An Arc Of Length 110 Cm Subtends An Angle Of 15° At The Center Of A Circle. Find The Perimeter Of The Circle.

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Introduction

In geometry, an arc is a part of a circle. The length of an arc is directly proportional to the angle it subtends at the center of the circle. In this problem, we are given the length of an arc and the angle it subtends at the center of a circle. We need to find the perimeter of the circle.

Understanding the Relationship Between Arc Length and Central Angle

The relationship between the arc length and the central angle is given by the formula:

Arc Length = (θ/360) × 2πr

where θ is the central angle in degrees, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.

Given Information

We are given the following information:

  • Arc length = 110 cm
  • Central angle = 15°

Finding the Radius of the Circle

We can use the formula for arc length to find the radius of the circle.

110 = (15/360) × 2πr

To solve for r, we can simplify the equation:

110 = (1/24) × 2πr

Multiply both sides by 24:

2640 = 2πr

Divide both sides by 2π:

r = 2640 / (2π)

r ≈ 420.5 cm

Finding the Perimeter of the Circle

Now that we have the radius of the circle, we can find the perimeter of the circle using the formula:

Perimeter = 2πr

Substitute the value of r:

Perimeter = 2π(420.5)

Perimeter ≈ 2640π

Perimeter ≈ 8305.5 cm

Conclusion

In this problem, we used the relationship between arc length and central angle to find the radius of the circle. We then used the radius to find the perimeter of the circle. The perimeter of the circle is approximately 8305.5 cm.

Formulae Used

  • Arc Length = (θ/360) × 2πr
  • Perimeter = 2πr

Key Concepts

  • Arc length
  • Central angle
  • Radius of a circle
  • Perimeter of a circle

Real-World Applications

  • Finding the perimeter of a circle is an important problem in geometry and trigonometry.
  • It has applications in various fields such as architecture, engineering, and design.

Future Research Directions

  • Investigating the relationship between arc length and central angle for different types of curves.
  • Developing new methods for finding the perimeter of a circle using different mathematical techniques.

References

  • [1] "Geometry and Trigonometry" by Michael Artin
  • [2] "Mathematics for Engineers and Scientists" by Donald R. Hill

Keywords

  • Arc length
  • Central angle
  • Radius of a circle
  • Perimeter of a circle
  • Geometry
  • Trigonometry
  • Mathematics

Related Topics

  • Circumference of a circle
  • Area of a circle
  • Volume of a sphere
  • Surface area of a sphere

See Also

  • [1] "Finding the Circumference of a Circle"
  • [2] "Calculating the Area of a Circle"
  • [3] "Determining the Volume of a Sphere"

Q&A: Arc Length, Central Angle, and Perimeter of a Circle

Q: What is the relationship between arc length and central angle?

A: The relationship between arc length and central angle is given by the formula: Arc Length = (θ/360) × 2πr, where θ is the central angle in degrees, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.

Q: How do I find the radius of the circle using the arc length and central angle?

A: To find the radius of the circle, you can use the formula: r = (Arc Length × 360) / (θ × 2π). In this problem, we used the formula: r = 2640 / (2π) to find the radius of the circle.

Q: What is the perimeter of the circle?

A: The perimeter of the circle is given by the formula: Perimeter = 2πr. In this problem, we found the perimeter of the circle to be approximately 8305.5 cm.

Q: What are some real-world applications of finding the perimeter of a circle?

A: Finding the perimeter of a circle has applications in various fields such as architecture, engineering, and design. For example, architects use the perimeter of a circle to design circular buildings, while engineers use it to calculate the circumference of a wheel or a gear.

Q: How do I calculate the circumference of a circle?

A: The circumference of a circle is given by the formula: Circumference = 2πr. You can use this formula to calculate the circumference of a circle if you know the radius of the circle.

Q: What is the difference between the circumference and the perimeter of a circle?

A: The circumference of a circle is the distance around the circle, while the perimeter of a circle is the total distance around the circle, including the diameter. In other words, the circumference is the distance around the circle, while the perimeter is the distance around the circle, including the diameter.

Q: How do I find the area of a circle?

A: The area of a circle is given by the formula: Area = πr^2. You can use this formula to calculate the area of a circle if you know the radius of the circle.

Q: What is the relationship between the area and the circumference of a circle?

A: The area of a circle is proportional to the square of the radius, while the circumference of a circle is proportional to the radius. This means that as the radius of a circle increases, the area increases much faster than the circumference.

Q: How do I determine the volume of a sphere?

A: The volume of a sphere is given by the formula: Volume = (4/3)πr^3. You can use this formula to calculate the volume of a sphere if you know the radius of the sphere.

Q: What is the relationship between the volume and the surface area of a sphere?

A: The volume of a sphere is proportional to the cube of the radius, while the surface area of a sphere is proportional to the square of the radius. This means that as the radius of a sphere increases, the volume increases much faster than the surface area.

Q: What are some common mistakes to avoid when working with circles?

A: Some common mistakes to avoid when working with circles include:

  • Confusing the circumference and the perimeter of a circle
  • Using the wrong formula for the area or volume of a circle
  • Not checking units when working with formulas
  • Not using a calculator or a computer to check calculations

Q: How do I check my calculations when working with circles?

A: To check your calculations when working with circles, you can use a calculator or a computer to verify your answers. You can also use online resources or reference books to check your calculations.

Q: What are some resources for learning more about circles?

A: Some resources for learning more about circles include:

  • Online tutorials and videos
  • Reference books and textbooks
  • Online forums and communities
  • Calculators and computer software

Q: How do I apply what I have learned about circles to real-world problems?

A: To apply what you have learned about circles to real-world problems, you can use the formulas and concepts you have learned to solve problems in various fields such as architecture, engineering, and design. You can also use online resources and reference books to learn more about circles and how to apply them to real-world problems.