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In Part A we found that we can determine the volume of rectangular prisms or boxes by multiplying the dimensions (length • width • height). Another way to determine the volume is to find the area of the base of the prism and multiply the area of the base by the height. This second method is sometimes referred to as the cross-section method and is a useful approach to finding the volume of other figures with parallel and congruent cross sections, such as triangular prisms and cylinders.
Notice in the figures above that each cross section is congruent to a base. Note 3
The formula for the volume of prisms is V = A • h, where A is the area of the base of the prism, and h is the height of the prism. Does it matter which face is the base in each of the following solids? Explain.
Find the volume of the figures above using the cross-section method.
How do we go about finding the volume of a figure that is not a prism-like solid? The figure below has two bases, and every cross section of the solid is a circular region that is parallel to a base. The circular cross sections, however, are not all congruent. To find the exact volume of this solid involves using methods from calculus, but can you find an approximate volume with the information about the cross sections indicated here?
Find the approximate volume of the prism-like vase above, given the areas of some of its cross sections.
In this part you will use two methods to compare the volumes of different solids. Note 4
Be sure to fill the bottom of the cylinder completely with the flattened sphere. Note 5
Problem B4
What is the relationship between the volume of the sphere and the volume of the cylinder?
Try to avoid as much measurement error as possible by lining up the top of the sphere and the top of the cylinder.
Problem B5
What is the relationship between the volume of the cone and the volume of the cylinder?
Sometimes this method of determining the relationship between the volume of a cone and cylinder is not very accurate because the cone does not hold its shape.
Alternate Experiment
Problem B6
If a cone, cylinder, and sphere have the same radius and the same height, what is the relationship among the volumes of the three shapes?
Problem B7
Using the illustration above, write the formulas you could use to find the volume of the following:
How are the formulas connected to your physical discoveries?
Problem B8
Are there similar relationships between other three-dimensional solids such as rectangular prisms and pyramids? In this final activity, compare the volumes of pairs of solids (PDF file). Record what is the same for both solids (e.g., height) and note how the volumes of the two solids are related. Try to generalize the relationships among volumes for similar three-dimensional solids. Fill in the table.
Note 6
Pair |
Solids |
What’s the same? |
How are the volumes related? |
1 | A, B | ||
2 | C, D | ||
3 | E, F | ||
4 | G, H | ||
5 | I, J | ||
6 | I, K |
Problem B9
Based on your findings in the previous problem, can you make any generalizations about how the volumes of some three-dimensional solids are related?
Problem B10
Write formulas for the volume of a square pyramid and a triangular pyramid. How are the volumes of pyramids and cones related?
Video Segment Boston’s Big Dig is the most expensive public works project in the history of the United States. In this segment, Michael Bertoulin explains how engineers calculate the volume of irregular shapes by breaking them down into smaller, regular shapes. As you’ll see, calculating volume is only one in a series of engineering and technological challenges engineers have to overcome. You can find this segment on the session video approximately 22 minutes and 18 seconds after the Annenberg Media logo. |
Note 3
The cross section method can be used to find the volume of all prisms (e.g., rectangular, triangular, hexagonal, octagonal) as well as other solids that have congruent parallel bases. In the cross-section method, we find the area of the base (a cross section) of the solid and then multiply that area by the height of the figure. Imagine that you are stacking layer after layer of the base shape on top of itself to build a tower in the shape of the base. This method also works for curved solids such as cylinders that have parallel congruent bases.
Note 4
Whereas we can use the cross-section method to find the volume of a cylinder, how do we determine the volume of a cone and a sphere? Are the volumes of these shapes in any way related to the volume of a cylinder? They are when the radii of the three solids are identical and when their heights are the same.
Note 5
This part of the session presents two different methods for determining the actual relationships between the volumes. If you’re using manipulatives, it is important to be very careful to make sure the heights and diameters of the cylinder, sphere, and cone are congruent. Sometimes when the clay sphere is flattened into the cylinder, there are holes and gaps, so it appears that the volume of the sphere is greater than it really is.
Furthermore, when forming a cone shape that fits into the cylinder, use stiff paper that doesn’t have a lot of give to it. Otherwise, you may again have inaccuracies in the relationship between solids.
It may be easier to observe the relationships between volume using plastic cylinders, cones, and spheres, as mentioned in the Alternate Experiment. Use water to fill the solids (color it with a drop or two of vegetable food coloring).
Note 6
This activity asks you to explore the relationship between prisms and pyramids and other cones and cylinders, many of which have the same height and have the same size base. You can use plastic solids and fill them with water, or use paper solids and fill them with rice or sand.
Problem B1
For the solids in parts (a) and (b), it does not matter which face is the base. In the third solid, it does matter, because the cross-section method only works with solids that have two parallel congruent bases and where the cross section is congruent to the base. The only face that can be a base here is a pentagon (made from the rectangle and triangle).
Problem B2
Problem B3
We can approximate the volume by selecting several cross sections at random, determining their areas, averaging the measures, and multiplying the average area by the height. Using the described method, find the area of each cross section and average them. The areas are 100π, 196π, 324π, 625π, and 400π. The average is 329π, so a good guess at the volume is 329π • 100, or, using an approximate value for π, we get V 329 • 3.14 • 100 103,306 cubic units.
Problem B4
You should find the sphere takes up two-thirds of the volume of the cylinder.
Problem B5
You should find the cone takes up one-third of the volume of the cylinder.
Problem B6
The volume of a cylinder is three times the volume of a cone with equal height and radius. The volume of a sphere is two times the volume of a cone with equal height and radius.
So the ratio of volumes is 3:1:2. In other words, it takes three times as much rice to fill the cylinder as it does to fill the cone, and twice as much to fill the sphere as it does to fill the cone.
Problem B7
Pair | Solids | What’s the same? | How are the volumes related? |
1 | A, B | Same base and height | Volume B is one-third of A. |
2 | C, D | Same base and height | Volume D is one-third of C. |
3 | E, F | Same base and height | Volume F is one-third of E. |
4 | G, H | Same base, height of H is twice the height of G | Volume H is two-thirds of G. |
5 | I, J | Same height, base J is half the area of base I | Volume J is one-sixth of I. |
6 | I, K | Same base and height | Volume K is one-third of I. |
The relationship between a prism and a pyramid parallels the relationship you discovered between a cone and cylinder: The volume of a pyramid is one third the volume of a prism with the same-sized base and the same height. If, however, the heights of the two solids are not the same, as with Solids G and H, or if the bases are not identical, as with Solids I and J, then the relationships differ.
Problem B9
If a solid comes to a point (a cone or a pyramid), its volume is one-third of the equivalent complete solid (cylinder or prism). Notice that the base and the height of the two corresponding solids must be the same in order for the relationship to hold.
Problem B10
The volume of a pyramid is V = (1/3)Bh, where B is the area of the base. This is related to a cone’s volume, since the cone’s base is a circle with area π • r^{2}. For a square pyramid, the area of the base is s^{2} (where s is the length of a side of the square base). So the volume is (1/3)s^{2}h. For a triangular pyramid, the area of the base is (1/2)ab (base and height of the triangle — here we use a for altitude so we don’t confuse the height of the triangle and the height of the pyramid). We get a volume formula of (1/6)abh.