# The Volume of a Cone

Warmup

The cone and cylinder have the same height, and the radii of their bases are equal.

(1.) Which figure has a larger volume?

(2.) Do you think the volume of the smaller one is more or less than the volume of the larger one? Explain your reasoning.

Activity #1

Sketch A Cone.

Here is a method for quickly sketching a cone:

• Draw an oval.
• Draw a point centered above the oval.
• Connect the edges of the oval to the point.

(1.) Which parts of your drawing would be hidden behind the object?

• Sketch two different sized cones. The oval doesn’t have to be on the bottom!
• For each drawing, label the cone’s radius with r and height with h.
• Make the hidden parts dashed lines.

Activity #2

Construct a Cone Using an Applet.

The 2-D plane in the applet below is the one that has the grid. It is the active plane because its tools are displayed on the menu bar. When you click on the section that does not have a grid, you immediately activate the 3-D plane. All the 2-D plane tools will replaced by 3-D tools.

• Click on the 3-D section of the applet.
• Click on the cone tool, .
• To create a vertical cone, click on a point on the vertical axis, either above or below the plane.
• Click on a second point on the same axis.
• Enter a height for your cone. Use a value not more than 5 for the height.
• Then click the Select tool .
• Now you use your cursor to rotate/spin the cone around.
• Repeat the steps above, and this time create a horizontal cone.

Activity #3

Explore the Volume of Cone.

In the applet below, the vertical slider controls the height of the cone, while the horizontal slider controls the radius.

• Check the “Formula for Volume” box in the applet to see the formula for finding the volume of a cone.
• Check the “Solution for volume” box to see the volume of the cone with the dimension seen on the two slider.
• Uncheck the “Solution for volume” box.
• Use the sliders to find the volume of a cone with the following dimensions; (Assume π to be 3.14.)

(1.) height = 4.4 and radius = 2.1

(2.) height = 5 and radius = 3

(3). height = 6.8 and radius = 4.6

(4.) height = 8 and radius = 5

(5.) height = 10 and radius = 4.2

Challenge #1

Here is a cylinder and cone that have the same height and the same base area.

What is the volume of each figure? Express your answers in terms of π.

Challenge #2

The volume of this cone is 36π cubic units.

What is the volume of a cylinder that has the same base area and the same height?

Challenge #3

A grain silo has a cone shaped spout on the bottom in order to regulate the flow of grain out of the silo. The diameter of the silo is 8 feet. The height of the cylindrical part of the silo above the cone spout is 12 feet while the height of the entire silo is 16 feet.

(1.) How many cubic feet of grain are held in the cone spout of the silo?

(2.) How many cubic feet of grain can the entire silo hold?

Quiz Time