Question

Total surface area of the cone is equal to 3 times of it's base area find vertical angle of the cone .

Answer 1

Radius of the base = r.
Height = h.
Slanting height = s.

$tsa = total \: surface \: area \\ = \pi \: r \: s \: + \pi \: {r}^{2} \\ = 3 \: \times \pi \: {r}^{2} \: \: \: ..... \: given \\ so \: \: s = 2 \: r \\ half \: the \: cone \: angle \: = {sin}^{ - 1} \frac{r}{s} \\ = {sin}^{ - 1} \frac{1}{2} = {30}^{0} .$
The angle between vertical and slanting height/ edge = 30°.

Answer 2

heya ..

total surface area of cone is =πrl+πr^2.

according to ur question...
.total surface area of cone=3 area of the base of cone ..

so,,,, πrl+πr^2=3πr^2

πrl=2πr^2.

l=2r .

verticle angle of cone =sin-1r/2r

sin-1 1/2=30°

hence the verticle angle of cone if 30°

hope it help you.

@rajukumar.