The radius of the base of a right circular cone of semi-vertical angle α is r. Show that its volume is 13πr3cotα and curved surface area is πr2cosecα
Let height of cone be 'h', then in △OBA
tanα=OAOB=rh⇒h=rtanα=rcotα
Volume =13πr2h=13πr2.rcotα=13πr3cotα
CSA=πrl=πr√r2+h2=πr=√r2+(rcotα)2
=πr√r2+r2cot2α=πr√r2(1+cot2α)=πr√r2csc2α
=πr.rcscα=πr2cscα