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Volume of Right Cones


The volume of a cone has the same relationship to a cylinder as we just saw that a pyramid has with a prism.  

That is:

Volume of Right Cone

$\text{Volume of Right Cone }=\frac{1}{3}\times\text{Area of Base }\times\text{Height of cylinder}$Volume of Right Cone =13×Area of Base ×Height of cylinder

$V=\frac{1}{3}\pi r^2h$V=13πr2h

The mathematical derivation of the formula for the volume of a cone is beyond this level of mathematics, so for now it is suffice to know the rule and how to use it.  

Where are cones?

Finding out the amount of snowcone that would fit inside one of those conical (cone-like) cups, you can use the volume equation to find the volume inside the cone, right? 

Funnels are in the shapes of cones. If you want to find out how much *volume* of water/liquid flows out of it per unit time (say liters per second for example), you will have to know what the volume of a cone is. you can find this volume flow rate, you can even write down an exact equation to describe the flow rate at any given time, that begins for instance with water filled to the brim, showing a slow flow rate (this is at early times), and for later times your equation will show that you have a faster slow. 

Traffic cones are cones. Maybe you need to work out the volume of plastic it takes to make one, ten or a thousand.

A whole lot of things are approximately cones, reactor cooling tower caps, thimbles, and so on. These actually are cones, they just have been truncated (i.e. a small cone at the tip has been subtracted from a larger cone), icicles, stalactites/stalagmites in caves, and paper-made megaphones. 

The front of pens have a cone shape to them.


Worked Examples


Find the volume of the cone shown.

Round your answer to two decimal places.

A cone is depicted with a vertical height labeled as 6 cm from the apex to the center of the base. The base radius is labeled as 2 cm. A right-angle indicator is shown where the height meets the base, suggesting the height is perpendicular to the base.


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