Finding the Radius of a Circle Given Arc Length and Central Angle

In geometry, the relationship between the arc length, the central angle, and the radius of a circle is a fundamental concept in trigonometry and calculus. This article explains how to find the radius of a circle when given the length of an arc and the central angle in radians. It covers multiple methods to arrive at the solution, ensuring clarity and understanding of the underlying principles.

Understanding the Concept

The arc length ( L ) of a circle is directly proportional to the central angle ( theta ) in radians and the radius ( r ) of the circle. The formula that describes this relationship is:

[ L r theta ]

Where: L is the length of the arc. r is the radius of the circle. θ is the central angle in radians.

Given Values

For this example, we have the following given values:

The length of the arc, L 50 cm. The central angle, θ 2.5 radians.

Using the Formula Directly

To find the radius, we rearrange the formula to solve for ( r ):

[ r frac{L}{theta} ]

Substituting the given values:

[ r frac{50 text{ cm}}{2.5 text{ radians}} 20 text{ cm} ]

Hence, the radius of the circle is 20 cm.

Using Proportional Reasoning

Another method involves recognizing that 1 radian corresponds to an arc length equal to the radius of the circle. Therefore, for an angle of 2.5 radians, the arc length is 2.5 times the radius:

[ 50 text{ cm} 2.5 times r ]

Solving for ( r ):

[ r frac{50 text{ cm}}{2.5} 20 text{ cm} ]

Alternative Methods

1. **Literal Proportion Method:**

Using the arc length formula and proportionality, we can derive:

[ frac{L}{2pi r} frac{theta}{2pi} ]

Solving for ( r ):

[ r frac{L}{theta} frac{50 text{ cm}}{2.5 text{ radians}} 20 text{ cm} ]

2. **Circumference Approach:**

Given that 2π radians correspond to the full circumference, we can find the radius by first finding the full circumference:

[ text{Circumference} frac{2.5 text{ radians}}{2pi} times 2pi r 50 text{ cm} ]

Solving for the circumference:

[ text{Circumference} frac{50 text{ cm} times 2pi}{2.5} 40pi text{ cm} ]

The circumference formula is:

[ text{Circumference} 2pi r ]

Solving for ( r ):

[ r frac{40pi text{ cm}}{2pi} 20 text{ cm} ]

Conclusion

The radius of the circle, given an arc length of 50 cm and a central angle of 2.5 radians, is 20 cm. This solution can be verified using multiple methods, ensuring the accuracy of the result. Understanding these principles is crucial for solving problems related to circles and trigonometry.