# How to derive the formula for surface area of a sphere

$Let\;radius\;of\;sphere\;be\;R\;and\;\theta\;be\;the\;vertical\;angle\\* consider\;the\;figure,\\*Now,\;select\;the\;thin\;strip\;at\;an\;angle\;of\;d\theta\\* \Rightarrow thickness\;will\;be=Rd\theta\\* Now,\;we\;can\;think\;it\;as\;a\;cylinder\;with \; radius=R \sin\theta,\;height=rd\theta\\* \Rightarrow surface\;area,A=2\pi r h\\* \Rightarrow dA=2\pi R\sin\theta Rd\theta\\* Integrate\;both\;side\;\\* A\rightarrow 0\;to\;A\;and\; \theta\rightarrow 0\;to\;\pi\\*\Rightarrow \int_{0}^{A} dA=\int_{0}^{\pi} 2\pi R^2 \sin\theta d\theta\\* \Rightarrow A=2\pi R^2[-\cos \pi - (-\cos 0)]\\*\because \cos \pi =-1\;and\;\cos 0=1 \\*A=2\pi R^2(2)\\*\Rightarrow A=4\pi R^2\\* So,\;Surface\;area\;of\;Sphere\;is\;4\pi R^2$

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