Pyramids and Cones

The formula for Pyramids and Cones is almost exactly like the formula for Prisms and Cylinders.  One of the major differences is the 1/3.  It's an important difference because in coming to a point for the height, it limits the volume compared to a similar prism or cylinder.  Here are some examples.

The same four steps apply to these shapes.

1. What shape is it?  

It's a cone.  The base is a circle, so we'll need area for that.

2. Write the formulas.

The general formula for cones and pyramids is V=(1/3)B*h

                                                                     V = (1/3) * B * h

specific to cones                                            V = (1/3) * (pi * r^2) * h

3. Plug the numbers in:

The base lists a diameter 2, not a radius, so that radius is 1 (2/2=1).  Height is 3.

                                                                    V = (1/3) * (3.14 * 1^2) * 3

4. Solve

3 divided by 3 is 1.  1^2 is 1.                        V = 1 * 3.14 * 1

                                                                    V = 3.14 cubic feet.

Let's try one that's more difficult.

1. What shape is it?

Cone, same as the first.

2. Write the formulas.

                                                                  V = (1/3) * B * h

Same as the first.                                        V = (1/3) * (pi * r^2) * h

3. Plug the numbers in.

Again the base is a diameter (16), so radius is 8.  V = (1/3) * (3.14 * 8^2) * 16

4. Solve

8^2 = 64.  64 * 3.14 = 200.96                 V = (1/3) * (200.96) * 16
                                                                 V = (1/3) * 3215.36
                                                                 V = 1071.79 cubic meters

Let's try this one.

1. What shape is it?

Pyramid with a square base.  I know it's square because of the 7x7.  The height is defined by the line in the middle.  It could also be listed outside the diagram as a separate line.  In this case, it's 6.

2. Write the formulas

general formula                                     V = (1/3) * B * h

specific formula                                    V = (1/3) * (b * h) * h

3. Plug the numbers in

                                                            V = (1/3) * (7 * 7) * 6

4. Solve

                                                            V = (1/3) * (49) * 6

6 divided by 3 is 2.                               V = 49 * 2

                                                            V = 98 cubic meters.

1. What shape is it?

Another square (rectangular) pyramid.

2. Write the formulas.

general formula                                    V = (1/3) * B * h

specific formula                                   V = (1/3) * (b * h) * h

3. Plug the numbers in

                                                           V = (1/3) * (6 * 6) * 8

4. Solve

                                                           V = (1/3) * (36) * 8

                                                           V = 12 * 8

                                                           V = 96 cubic yards

1. What shape is it?

This one's a triangular pyramid.  There are no rectangles or squares on the shape.  The base has a right triangle, so its base and height are 4 and 3.  The height of the pyramid is the line through the middle, and is 4.

2. Write the formulas

general formula                                   V = (1/3) * B * h

specific formula                                  V = (1/3) *(0.5 * b * h) * h

3. Plug the numbers in

                                                          V = (1/3) * (0.5 * 3 * 4) * 4

4. Solve

                                                          V = (1/3) * (6) * 4

                                                          V = 2 * 4

                                                          V = 8 cubic meters