# Expressing a radius in terms of the height in an hourglass

I am in high school and I have a question that I haven't been able to solve for a very long time. Please help?

A sand timer consists of two cones joined at the apex. Each cone has height h, radius r and an angle at the apex of 60°.

Express the radius of the top cone in terms of its height. Give your answer in exact form.

I am stuck and don't know how to complete this question. Could you please help?

Thank you! ^^

• How sand can go from top cone to the bottom cone if there is no passage between them... – Peđa Terzić May 6 '12 at 5:42
• @pedja: The question didn't say it was a working sand timer. ;) – Isaac May 12 '12 at 23:36

## 1 Answer

Draw two triangles representing a "side view" of the two cones; then draw the line through the middle. This divides each of the triangles into two right triangles with top angle of $30^{\circ}$. They are all the same, so let's concentrate on one in the bottom.

The height of the triangle is $h$. The foot of the triangle is $r$.

Because that line we drew divided the angle at the apex into two equal parts, the angle on top of our triangle is of $30^{\circ}$.

So you have a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, and you want to express the length of one of the sides in terms of the length of the other side.