Learning Math: Measurement
Volume Part B: Volume Formulas (60 minutes)
Session 8, Part B
In This Part
- Cross Section Method
- Cylinders, Cones, and Spheres
Cross Section Method
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.
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.
Take It Further
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.
Cylinders, Cones, and Spheres
In this part you will use two methods to compare the volumes of different solids.
- Using modeling dough, make a sphere with a diameter between 3 and 5 cm.
- Using a strip of transparent plastic, make a cylinder with an open top and bottom that fits snugly around your sphere. Trim the height of the cylinder to match the height of the sphere. Tape the cylinder together so that it remains rigid.
- Now flatten the sphere so that it fits snugly in the bottom of the cylinder. Mark the height of the flattened sphere on the cylinder.
Be sure to fill the bottom of the cylinder completely with the flattened sphere.
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.
- Next, roll a piece of stiff paper into a cone shape so that the tip touches the bottom of your cylinder.
- Tape the cone shape along the seam and trim it to form a cone with the same height as the cylinder.
- Fill the cone to the top with rice and empty the contents into the cylinder. Repeat this as many times as needed to completely fill the cylinder.
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.
- Take a plastic cone, sphere, and cylinder with the same height and radius. Using water or rice, experiment with filling the solids to determine relationships among their volumes.
If your plastic solids are small, fill with water for a more precise approximation. Larger models are easier to work with and can be filled with either material.
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?
Using the illustration above, write the formulas you could use to find the volume of the following:
- A cylinderUse the cross-section method. Remember, the height is twice the radius in this case.
- A cone
Use the formula for a cylinder and what you know about the ratios.
- A sphereUse the formula for a cylinder and what you know about the ratios.
How are the formulas connected to your physical discoveries?
Take It Further
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.
Based on your findings in the previous problem, can you make any generalizations about how the volumes of some three-dimensional solids are related?
Write formulas for the volume of a square pyramid and a triangular pyramid. How are the volumes of pyramids and cones related?
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.
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.
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.
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).
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.
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).
- A horizontal cross section has area 9 • 5 = 45 square units, and the height is 2. So the volume is 90 cubic units.
- A horizontal cross section has area 8 • 11 = 88 square units, and the height is 15. So the volume is 1,320 cubic units.
- It is 1,980 cubic units. Find the area of the base (pentagon) by first finding the area of the rectangle (8 • 12 = 96) plus the area of the triangle (6 • 12 • 1/2 = 36) so the area of the base is 96 + 36 = 132 square units. Multiply the cross-section area (same as the base area; i.e., 132) by height (15) and get 1,980 cubic units.
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.
You should find the sphere takes up two-thirds of the volume of the cylinder.
You should find the cone takes up one-third of the volume of the cylinder.
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.
- The area of the base is πr2. The formula for the volume of a cylinder is V = πr2h. Since, in our case, h is equal to 2r, we have V = 2πr3.
- Based on the ratio observations in an earlier problem, the formula for the volume of a cone is V = (1/3) πr2 h. Since in this particular case h = 2r, we have V = (2/3)πr3.
- Based on the ratios observed earlier, the volume of a sphere is two-thirds of the volume of a cylinder, so the formula is V = (2/3) πr2h. Since the height of our sphere is twice the radius, we have V = (2/3) πr2 (2r), or V = (4/3) πr3.
|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.
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.
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 • r2. For a square pyramid, the area of the base is s2 (where s is the length of a side of the square base). So the volume is (1/3)s2h. 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.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.