Pyramids and Cones
Recall that a pyramid is a solid with a polygon base and triangular lateral faces that meet in a vertex. Pyramids are named by their base shape.
You have seen the formula for the volume of a pyramid before.
Where does this formula come from? Recall that to find the volume of a prism or a cylinder, you need to find the area of the base and multiply by the height.
The difference between these two formulas is the division by 3. The key to understanding where the 3 comes from is remembering Cavalieri's principle and investigating a square based pyramid.
Let's take a look at some problems involving pyramids.
1. The two square pyramids below are each constructed within cubes of the same size. The pyramid on the left has a vertex at the center of a face of the cube. The pyramid on the right has a vertex at one of the vertices of the cube. Is the volume of each pyramid the same?
Because each pyramid is constructed within the same cube, the heights of the pyramids are the same and the areas of their bases are the same. When a plane parallel to the base of the pyramid is constructed through the center of each cube, the cross sections of each pyramid are the same.
Each cross section is a square. The length of the side of the square is half the length of an edge of the original cube, since the plane was constructed through the middle of the cube.
Because cross sections are the same area, heights are the same, and bases are the same, these pyramids must have the same volume due to Cavalieri's principle.
2. How many square based pyramids congruent to the one below would it take to fill the cube? Can you visualize this?
It will take exactly 3 congruent pyramids to fill the cube. The image below shows each pyramid being added to the cube.
Note: This is an informal argument for the formula for the volume of a pyramid. A rigorous derivation of the formula that considers pyramids of any base shape will be developed in calculus.
The apothem of a regular polygon is a perpendicular segment from the center point of the polygon to the midpoint of one of its sides (see image below). Find the volume of a pyramid with a height of 10 inches and a regular pentagon base with an apothem of 1 inch.
Find the volume of a cone with a height of 10 inches and a radius of 1 inch.
1. Explain the connections between a prism and a pyramid. Why do you divide by three when calculating the volume of a pyramid?
2. Explain the connections between a cone and a cylinder. Why do you divide by three when calculating the volume of a cone?
Find the volume of each solid based on its description.
3. A cone with a diameter of 4 inches and a height of 12 inches.
4. A pyramid with a height of 15 inches and a regular hexagon base with an apothem of 4 inches.
5. A cone with a radius of 8 centimeters and a height of 15 centimeters.
6. A square based pyramid with a vertex in the center of the square such that each triangular face has a base of 12 inches and a height of 10 inches.
An hourglass is created by placing two congruent cones inside of a cylinder with the same base area. The radius is 5 inches and the height of the cylinder is 20 inches.
7. Find the volume of one of the cones.
8. Find the volume of the cylinder.
9. Find the volume of the space between the cones and the cylinder.
To see the Review answers, open this PDF file and look for section 9.2.