Length of an Arc

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.


3. If α1, α2, α3 radians be the angles subtended by the arcs of lengths l1, l2, l3 at the centers of the circles whose radii are r1, r2, r3 respectively then show that the angle subtended at the centre by the arc of length (l1 + l2 + l3) of a circle whose radius is (r1 + r2 + r3) will be (r1 α1 + r2α2 + r3α3)/(r1 + r2 + r3) radian.

Solution:

According to the problem, the length of an arc l1 of a circle of radius r1 subtends an angle α1 at its center. Hence, using the formula, s = rθ we get,

l1 = r1α1.

Similarly, l2 = r2α2

and l3 = r3 α3.

Therefore, , l1 + l2 + l3 = r1α1 + r2α2 + r3α3.

Let an arc of length (l1 + l2 + l3) of a circle of radius (r1 + r2 + r3) subtend an angle α radian at its center.

Then, α = (l1 + l2 + l3)/(r1 + r2 + r3)

Now, put the value of l1 = r1α1, l2 = r2α2 and l3 = r3α3.

or, α = (r1α1 + r2α2 + r3α3)/(r1 + r2 + r3) radian. Proved.

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

 Measurement of Angles





11 and 12 Grade Math

From Length of an Arc to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Counting Numbers from 1 to 50 | Match the Number | Missing Numbers

    Apr 04, 25 03:46 PM

    Math Coloring Pages on Counting Number Oredr
    In counting numbers from 1 to 50, recognize the numbers, count and then join the numbers in the correct number order. Here we mainly need eye-hand coordination to draw the picture and maintain the num

    Read More

  2. Counting Eleven to Twenty with Numbers and Words |Numbers from 11 - 20

    Apr 04, 25 03:21 PM

    Counting eleven to twenty with numbers and words are explained below. One ten and one more is eleven. Eleven comes after ten. One ten and two more is twelve. Twelve comes after eleven.

    Read More

  3. 5th Grade BODMAS Rule Worksheet | PEMDAS | Order of operations|Answers

    Apr 03, 25 03:11 PM

    5th Grade BODMAS Rule Worksheet
    In 5th Grade BODMAS Rule Worksheet you will get different types of problems on mathematical expressions involving different operations, mathematical expression with 'brackets' and 'of' and simplifying…

    Read More

  4. Worksheet on Simplification | Simplify Expressions | BODMAS Questions

    Apr 03, 25 02:58 PM

    Worksheet on Simplification
    In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

    Read More

  5. Divisible by 2 Video |Test of Divisibility by 2 Trick| Rules| Examples

    Apr 03, 25 10:25 AM

    Divisible by 2
    A number is divisible by 2 if the digit at unit place is either 0 or multiple of 2. So a number is divisible by 2 if digit at its units place is 0, 2, 4, 6 or 8.

    Read More