Exploring Arc Lengths and Central Angles in Circles

Arc Length and Central Angles

Introduction

  • The video explains how to calculate the length of an arc subtended by a central angle in a circle.

  • It uses the relationship between the ratio of the arc length to the circumference and the ratio of the central angle to 360 degrees.

Steps to Calculate Arc Length

  • Objective: Find the length of an arc given a central angle and the circumference of the circle.

  • Step 1: Identify Given Values

    • Determine the central angle (in degrees) and the circumference of the circle.

  • Step 2: Set Up the Proportion

    • Use the formula: arc lengthcircumference=central angle360∘\frac{\text{arc length}}{\text{circumference}} = \frac{\text{central angle}}{360^{\circ}}

  • Step 3: Simplify the Proportion

    • Simplify the fraction representing the central angle over 360 degrees.

  • Step 4: Solve for Arc Length

    • Multiply both sides of the equation by the circumference to solve for the arc length.

  • Step 5: Simplify the Result

    • Simplify the expression to find the numerical value of the arc length.

Scenario 1: Acute Central Angle

  • A circle with a circumference of 18π18\pi is given.

  • A central angle of 10∘10^{\circ} is given.

  • The goal is to find the length of the arc subtended by this central angle.

Calculation

  • The ratio of the arc length (a) to the circumference is equal to the ratio of the central angle to 360 degrees:

a18π=10360\frac{a}{18\pi} = \frac{10}{360}

  • Simplifying the fraction:

10360=136\frac{10}{360} = \frac{1}{36}

  • Solving for 'a':

a=136⋅18πa = \frac{1}{36} \cdot 18\pi

a=18π36a = \frac{18\pi}{36}

a=π2a = \frac{\pi}{2}

  • Therefore, the arc length is π2\frac{\pi}{2}.

Scenario 2: Obtuse Central Angle

  • The same circle with a circumference of 18π18\pi is considered.

  • A central angle of 350∘350^{\circ} is given.

  • The goal is to find the length of the arc subtended by this obtuse angle.

Calculation

  • The ratio of the arc length (a) to the circumference is equal to the ratio of the central angle to 360 degrees:

a18π=350360\frac{a}{18\pi} = \frac{350}{360}

  • Simplifying the fraction:

350360=3536\frac{350}{360} = \frac{35}{36}

  • Solving for 'a':

a=3536⋅18πa = \frac{35}{36} \cdot 18\pi

a=35⋅18⋅π36a = \frac{35 \cdot 18 \cdot \pi}{36}

  • Dividing 18 and 36 by 18:

a=35π2a = \frac{35\pi}{2}

  • Therefore, the arc length is 35π2\frac{35\pi}{2} or 17.5π17.5\pi.

Verification

  • The sum of the two arc lengths should equal the circumference of the circle.

  • Arc length 1: π2=0.5π\frac{\pi}{2} = 0.5\pi

  • Arc length 2: 17.5π17.5\pi

  • Sum: 0.5π+17.5π=18π0.5\pi + 17.5\pi = 18\pi

  • The sum of the two central angles should equal 360 degrees.

  • Central angle 1: 10∘10^{\circ}

  • Central angle 2: 350∘350^{\circ}

  • Sum: 10∘+350∘=360∘10^{\circ} + 350^{\circ} = 360^{\circ}

Conclusion

  • The calculations and verification demonstrate the relationship between arc length, central angles, and the circumference of a circle.

  • The ratio of arc length to circumference is equal to the ratio of the central angle to 360 degrees.