Question

If a piece of wire 30 cm long is bent into the form of an arc of a circle, subtending an angle of 60° at its centre, then radius of the circle is :

Answer 1

$\orange{Let\:the\:name\:of\:the\:wire\:be\:AB}$

$\huge\mathcal \fcolorbox{red}{green}{Given}$

$Length\:of\:the\:wire\:=\:30cm$

$It\:is\:bent\:in\:an\:arc\:which\:subtends\:an\:angle = \theta = {60}^{ \degree} \: at \: the \: centre \: of \: the \:circle\:$

$which \: the \: arc \: AB\:belongs\:to.$

$\huge\mathcal{\fcolorbox{aqua}{azure}{\red{To\:find}}}$

$The\:radius\:of\:the\:circle$

$\boxed{\boxed{{\orange{\bold{{Solution}}}}}}$

$The \: lenght \: of \: the \: arc \: is \: 30 \: cm$

$It \: forms \: a \: part \: of \: the \: circle \: with \: radius \: = \: r$

$Also\:it\:subtends\:an\:angle \: \theta \: = \: {60}^{ \degree}$

$Now\:,\:for\: \theta\:=\: {60}^{ \degree} \: the \: arc \: lenght \: is \: 30 \: cm$

$\therefore\: \theta\:=\: {360}^{ \degree} \: the \: arc \: lenght \: is \: C \: = \: \frac{1}{ \theta } \times {360}^{ \degree} cm \: =\: circumference\:=\:C\:of\:the\:circle$

$So\:C\:=\: \frac{20}{ 60 \: \degree} \times {360}^{ \degree} cm \: = \: 180cm$

$But\:C\:=\:2\pi \: r$

$\therefore\:2\:\pi \: r \: = \: 180 \: cm$

$r \: = \: \frac{90}{\pi}$