The diameter of the base of a right circular cone is 10 cm and its height is 12 cm. What is the total surface area (in cm$$^2$$ ) of the cone?

the total surface area of the cone = $$\pi\times\ radius\left(radius+slant\ height\right)\ $$    Eq.(i)

The diameter of the base of a right circular cone is 10 cm and its height is 12 cm.

radius of of the base of a right circular cone = half of diameter = $$\frac{1}{2}\times\ 10$$ = 5 cm    Eq.(ii)

slant height = $$\sqrt{\ \left(radius\right)^2\ +\ \ \left(height\right)^2}$$

= $$\sqrt{(5)^2 + (12)^2}$$

= $$\sqrt{25 + 144}$$

= $$\sqrt{169}$$

= 13 cm    Eq.(iii)

Put Eq.(ii) and Eq.(iii) in Eq.(i),

= $$\pi\times\ 5\times\ \left(5+13\right)\ $$

= $$\pi\times5\times18$$

= $$90\pi$$