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I have a truncated cone with $\text {smaller perimeter} = 70\space cm$ and $\text{greater perimeter} = 64\space cm$. The edge length is $42\space cm$

How can I now calculate the angle $\alpha$?

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I presuming it is a right circular cone and $\alpha$ is the angle between the base and the slant (details which should have been included in the question). Hint: Think about a trapezoid formed with the axis of the cylinder, two parallel radii (one on each circular surface) and the slant between the ends of the radii. You should be able to calculate the length of the radii and the angles at the ends of the axis are right angles.

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