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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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3 Answers 3

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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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Assuming a perfectly right angled circular truncated cone, the first step is to realise that a vertical 2D cross section of a truncated cone gives a trapezoidal shape.

Applying the dimensions you've given and following the working in the picture below (and ignoring the typo where I forgot to omit the "cm" units for "14 cm"), the angle alpha for your truncated cone works out to be 85.91°.

enter image description here

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HINT:

Let the constants mentioned in the question be $(P,p,L)$ respectively (Perimeters, slant length). We have

$$ \sin \alpha= \frac{R-r}{L}=\frac{P-p}{2 \pi L}.$$

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