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Feb 24, 2015

A sphere ,cylinder and a cone of same radius and same height find the ratio of their CSA

A sphere ,cylinder and a cone of same radius and same height find the ratio of their CSA.



Raghunath Reddy

Since, the height of a sphere is the diameter, the cone and cylinder have height 2r. 

Curved surface area of Sphere= 4πr²

Curved surface area of cylinder = 2πr(2r) = 4πr²

Curved surface area of cone = πrl

where, l = √(r2 + h2 ) = √( r2 + (2r)2) = √(5r2) = r√5  

⇒ Curved surface area of cone = π√5r2 


Ratio of CSA 's a sphere ,cylinder and a cone = 4πr2:2πrh : πrl

= 4πr2:4πr2 : πr2√5

= 4 : 4 : √5

Parthasaradhi M

Member since Apr 1, 2017

\\Solution:\\\\ Given:\\ Radius\ of\ sphere=Radius\ of\ cylinder=Radius\ of\ cone=r\\\\ Since, the\ height\ of\ a\ sphere\ is\ its\ diameter, then\ its\ height= 2r.\\ \\ Height\ of\ sphere=Height\ of\ cylinder=Height\ of\ cone=2r\\\\ Now,\\ Ratio\ of\ curved\ surface\ area's\ of\ sphere,\ cylinder\ and\ cone\\\\ =4\pi r^2:2\pi rh:\pi rl\\\\ =4\pi r^2:2\pi rh:\pi r \sqrt{r^2+h^2}\\\\ =4\pi r^3:2\pi r(2r):\pi r \sqrt{r^2+(2r)^2}\\\\ =4\pi r^2:2\pi r(2r):\pi r \sqrt{r^2+(2r)^2}\\\\ =4\pi r^2:4\pi r^2:\pi r \sqrt{5r^2}\\\\ =4\pi r^2:4\pi r^2:\pi r \times r \sqrt{5r}\\\\ =4\pi r^2:4\pi r^2:\pi r^2 \sqrt{5}\\\\ =4:4: \sqrt{5}\\\\


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