Calculating the Side Length of the Rhombus

Oct 13, 2025 by leolavanderia.com.br 132 views

Introduction

A garden is a serene and beautiful place where nature and mathematics come together in perfect harmony. In this article, we will delve into the fascinating world of geometry and explore a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. Our goal is to find the distance around the perimeter of this unique garden.

Understanding the Rhombus

A rhombus is a quadrilateral with all sides of equal length. In this case, the garden is formed from 4 identical 30°-60°-90° triangles, which means each side of the rhombus is a combination of two 30°-60°-90° triangles. To understand the properties of a rhombus, let's recall the characteristics of a 30°-60°-90° triangle:

The side opposite the 30° angle is half the length of the hypotenuse.
The side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
The hypotenuse is twice the length of the side opposite the 30° angle.

Calculating the Side Length of the Rhombus

Since the garden is formed from 4 identical 30°-60°-90° triangles, we can use the properties of these triangles to find the side length of the rhombus. Let's denote the side length of the rhombus as 's'. Since each side of the rhombus is a combination of two 30°-60°-90° triangles, we can write:

s = 2 × (side opposite the 30° angle)

Using the property of a 30°-60°-90° triangle, we know that the side opposite the 30° angle is half the length of the hypotenuse. Therefore:

s = 2 × (0.5 × hypotenuse)

Since the hypotenuse is twice the length of the side opposite the 30° angle, we can write:

s = 2 × (0.5 × 2 × (side opposite the 30° angle))

Simplifying the equation, we get:

s = 2 × (side opposite the 30° angle)

Now, let's use the fact that the shorter distance across the middle of the garden measures 30 feet. This distance is equal to the sum of the two sides opposite the 30° angles in each of the two 30°-60°-90° triangles that form the rhombus. Therefore:

30 = 2 × (side opposite the 30° angle)

Dividing both sides by 2, we get:

15 = side opposite the 30° angle

Now that we know the length of the side opposite the 30° angle, we can find the side length of the rhombus:

s = 2 × (side opposite the 30° angle) s = 2 × 15 s = 30

Calculating the Perimeter of the Rhombus

Now that we know the side length of the rhombus, we can calculate its perimeter. The perimeter of a rhombus is equal to the sum of its four sides. Since all sides of a rhombus are equal, we can write:

Perimeter = 4 × s Perimeter = 4 × 30 Perimeter = 120

Conclusion

In this article, we explored a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. Using the properties of 30°-60°-90° triangles, we calculated the side length of the rhombus and found it to be 30 feet. Finally, we calculated the perimeter of the rhombus and found it to be 120 feet.

Final Thoughts

The garden in the shape of a rhombus is a unique and fascinating example of how mathematics can be applied to real-world problems. By understanding the properties of 30°-60°-90° triangles, we can design and calculate the dimensions of complex geometric shapes like the rhombus. This article has shown that with a little bit of mathematical knowledge and creativity, we can create beautiful and intricate designs that reflect the harmony between nature and mathematics.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Joy of Geometry" by Paul Lockhart

Additional Resources

[1] "Geometry Tutorials" by Khan Academy
[2] "Mathematics Tutorials" by MIT OpenCourseWare
[3] "Geometry Software" by GeoGebra

Introduction

In our previous article, we explored a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the middle of the garden measures 30 feet. We calculated the side length of the rhombus and found it to be 30 feet, and finally, we calculated the perimeter of the rhombus and found it to be 120 feet. In this article, we will answer some of the most frequently asked questions about this unique garden.

Q&A

Q: What is a rhombus?

A: A rhombus is a quadrilateral with all sides of equal length. In this case, the garden is formed from 4 identical 30°-60°-90° triangles, which means each side of the rhombus is a combination of two 30°-60°-90° triangles.

Q: What is a 30°-60°-90° triangle?

A: A 30°-60°-90° triangle is a special right triangle with angles measuring 30°, 60°, and 90°. The side opposite the 30° angle is half the length of the hypotenuse, the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle, and the hypotenuse is twice the length of the side opposite the 30° angle.

Q: How did you calculate the side length of the rhombus?

A: We used the properties of 30°-60°-90° triangles to calculate the side length of the rhombus. Since each side of the rhombus is a combination of two 30°-60°-90° triangles, we can write:

s = 2 × (side opposite the 30° angle)

Using the property of a 30°-60°-90° triangle, we know that the side opposite the 30° angle is half the length of the hypotenuse. Therefore:

s = 2 × (0.5 × hypotenuse)

Since the hypotenuse is twice the length of the side opposite the 30° angle, we can write:

s = 2 × (0.5 × 2 × (side opposite the 30° angle))

Simplifying the equation, we get:

s = 2 × (side opposite the 30° angle)

30 = 2 × (side opposite the 30° angle)

Dividing both sides by 2, we get:

15 = side opposite the 30° angle

Now that we know the length of the side opposite the 30° angle, we can find the side length of the rhombus:

s = 2 × (side opposite the 30° angle) s = 2 × 15 s = 30

Q: How did you calculate the perimeter of the rhombus?

A: Now that we know the side length of the rhombus, we can calculate its perimeter. The perimeter of a rhombus is equal to the sum of its four sides. Since all sides of a rhombus are equal, we can write:

Perimeter = 4 × s Perimeter = 4 × 30 Perimeter = 120

Q: What is the significance of the 30°-60°-90° triangle in this problem?

A: The 30°-60°-90° triangle is a special right triangle with unique properties that make it useful in this problem. The properties of the 30°-60°-90° triangle allow us to calculate the side length of the rhombus and its perimeter.

Q: Can you provide more examples of how to use the 30°-60°-90° triangle in real-world problems?

A: Yes, the 30°-60°-90° triangle has many applications in real-world problems, such as:

Calculating the height of a building or a tree
Determining the length of a shadow
Finding the distance between two points on a map
Calculating the area of a triangle or a rectangle

These are just a few examples of how the 30°-60°-90° triangle can be used in real-world problems.

Conclusion

In this article, we answered some of the most frequently asked questions about the garden in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. We hope that this Q&A article has provided you with a better understanding of the properties of the 30°-60°-90° triangle and its applications in real-world problems.

Final Thoughts

The 30°-60°-90° triangle is a special right triangle with unique properties that make it useful in many real-world problems. By understanding the properties of the 30°-60°-90° triangle, we can solve complex problems and make informed decisions in our daily lives.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "The Joy of Geometry" by Paul Lockhart

Additional Resources

[1] "Geometry Tutorials" by Khan Academy
[2] "Mathematics Tutorials" by MIT OpenCourseWare
[3] "Geometry Software" by GeoGebra