Find the length of one side

Oct 13, 2025 by recoveryshake.com 14 views

Introduction

A regular decagon is a polygon with 10 equal sides and 10 equal angles. When a regular decagon is inscribed in a circle, it means that all the vertices of the decagon touch the circle. In this article, we will explore the perimeter of a regular decagon inscribed in a circle with a radius of 1 meter.

Understanding the Problem

To find the perimeter of the regular decagon, we need to understand the relationship between the decagon and the circle. Since the decagon is inscribed in the circle, the radius of the circle is also the distance from the center of the circle to any vertex of the decagon.

Breaking Down the Problem

To find the perimeter of the decagon, we can break down the problem into smaller steps:

Find the central angle: The central angle is the angle formed by two radii that intersect at the center of the circle. Since the decagon is regular, the central angle is equal to 360°/10 = 36°.
Find the length of one side: To find the length of one side of the decagon, we can use the formula for the length of a chord in a circle: c = 2r sin(θ/2), where c is the length of the chord, r is the radius of the circle, and θ is the central angle.
Find the perimeter: Once we have the length of one side, we can find the perimeter of the decagon by multiplying the length of one side by the number of sides.

Calculating the Central Angle

The central angle is equal to 360°/10 = 36°.

Calculating the Length of One Side

Using the formula for the length of a chord in a circle, we can find the length of one side of the decagon:

c = 2r sin(θ/2) c = 2(1) sin(36°/2) c = 2(1) sin(18°) c ≈ 1.682

Calculating the Perimeter

Now that we have the length of one side, we can find the perimeter of the decagon by multiplying the length of one side by the number of sides:

Perimeter = 10c Perimeter ≈ 10(1.682) Perimeter ≈ 16.82

Conclusion

The perimeter of a regular decagon inscribed in a circle with a radius of 1 meter is approximately 16.82 meters.

Additional Information

The formula for the length of a chord in a circle is c = 2r sin(θ/2), where c is the length of the chord, r is the radius of the circle, and θ is the central angle.
The central angle is equal to 360°/n, where n is the number of sides of the polygon.
The perimeter of a regular polygon is equal to the number of sides multiplied by the length of one side.

References

[1] "Geometry" by Michael Artin
[2] "Trigonometry" by I.M. Gelfand
[3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored the perimeter of a regular decagon inscribed in a circle with a radius of 1 meter. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the formula for the perimeter of a regular decagon inscribed in a circle?

A: The formula for the perimeter of a regular decagon inscribed in a circle is P = 10c, where c is the length of one side of the decagon. To find the length of one side, we can use the formula c = 2r sin(θ/2), where r is the radius of the circle and θ is the central angle.

Q: How do I find the central angle of a regular decagon inscribed in a circle?

A: The central angle of a regular decagon inscribed in a circle is equal to 360°/10 = 36°.

Q: What is the relationship between the radius of the circle and the length of one side of the decagon?

A: The radius of the circle is also the distance from the center of the circle to any vertex of the decagon. This distance is used to find the length of one side of the decagon using the formula c = 2r sin(θ/2).

Q: Can I use the formula c = 2r sin(θ/2) to find the length of one side of any polygon inscribed in a circle?

A: Yes, you can use the formula c = 2r sin(θ/2) to find the length of one side of any polygon inscribed in a circle, as long as you know the radius of the circle and the central angle.

Q: What is the significance of the central angle in finding the perimeter of a regular decagon?

A: The central angle is used to find the length of one side of the decagon, which is then used to find the perimeter of the decagon.

Q: Can I use a calculator to find the perimeter of a regular decagon inscribed in a circle?

A: Yes, you can use a calculator to find the perimeter of a regular decagon inscribed in a circle. Simply enter the values of the radius and the central angle into the formula c = 2r sin(θ/2) and then multiply the result by 10 to find the perimeter.

Q: What is the relationship between the perimeter of a regular decagon and the number of sides?

A: The perimeter of a regular decagon is equal to the number of sides (10) multiplied by the length of one side.

Conclusion

In this article, we have answered some frequently asked questions related to the perimeter of a regular decagon inscribed in a circle with a radius of 1 meter. We hope that this article has provided you with a better understanding of the topic.

Additional Information

The formula for the perimeter of a regular decagon inscribed in a circle is P = 10c, where c is the length of one side of the decagon.
The formula for the length of one side of a decagon inscribed in a circle is c = 2r sin(θ/2), where r is the radius of the circle and θ is the central angle.
The central angle of a regular decagon inscribed in a circle is equal to 360°/10 = 36°.

References

[1] "Geometry" by Michael Artin
[2] "Trigonometry" by I.M. Gelfand
[3] "Mathematics for the Nonmathematician" by Morris Kline