How do you find the radius with only the slant height? Please show work.

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The radius of the circle is 12cm, it is then made into a cone

Question:

A cone is made by joining the straight edges and securing them with tape. Calculate the angle of inclination, to the nearest tenth of a degree, for the sides of the cone.

Answer:

Angle of inclination is 41.4 degree.

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2 Answers

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We need to find the perimeter of the circle sector first:

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Let $l$ denote the slant height of the cone. Let $r$ denote the radius of the cone. Let $\varphi$ denote the angle of the sector of the circle. Let $\theta$ denote the angle of inclination.

Strategy:

1. The lateral surface area of the cone is equal to the area of the sector. The lateral surface area of the cone is $$L = \pi rl$$ Since the sector has radius $l$, its area is $$A = \frac{\varphi}{360^\circ} \pi l^2$$ where $\varphi$ is measured in degrees. Since $l$ is given, you can set $L = A$ to solve for $r$.
2. To determine the angle of inclination, solve the equation $$\cos\theta = \frac{r}{l}$$ for $\theta$.

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