The volume of a right circular cone is 660 cm³ and diameter of its base is 12 cm. Calculate:

(i) the height of the cone

(ii) the slant height of the cone

(ii) the total surface area of the cone

Given, radius, r = $\dfrac{\text{diameter}}{2} = \dfrac{12}{2} = 6 \text{ cm.}$

Volume of cone = 660 cm³

(i) By formula,

Volume of cone = $\dfrac{1}{3}$ πr²h

$\Rightarrow 660 = \dfrac{1}{3} \times \dfrac{22}{7} \times 6^2 \times \text{h} \\[1em] \Rightarrow 660 = \dfrac{22}{21} \times 36 \times \text{h} \\[1em] \Rightarrow \text{h} = \dfrac{660 \times 21}{22 \times 36} \\[1em] \Rightarrow \text{h} = \dfrac{13860}{792} \\[1em] \Rightarrow \text{h} = 17.5 \text{ cm}.$

Hence, the height of the cone is 17.5 cm.

(ii) By formula,

Slant height (l) = $\sqrt{\text{h}^2 + \text{r}^2}$

$= \sqrt{(17.5)^2 + 6^2} = \sqrt{306.25 + 36} = \sqrt{342.25} = 18.5 \text{ cm.}$

Hence, slant height of the cone is 18.5 cm.

(iii) Total surface area = πr(l + r)

$= \dfrac{22}{7} \times 6 \times (18.5 + 6) \\[1em] = \dfrac{132}{7} \times (24.5) \\[1em] = \dfrac{3234}{7} \\[1em] = 462 \text{ cm}^2$

Hence, total surface area of the cone is 462 cm².