2) can be calculated using the following formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Calculating the Length of Each Side

Using the coordinates of the vertices, we can calculate the length of each side by applying the distance formula. Let's start by calculating the length of each side:

Side 1: (10.5,8) to (145,5)

To calculate the length of side 1, we need to calculate the distance between the points (10.5,8) and (145,5).

Distance = √((145 - 10.5)^2 + (5 - 8)^2) Distance = √((134.5)^2 + (-3)^2) Distance = √(18022.25 + 9) Distance = √18031.25 Distance ≈ 134.5

Side 2: (145,5) to (10.5,2)

To calculate the length of side 2, we need to calculate the distance between the points (145,5) and (10.5,2).

Distance = √((10.5 - 145)^2 + (2 - 5)^2) Distance = √((-134.5)^2 + (-3)^2) Distance = √(18022.25 + 9) Distance = √18031.25 Distance ≈ 134.5

Side 3: (10.5,2) to (5.5,2)

To calculate the length of side 3, we need to calculate the distance between the points (10.5,2) and (5.5,2).

Distance = √((5.5 - 10.5)^2 + (2 - 2)^2) Distance = √((-5)^2 + 0) Distance = √25 Distance = 5

Side 4: (5.5,2) to (1.5,5)

To calculate the length of side 4, we need to calculate the distance between the points (5.5,2) and (1.5,5).

Distance = √((1.5 - 5.5)^2 + (5 - 2)^2) Distance = √((-4)^2 + 3^2) Distance = √(16 + 9) Distance = √25 Distance = 5

Side 5: (1.5,5) to (5.5,8)

To calculate the length of side 5, we need to calculate the distance between the points (1.5,5) and (5.5,8).

Distance = √((5.5 - 1.5)^2 + (8 - 5)^2) Distance = √((4)^2 + 3^2) Distance = √(16 + 9) Distance = √25 Distance = 5

Side 6: (5.5,8) to (10.5,8)

To calculate the length of side 6, we need to calculate the distance between the points (5.5,8) and (10.5,8).

Distance = √((10.5 - 5.5)^2 + (8 - 8)^2) Distance = √((5)^2 + 0) Distance = √25 Distance = 5

Conclusion

In this article, we calculated the length of each side of a hexagon using the coordinates of its vertices. We used the distance formula to calculate the length of each side, and we found that each side has a length of approximately 134.5, 134.5, 5, 5, 5, and 5, respectively. This demonstrates the concept of a hexagon and how to calculate the length of each side using the coordinates of its vertices.

Discussion

The calculation of the length of each side of a hexagon is an important concept in mathematics, particularly in geometry. It has many practical applications in fields such as architecture, engineering, and computer science. The ability to calculate the length of each side of a hexagon is essential for designing and building structures that require precise measurements.

References

[1] "Geometry" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.