Length of Arc

The length of arc is denoted by 'S'.
S = $r \times \Theta$
where r is the radius of the circle and $\Theta$ is the angle formed by two radii which is in radian.

Examples on length of arc

1) Find the arc length when the radius is 5 cm and angle subtended between the two radii is $72^{0}$.
Solution : As we know that,
S = $r \times \Theta$
r = 5 cm and $ \Theta = 72^{0} = \left ( 72\times \frac{\pi}{180} \right )^{c} = \left ( \frac{2\pi}{5} \right )^{c}$

∴ S = $5 \times \left ( \frac{2\pi}{5} \right )^{c}$
⇒ S = 2$\pi$
⇒ S = 2$\times$ 3.14 = 6.28 cm

2) If the arcs of same length in two circles subtend angles of $60^{0}$ and $75^{0}$ at their centers. Find the ratio of their radii.
Solution : $ 60^{0} = \left ( 60\times \frac{\pi}{180} \right )^{c} = \left ( \frac{\pi}{3} \right )^{c}$

$ 75^{0} = \left ( 75\times \frac{\pi}{180} \right )^{c} = \left ( \frac{5\pi}{12} \right )^{c}$

∴ $\frac{\pi}{3} = \frac{S}{r1}$ and $\frac{5\pi}{12} = \frac{S}{r2}$

⇒ S = $\frac{\pi}{3}\times r1$ and S = $\frac{5\pi}{12}\times r2$
⇒ 4r1 = 5r2
⇒ r1 : r2 = 5 : 4

2)A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 88 m when it has traced out $72^{0}$ at the center, find the length of the rope.
Solution :
Let the post be at 'P' and PA be the length of the rope in tight position. Suppose the horse moves along the arc AB so that $\angle APB = 72^{0}$ and arc AB = 88 m.Let 'r' be the length of the rope .
So, PA = r m
$\Theta = 72^{0} = \left ( 72\times \frac{\pi}{180} \right )^{c} = \left ( \frac{2\pi}{5} \right )^{c}$ and S = 88 m

∴ $\Theta = \frac{arc}{radius}$

⇒ $\frac{2\pi}{5} = \frac{88}{r}$

⇒ r = 88$ \times \frac{5}{2\pi}$

∴ r = 70 m


     
3) A circular wire of radius 7.5 cm is cut and bent so as to lie along the circumference of a hoop whose radius is 120 cm. Find in degrees the angle which is subtended at the center of the hoop.
Solution : Radius of circular wire = 7.5 cm
∴ length of the circular wire = 2$\pi$ r = 2$\pi \times$ 7.5 = 15$\pi$
Radius of the hoop = 120 cm
Let the angle subtended by the wire at the center of the hoop be $\Theta$
$\Theta = \frac{arc}{radius}$

⇒ $\Theta = \left ( \frac{15\pi}{120} \right )^{c} = \left ( \frac{\pi}{8} \right )^c = \left ( \frac{\pi}{8} \times \frac{180}{\pi} \right )^{0} = 22^{0}30^{'}$


11th grade math

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