The area of the lateral surface of the cone is twice the area of the base. Find the angle between the generatrix of the cone and the plane of the base.
The lateral surface area of the cone is:
Sside = пRl (R – base radius, l – generatrix).
The area of the base (circle) is equal to:
Sosn = nR².
By the condition Sbok = 2Sosn; пRl = 2пR² (divided by R and п).
Hence l = 2R.
That is (according to the figure) MВ = 2MO.
In a right-angled triangle MOВ, the angle of OMВ is 30 ° (in a right-angled triangle opposite 30 ° there is a leg, half the size of the hypotenuse).
This means that the angle MВO = 180 ° – (90 ° + 30 °) = 60 °.
Answer: the angle between the generatrix and the plane of the base is 60 °.
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