7. Geometry

Lesson

Imagine that a cone and a cylinder have the same radius and the same height. How many times greater is the volume of the cylinder than the cone?

Test your conjecture with the applet below. Click the button to pour the water from the cone to the cylinder. Then, refill the water in the cone and repeat until the cylinder is full.

Then, consider the following:

- How many cones of water did it take to fill the cylinder?
- What fraction of the cylinder does one cone fill?
- Recall that the volume of a cylinder can be found using the formula $V=\pi r^2h$
`V`=π`r`2`h`. Using formula for the volume of a cylinder and your answers to the questions above, derive the equation for the volume of a cone.

Notice that if a cone and a cylinder have equal radius and height, the cone will fill the cylinder exactly three times. This means that the volume of a cone is $\frac{1}{3}$13 the volume of the cylinder. This gives us the following formula for the volume of a cone:

Volume of a cone

The volume, V, of a cone can be calculated using the formula

$V=\frac{1}{3}\pi r^2h$`V`=13π`r`2`h`

where $r$`r` is the radius and $h$`h` is the height of the cone.

We can apply this formula to find the volume of any cone, or to approximate the volume of cone-shaped objects.

Find the volume of the cone shown.

Round your answer to two decimal places.

Solve real-world and other mathematical problems involving volume of cones, spheres, and pyramids and surface area of spheres.