The diagram shows an inverted hollow paper cone with a base of radius $r$ cm and a height of $h$ cm. The cone is filled with water, represented by the shaded region, to a depth of $\frac{1}{3}h$ cm. The internal surface area of the cone is 106 cm^2. Find the surface area of the inside of the cone that is in contact with the water.

1. Identify the linear scale factor of the smaller cone to the whole cone.

The water forms a smaller cone similar to the whole cone.

Since its height is $\dfrac{1}{3}h$ and the whole cone has height $h$,
$\text{linear scale factor}=\frac{\frac{1}{3}h}{h}=\frac{1}{3}.$

2. Use the area scale factor to find the curved surface area in contact with water.

For similar figures, surface areas are in the ratio of the squares of corresponding lengths.

$\text{area scale factor}=\left(\frac{1}{3}\right)^2=\frac{1}{9}.$

So the internal surface area in contact with the water is
$106\times \frac{1}{9}=\frac{106}{9}\text{ cm}^2.$