Calculating the Length of the Sides of the Rhombus

Oct 13, 2025 by wbrum.org 53 views

Introduction

A rhombus is a type of quadrilateral with all sides of equal length. In this article, we will 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 rhombus-shaped garden.

Understanding the Rhombus-Shaped Garden

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. Each triangle has a 30° angle, a 60° angle, and a 90° angle. The shorter distance across the middle of the garden measures 30 feet, which is the base of the rhombus.

Properties of a 30°-60°-90° Triangle

A 30°-60°-90° triangle is a special right triangle with a 30° angle, a 60° angle, and a 90° angle. The sides of this triangle are in the ratio 1:√3:2. The side opposite the 30° angle is 1 unit, the side opposite the 60° angle is √3 units, and the hypotenuse (the side opposite the 90° angle) is 2 units.

Calculating the Length of the Sides of the Rhombus

Since the garden is formed from 4 identical 30°-60°-90° triangles, each triangle has a base of 30 feet. The height of each triangle is √3 times the base, which is 30√3 feet. The hypotenuse of each triangle is 2 times the base, which is 60 feet.

Finding the Perimeter of the Rhombus

The perimeter of a rhombus is the sum of the lengths of all its sides. Since all sides of a rhombus are equal, we can find the perimeter by multiplying the length of one side by 4.

Step 1: Find the Length of One Side of the Rhombus

To find the length of one side of the rhombus, we need to find the length of the hypotenuse of one of the 30°-60°-90° triangles. The hypotenuse of each triangle is 60 feet.

Step 2: Multiply the Length of One Side by 4

Now that we have the length of one side of the rhombus, we can multiply it by 4 to find the perimeter.

Calculating the Perimeter

The perimeter of the rhombus is 4 times the length of one side. Since the length of one side is 60 feet, the perimeter is:

4 × 60 = 240 feet

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. We used the properties of a 30°-60°-90° triangle to find the length of the sides of the rhombus and then calculated the perimeter by multiplying the length of one side by 4. The perimeter of the rhombus-shaped garden is 240 feet.

Additional Information

A rhombus is a type of quadrilateral with all sides of equal length.
A 30°-60°-90° triangle is a special right triangle with a 30° angle, a 60° angle, and a 90° angle.
The sides of a 30°-60°-90° triangle are in the ratio 1:√3:2.
The perimeter of a rhombus is the sum of the lengths of all its sides.

References

[1] "Rhombus." Wikipedia, Wikimedia Foundation, 2023.
[2] "30°-60°-90° Triangle." Wikipedia, Wikimedia Foundation, 2023.
[3] "Perimeter of a Rhombus." Math Open Reference, 2023.
A Rhombus-Shaped Garden: Unraveling the Mystery of Its Perimeter ===========================================================

Q&A: A Rhombus-Shaped Garden

Q: What is a rhombus?

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

Q: What is a 30°-60°-90° triangle?

A: A 30°-60°-90° triangle is a special right triangle with a 30° angle, a 60° angle, and a 90° angle. The sides of this triangle are in the ratio 1:√3:2.

Q: How do you find the length of the sides of a rhombus?

A: To find the length of one side of the rhombus, we need to find the length of the hypotenuse of one of the 30°-60°-90° triangles. The hypotenuse of each triangle is 60 feet.

Q: How do you calculate the perimeter of a rhombus?

A: The perimeter of a rhombus is the sum of the lengths of all its sides. Since all sides of a rhombus are equal, we can find the perimeter by multiplying the length of one side by 4.

Q: What is the perimeter of the rhombus-shaped garden?

A: The perimeter of the rhombus-shaped garden is 240 feet.

Q: What are some additional properties of a rhombus?

A: A rhombus is a type of quadrilateral with all sides of equal length. The perimeter of a rhombus is the sum of the lengths of all its sides. The diagonals of a rhombus bisect each other at right angles.

Q: What are some additional properties of a 30°-60°-90° triangle?

A: A 30°-60°-90° triangle is a special right triangle with a 30° angle, a 60° angle, and a 90° angle. The sides of this triangle are in the ratio 1:√3:2. The hypotenuse of a 30°-60°-90° triangle is twice the length of the shorter leg.

Q: How do you find the length of the shorter leg of a 30°-60°-90° triangle?

A: To find the length of the shorter leg of a 30°-60°-90° triangle, we need to divide the length of the hypotenuse by 2.

Q: What is the relationship between the length of the shorter leg and the length of the hypotenuse of a 30°-60°-90° triangle?

A: The length of the shorter leg of a 30°-60°-90° triangle is half the length of the hypotenuse.

Q: What is the relationship between the length of the longer leg and the length of the hypotenuse of a 30°-60°-90° triangle?

A: The length of the longer leg of a 30°-60°-90° triangle is √3 times the length of the shorter leg.

Q: How do you find the length of the longer leg of a 30°-60°-90° triangle?

A: To find the length of the longer leg of a 30°-60°-90° triangle, we need to multiply the length of the shorter leg by √3.

Conclusion

In this article, we explored a garden designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. We used the properties of a 30°-60°-90° triangle to find the length of the sides of the rhombus and then calculated the perimeter by multiplying the length of one side by 4. We also answered some frequently asked questions about rhombuses and 30°-60°-90° triangles.

Additional Information

A rhombus is a type of quadrilateral with all sides of equal length.
A 30°-60°-90° triangle is a special right triangle with a 30° angle, a 60° angle, and a 90° angle.
The sides of a 30°-60°-90° triangle are in the ratio 1:√3:2.
The perimeter of a rhombus is the sum of the lengths of all its sides.

References

[1] "Rhombus." Wikipedia, Wikimedia Foundation, 2023.
[2] "30°-60°-90° Triangle." Wikipedia, Wikimedia Foundation, 2023.
[3] "Perimeter of a Rhombus." Math Open Reference, 2023.