This unit has two parts:

1. Making a cone with the same height and base diameter as a given cylinder, and
2. Figuring out how many cones it will take to fill the cylinder.

Before trying this unit, be sure you are familiar with the formula for the volume of a cylinder. It is V = 𝜋r²h . It is also in Rice Cakes Box, lesson 19 in Breaking Away from the Algebra & Geometry Book.

Given a cylinder, design and build a cone with the same height and base diameter. First, measure the height h and diameter d of your cylinder. The diameter d = 2 times the radius r, d = 2*r.

In the diagrams, h is the height of the cylinder and the cone, and r is the radius of their bases, which are equal. The circumference C of a circle is C = 𝜋d or C = 2𝜋r. And using the Pythagorean Theorem, we can find the slant height s of the cone, s = √(h² + r²). After you measure the height and diameter of your cylinder, you will need a calculator to get a value for the slant height.

But the trick is to figure out how to design a 2-D net for the cone. Did you know that the 2-D net for a cone is a sector of a circle? Here, the circle we are talking about has radius s (the slant height of the cone). So we know the radius of the sector is s, not r. But the big question is, how big is the angle of the sector? That is, what is angle A in the figure below? The amount of the circumference of the sector is the same as the whole circumference of the cone’s base, namely, 2𝜋r.

Do you see it? (A°/360°)*2𝜋s is a part of the circumference of the circle with radius s that is the base of the cone. It is equal to 2𝜋r, which is the circumference of the cylinder.

How many filled cones does it take to fill the cylinder? You may use rice to find out! When you know the answer, write the formula for the volume of a cylinder and the volume of a cone! How cool is that?