Angle of deviation

The surface of the rotating cone is 30 cm2 (with circle base), its surface area is 20 cm2. Calculate the deviation of the side of this cone from the plane of the base.

Result

A =  60 °

Solution:

S1=30 cm2 S2=20 cm2 S1=S2+πr2 r=(S1−S2)/π=(30−20)/3.1416≐1.7841 cm S2=πrs s=S2/(π⋅ r)=20/(3.1416⋅ 1.7841)≐3.5682 cos⁡A=r/s A=180∘π⋅arccos⁡(r/s)=180∘π⋅arccos⁡(1.7841/3.5682)=60=60∘=60∘S_{1}=30 \ \text{cm}^2 \ \\ S_{2}=20 \ \text{cm}^2 \ \\ S_{1}=S_{2} + \pi r^2 \ \\ r=\sqrt{ (S_{1}-S_{2})/\pi }=\sqrt{ (30-20)/3.1416 } \doteq 1.7841 \ \text{cm} \ \\ S_{2}=\pi r s \ \\ s=S_{2} / (\pi \cdot \ r)=20 / (3.1416 \cdot \ 1.7841) \doteq 3.5682 \ \\ \cos A=r/s \ \\ A=\dfrac{ 180^\circ }{ \pi } \cdot \arccos(r/s)=\dfrac{ 180^\circ }{ \pi } \cdot \arccos(1.7841/3.5682)=60=60 ^\circ =60^\circ

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