The examples will help us to understand to how find the length of an arc using the formula of ‘s is equal to r theta’.

Worked-out problems on length of an arc:

**1.** In a circle of radius 6 cm, an arc of certain length subtends 20° 17’ at the center. Find in sexagesimal unit the angle subtended by the same arc at the center of a circle of radius 8 cm.

**Solution:**

Let an arc of length be m cm subtends 20° 17’ at the center of a circle of radius 6 cm and α° at the center of a circle of radius 8 cm.

Now, 20° 17’ = {20 (17/60)}°

= (1217/60)°

= 1217π/(60 × 180) radian [since, 180° = π radian]

And α° = πα/180 radian

We know, the formula, s = rθ then we get,

When the circle of radius is 6 cm; m = 6 × [(1217π)/(60 × 180)] ………… (i)

And when the circle of radius 8 cm; m = 8 × (πα)/180 …………… (ii)

Therefore, from (i) and (ii) we get;

8 × (πα)/180 = 6 × [(1217π)/(60 × 180)]

or, α = [(6/8) × (1217/60)]°

or, α = (3/4) × 20° 17’ [since, (1217/60)° = 20° 17’]

or, α = 3 × 5°4’ 15”

or, α = 15° 12’ 45”.

Therefore, the required angle in sexagesimal unit = 15° 12’ 45”.

**2.** Aaron is running along a circular track at the rate of 10 mile per hour traverses in 36 seconds an arc which subtends 56° at the center. Find the diameter of the circle.

**Solution:**

One hour = 3600 seconds

One mile = 5280 feet

Therefore, 10 miles = (5280 × 10) feet = 52800 feet

In 3600 seconds Aaron goes 52800 feet

In 1 second Aaron goes 52800/3600 feet = 44/3 feet

Therefore, in 36 seconds the Aaron goes (44/3) × 36 feet = 528 feet.

Clearly, an arc of length 528 feet subtends 56° = 56 × π/180 radian at the center of the circular track. If ‘y’ feet is the radius of the circular track then using the formula s = rθ we get,

y = s/θ

y = 528/[56 × (π/180)]

y = (528 × 180 × 7)/(56 × 22) feet

y = 540 feet

y = (540/3) yards [since, we know that 3 foot = 1 yard]

y = 180 yards

Therefore, the required diameter = 2 × 180 yards = 360 yards.

According to the problem, the length of an arc l

l

Similarly, l

and l

Therefore, , l

Let an arc of length (l

Then, α = (l

Now, put the value of l

or, α = (r

To solve more problems on length of an arc follow the proof on 'Theta equals s over r'.

**●** **Measurement of Angles**

**Sign of Angles****Trigonometric Angles****Measure of Angles in Trigonometry****Systems of Measuring Angles****Important Properties on Circle****S is Equal to R Theta****Sexagesimal, Centesimal and Circular Systems****Convert the Systems of Measuring Angles****Convert Circular Measure****Convert into Radian****Problems Based on Systems of Measuring Angles****Length of an Arc****Problems based on S R Theta Formula**

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