Ratio of Surface Areas of a Cone and a Sphere

Geometry Level pending

Given a sphere of radius R R , and a right circular cone of base radius R R and height 2 R 2 R . Find the ratio of the total surface area (i.e. include the base area) of the cone to the surface area of the sphere. In the answers, ϕ \phi is the golden ratio, ϕ = 1 + 5 2 \phi = \dfrac{1+\sqrt{5}}{2} .

2 3 ϕ \frac{2}{3} \phi 1 2 ϕ \frac{1}{2} \phi 1 2 ϕ 2 \frac{1}{2} \phi^2 ϕ \phi

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Slant height of the cone is l = R 2 + ( 2 R ) 2 = R 5 l=\sqrt {R^2+(2R)^2}=R\sqrt 5 .

So total surface area of the cone is π R ( l + R ) = 2 π R 2 ϕ πR(l+R)=2πR^2\phi .

Surface area of the sphere is 4 π R 2 4πR^2 , and the required ratio is ϕ 2 \boxed {\dfrac{\phi}{2}} .

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