Consider Circle \[$ T \$\] With A Radius Of 24 In. And \[$\theta=\frac{5 \pi}{6}\$\] Radians.What Is The Length Of The Minor Arc \[$ SV \$\]?A. \[$ 20 \pi \$\] In. B. \[$ 28 \pi \$\] In. C. \[$ 40 \pi

Introduction

In mathematics, circles are fundamental geometric shapes that have been studied for centuries. A circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore the concept of minor arcs in a circle and how to calculate their length.

What is a Minor Arc?

A minor arc is a portion of a circle that is less than a semicircle. It is a curved line segment that connects two points on the circle. The length of a minor arc is a measure of the distance along the arc. In this article, we will focus on calculating the length of a minor arc given the radius of the circle and the angle subtended by the arc at the center of the circle.

Given Information

We are given a circle with a radius of 24 in. and an angle of $\theta = \frac{5 \pi}{6}$ radians. We need to find the length of the minor arc $SV$.

Calculating the Length of the Minor Arc

To calculate the length of the minor arc, we can use the formula:

$$L = \frac{\theta}{2\pi} \times 2\pi r$$

where $L$ is the length of the arc, $\theta$ is the angle subtended by the arc at the center of the circle, and $r$ is the radius of the circle.

Substituting the Given Values

We are given that $\theta = \frac{5 \pi}{6}$ radians and $r = 24$ in. Substituting these values into the formula, we get:

$$L = \frac{\frac{5 \pi}{6}}{2\pi} \times 2\pi \times 24$$

Simplifying the Expression

To simplify the expression, we can cancel out the $2\pi$ terms:

$$L = \frac{5 \pi}{6} \times 24$$

Multiplying the Numerator and Denominator

To multiply the numerator and denominator, we can multiply the numbers and keep the $\pi$ term separate:

$$L = \frac{5 \times 24}{6} \times \pi$$

Simplifying the Fraction

To simplify the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 6:

$$L = \frac{5 \times 4}{1} \times \pi$$

Multiplying the Numerator and Denominator

To multiply the numerator and denominator, we can multiply the numbers:

$$L = 20 \times \pi$$

Multiplying the Numerator and Denominator

To multiply the numerator and denominator, we can multiply the numbers:

$$L = 20 \pi$$

Conclusion

In this article, we calculated the length of a minor arc in a circle given the radius of the circle and the angle subtended by the arc at the center of the circle. We used the formula $L = \frac{\theta}{2\pi} \times 2\pi r$ to calculate the length of the arc. We substituted the given values into the formula and simplified the expression to get the final answer.

Final Answer

The length of the minor arc $SV$ is $20 \pi$ in.

Discussion

This problem is a classic example of how to calculate the length of a minor arc in a circle. The formula $L = \frac{\theta}{2\pi} \times 2\pi r$ is a fundamental concept in mathematics and is used to calculate the length of arcs in various geometric shapes.

Related Problems

• Calculating the length of a major arc in a circle
• Calculating the length of a semicircle in a circle
• Calculating the length of a circle in a circle

References

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

• Minor arc
• Circle
• Radius
• Angle
• Length
• Formula
• Calculation
• Geometry
• Mathematics
  Q&A: Calculating the Length of a Minor Arc in a Circle =====================================================

Introduction

In our previous article, we explored the concept of minor arcs in a circle and how to calculate their length. In this article, we will answer some frequently asked questions related to calculating the length of a minor arc in a circle.

Q: What is the formula for calculating the length of a minor arc in a circle?

A: The formula for calculating the length of a minor arc in a circle is:

$$L = \frac{\theta}{2\pi} \times 2\pi r$$

where $L$ is the length of the arc, $\theta$ is the angle subtended by the arc at the center of the circle, and $r$ is the radius of the circle.

Q: What is the difference between a minor arc and a major arc?

A: A minor arc is a portion of a circle that is less than a semicircle, while a major arc is a portion of a circle that is greater than a semicircle.

Q: How do I calculate the length of a major arc in a circle?

A: To calculate the length of a major arc in a circle, you can use the formula:

$$L = \frac{\theta}{2\pi} \times 2\pi r$$

However, since the arc is greater than a semicircle, you will need to subtract the length of the semicircle from the total length of the circle.

Q: Can I use the formula to calculate the length of a semicircle in a circle?

A: Yes, you can use the formula to calculate the length of a semicircle in a circle. Since the angle subtended by a semicircle is $\pi$ radians, you can substitute this value into the formula:

$$L = \frac{\pi}{2\pi} \times 2\pi r$$

Simplifying the expression, you get:

$$L = r$$

Q: What if I don't know the radius of the circle? Can I still calculate the length of the minor arc?

A: Yes, you can still calculate the length of the minor arc even if you don't know the radius of the circle. You can use the formula:

$$L = \frac{\theta}{2\pi} \times 2\pi R$$

where $R$ is the distance from the center of the circle to the point where the arc intersects the circle.

Q: Can I use the formula to calculate the length of an arc in a circle with a non-uniform radius?

A: No, the formula is only applicable to circles with a uniform radius. If the circle has a non-uniform radius, you will need to use a different formula or method to calculate the length of the arc.

Q: What if I have a circle with a radius of 10 in. and an angle of $\frac{3 \pi}{4}$ radians? How do I calculate the length of the minor arc?

A: To calculate the length of the minor arc, you can use the formula:

$$L = \frac{\theta}{2\pi} \times 2\pi r$$

Substituting the given values, you get:

$$L = \frac{\frac{3 \pi}{4}}{2\pi} \times 2\pi \times 10$$

Simplifying the expression, you get:

$$L = \frac{3 \pi}{4} \times 20$$

Multiplying the numerator and denominator, you get:

$$L = \frac{3 \times 20}{4} \times \pi$$

Simplifying the fraction, you get:

$$L = \frac{60}{4} \times \pi$$

Multiplying the numerator and denominator, you get:

$$L = 15 \times \pi$$

Multiplying the numerator and denominator, you get:

$$L = 15 \pi$$

Conclusion

In this article, we answered some frequently asked questions related to calculating the length of a minor arc in a circle. We provided formulas and examples to help you understand how to calculate the length of a minor arc in a circle.

Final Answer

The length of the minor arc is $15 \pi$ in.

Discussion

This problem is a classic example of how to calculate the length of a minor arc in a circle. The formula $L = \frac{\theta}{2\pi} \times 2\pi r$ is a fundamental concept in mathematics and is used to calculate the length of arcs in various geometric shapes.

Related Problems

• Calculating the length of a major arc in a circle
• Calculating the length of a semicircle in a circle
• Calculating the length of a circle in a circle

References

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

• Minor arc
• Circle
• Radius
• Angle
• Length
• Formula
• Calculation
• Geometry
• Mathematics