A Circle Has A Central Angle Measuring $\frac{7 \pi}{10}$ Radians That Intersects An Arc Of Length 33 Cm. What Is The Length Of The Radius Of The Circle? Round Your Answer To The Nearest Whole Cm. Use 3.14 For $\pi$.A. 11 Cm B.

Mar 11, 2025 by ADMIN 233 views

A Circle's Central Angle and Arc Length: Finding the Radius

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Understanding the Problem

A circle is a fundamental geometric shape with a wide range of applications in mathematics, physics, and engineering. In this article, we will explore the relationship between a circle's central angle, arc length, and radius. We will use the given information to find the length of the radius of the circle.

The Relationship Between Central Angle, Arc Length, and Radius

The central angle of a circle is the angle formed by two radii that intersect at the center of the circle. The arc length of a circle is the length of the curved line that makes up a portion of the circle. The relationship between the central angle, arc length, and radius of a circle is given by the formula:

\text{Arc Length} = \text{Radius} \times \text{Central Angle}

where the central angle is measured in radians.

Given Information

We are given that the central angle of the circle measures $\frac{7 \pi}{10}$ radians and that the arc length is 33 cm. We are also given that $\pi = 3.14$.

Finding the Radius

We can use the formula above to find the radius of the circle. Rearranging the formula to solve for the radius, we get:

\text{Radius} = \frac{\text{Arc Length}}{\text{Central Angle}}

Substituting the given values, we get:

\text{Radius} = \frac{33}{\frac{7 \pi}{10}}

Using the value of $\pi = 3.14$, we get:

\text{Radius} = \frac{33}{\frac{7 \times 3.14}{10}}

Simplifying the expression, we get:

\text{Radius} = \frac{33}{2.198}

Evaluating the expression, we get:

\text{Radius} = 15

Conclusion

We have found that the length of the radius of the circle is 15 cm. This is the final answer to the problem.

Discussion

The relationship between the central angle, arc length, and radius of a circle is a fundamental concept in geometry. The formula above can be used to find the radius of a circle given the central angle and arc length. This formula is a powerful tool for solving problems involving circles.

Example Use Case

The formula above can be used in a variety of real-world applications, such as:

Calculating the circumference of a circle given the central angle and arc length.
Finding the area of a circle given the central angle and arc length.
Determining the length of a chord of a circle given the central angle and arc length.

Limitations

The formula above assumes that the central angle is measured in radians. If the central angle is measured in degrees, the formula above will not give the correct result.

Future Work

In the future, it would be interesting to explore the relationship between the central angle, arc length, and radius of a circle in more detail. This could involve investigating the properties of circles with different central angles and arc lengths.

References

[1] "Geometry: A Comprehensive Introduction". By Dan Pedoe.
[2] "A First Course in Algebra". By John B. Fraleigh.

Code

import math
def find_radius(arc_length, central_angle):
"""
Find the radius of a circle given the arc length and central angle.
Parameters:
arc_length (float): The length of the arc.
central_angle (float): The central angle in radians.

Returns:
float: The radius of the circle.
&quot;&quot;&quot;
radius = arc_length / central_angle
return radius

arc_length = 33
central_angle = (7 * math.pi) / 10

radius = find_radius(arc_length, central_angle)

print("The radius of the circle is:", round(radius))

This code defines a function find_radius that takes the arc length and central angle as input and returns the radius of the circle. The function uses the formula above to calculate the radius. The code then uses this function to find the radius of the circle given the given values. The result is printed to the console.

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about the relationship between a circle's central angle, arc length, and radius.

Q: What is the relationship between the central angle, arc length, and radius of a circle?

A: The relationship between the central angle, arc length, and radius of a circle is given by the formula:

\text{Arc Length} = \text{Radius} \times \text{Central Angle}

where the central angle is measured in radians.

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

A: To find the radius of a circle given the central angle and arc length, you can use the formula:

\text{Radius} = \frac{\text{Arc Length}}{\text{Central Angle}}

Q: What if the central angle is measured in degrees instead of radians?

A: If the central angle is measured in degrees instead of radians, you will need to convert it to radians before using the formula. To convert degrees to radians, you can use the following formula:

\text{Radians} = \text{Degrees} \times \frac{\pi}{180}

Q: Can I use the formula to find the circumference of a circle given the central angle and arc length?

A: Yes, you can use the formula to find the circumference of a circle given the central angle and arc length. The circumference of a circle is given by the formula:

\text{Circumference} = 2 \times \pi \times \text{Radius}

Q: What if I want to find the area of a circle given the central angle and arc length?

A: To find the area of a circle given the central angle and arc length, you will need to use the formula for the area of a circle:

\text{Area} = \pi \times \text{Radius}^2

Q: Can I use the formula to find the length of a chord of a circle given the central angle and arc length?

A: Yes, you can use the formula to find the length of a chord of a circle given the central angle and arc length. The length of a chord of a circle is given by the formula:

\text{Chord Length} = 2 \times \text{Radius} \times \sin\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a tangent of a circle given the central angle and arc length?

A: To find the length of a tangent of a circle given the central angle and arc length, you will need to use the formula for the length of a tangent:

\text{Tangent Length} = \text{Radius} \times \tan\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a secant of a circle given the central angle and arc length?

A: Yes, you can use the formula to find the length of a secant of a circle given the central angle and arc length. The length of a secant of a circle is given by the formula:

\text{Secant Length} = \text{Radius} \times \sec\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a normal of a circle given the central angle and arc length?

A: To find the length of a normal of a circle given the central angle and arc length, you will need to use the formula for the length of a normal:

\text{Normal Length} = \text{Radius} \times \csc\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a bisector of a circle given the central angle and arc length?

A: Yes, you can use the formula to find the length of a bisector of a circle given the central angle and arc length. The length of a bisector of a circle is given by the formula:

\text{Bisector Length} = \text{Radius} \times \cot\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a median of a circle given the central angle and arc length?

A: To find the length of a median of a circle given the central angle and arc length, you will need to use the formula for the length of a median:

\text{Median Length} = \text{Radius} \times \sin\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a diameter of a circle given the central angle and arc length?

A: Yes, you can use the formula to find the length of a diameter of a circle given the central angle and arc length. The length of a diameter of a circle is given by the formula:

\text{Diameter Length} = 2 \times \text{Radius}

Q: What if I want to find the length of a chord of a circle given the central angle and arc length, and the chord is not a diameter?

A: To find the length of a chord of a circle given the central angle and arc length, and the chord is not a diameter, you will need to use the formula for the length of a chord:

\text{Chord Length} = 2 \times \text{Radius} \times \sin\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a tangent of a circle given the central angle and arc length, and the tangent is not a diameter?

A: Yes, you can use the formula to find the length of a tangent of a circle given the central angle and arc length, and the tangent is not a diameter. The length of a tangent of a circle is given by the formula:

\text{Tangent Length} = \text{Radius} \times \tan\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a secant of a circle given the central angle and arc length, and the secant is not a diameter?

A: To find the length of a secant of a circle given the central angle and arc length, and the secant is not a diameter, you will need to use the formula for the length of a secant:

\text{Secant Length} = \text{Radius} \times \sec\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a normal of a circle given the central angle and arc length, and the normal is not a diameter?

A: Yes, you can use the formula to find the length of a normal of a circle given the central angle and arc length, and the normal is not a diameter. The length of a normal of a circle is given by the formula:

\text{Normal Length} = \text{Radius} \times \csc\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a bisector of a circle given the central angle and arc length, and the bisector is not a diameter?

A: To find the length of a bisector of a circle given the central angle and arc length, and the bisector is not a diameter, you will need to use the formula for the length of a bisector:

\text{Bisector Length} = \text{Radius} \times \cot\left(\frac{\text{Central Angle}}{2}\right)

Q: Can I use the formula to find the length of a median of a circle given the central angle and arc length, and the median is not a diameter?

A: Yes, you can use the formula to find the length of a median of a circle given the central angle and arc length, and the median is not a diameter. The length of a median of a circle is given by the formula:

\text{Median Length} = \text{Radius} \times \sin\left(\frac{\text{Central Angle}}{2}\right)

Q: What if I want to find the length of a diameter of a circle given the central angle and arc length, and the diameter is not a chord?

A: To find the length of a diameter of a circle given the central angle and arc length, and the diameter is not a chord, you will need to use the formula for the length of a diameter:

\text{Diameter Length} = 2 \times \text{Radius}

Q: Can I use the formula to find the length of a chord of a circle given the central angle and arc length, and the chord is not a diameter, and the chord is not a tangent?

A: Yes, you can use the formula to find the length of a chord of a circle given the central angle and arc length, and the chord is not a diameter, and the chord is not a tangent. The length of a chord of a circle is given by the formula:

\text{Ch