Calculating the volume of a cone given the surface and $s$

Question

I've been struggling with this for so long and I never got a chance to ask my teacher how to solve it.

If the surface of the cone is $360\pi$ and $s = 26 \text{cm}$, calculate the volume of that cone.I found the solution but there is no explanation, somehow you need to get to squared binomial and I'm not sure why.

Formula for the cone volume: $V = \frac13\cdot\pi\cdot r^2\cdot H$

Formula for the cone surface: $P = \pi\cdot r\cdot(r+s)$

Answer 1

$SA = \pi rs + \pi r^2$. Thus: $360\pi = \pi 26r + \pi r^2 \to r^2 + 26r = 360 \to (r+13)^2 = 23^2 \to r = 10$. Then $h = \sqrt{s^2 - r^2} = \sqrt{26^2 - 10^2} = 24$. We can now find $V$, and $V = \dfrac{\pi r^2h}{3} = \dfrac{\pi 10^2\cdot 24}{3} = 800\pi$

Answer 2

First off, write the formula for the cone volume, and the formula for the cone surface. Edit your question with them, and I will edit my answer to help you further.

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EDIT:

Using the formula for $P$, can you extract what $r$ is? (hint: yes you can) Then, once you have $r$, you can calculate $V=\frac13 \pi r^2 H$, as long as you can calculate $H$ from $r$ and $s$ (which you can, using a very famous theorem).