The height of a right circular cone is 8 cm and the diameter of its base is 12 cm. Calculate :

(i) the slant height of the cone

(ii) the total surface area of the cone

(iii) the volume of the cone

Given, h = 8 cm and r = $\dfrac{\text{diameter}}{2} = \dfrac{12}{2} = 6$ cm

(i) Slant height, l = $\sqrt{\text{h}^2 + \text{r}^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm.}$

Hence, slant height of the cone is 10 cm.

(ii) Total surface area = πr(l + r)

$= \dfrac{22}{7} \times 6(10 + 6) \\[1em] = \dfrac{132}{7} \times (16) \\[1em] = \dfrac{2112}{7} \\[1em] = 301.7 \text{ cm}^2$

Hence, total surface area of the cone is 301.7 cm².

(iii) Volume of cone = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times \dfrac{22}{7} \times 6^2 \times 8 \\[1em] = \dfrac{22}{21} \times 36 \times 8 \\[1em] = \dfrac{6336}{21} \\[1em] = 301.7 \text{ cm}^3$

Hence, volume of the cone is 301.7 cm³.