Calculate the height of the cone from the frustum’s bottom and top radius and frustum height.

geometrylinear algebra

Given the following diagram of a cone frustum, with the top radius of 5cm, the bottom radius of 9cm and the height of the frustum being
8cm, what is the height of the original cone? Don’t use a method that involves using rise/run or gradient calculations in any way shape or form.

I have tried to attempt this many times, but I am extremely lost still – and don’t understand it. I have tried to do some scuffed calculation using squaring and fractions like this:

(OH is Original Cone Height)

$$
OH \equiv \frac{h + 8}{9} + \frac{h}{5}
$$

Which then must mean:
$$
OH \equiv \frac{h^2+8}{4}
$$

Which somehow gets me 18, which my teacher says is the full cone height.

How did I do this, and does this work for everything? If not, what is a good equation that matches the rule set of the question asked.

Thanks,
Henry.

Diagram of Frustum Image

Best Answer

enter image description here

The figure above summarizes the situation. We have two similar triangles, the big one with height $h$ and base $9$, and the small one with height $(h-8)$ and base $5$.

Therefore, by equal ratios of similar triangles corresponding sides, we have

$ \dfrac{h}{h-8} = \dfrac{9}{5} $

From which

$ 5 h = 9 h - 72 $, so that $h = \dfrac{72}{4} = 18 $

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