VOLUME OF CYLINDER, CONES AND SPHERES
INTRODUCTION
The objective for this lesson on Volumes of Cylinders, Cones, and Spheres is, the student will solve real world and mathematical problems involving volumes of cylinders, cones, and spheres.
The skills students should have in order to help them in this lesson include area of a rectangle, area of a circle and volume of rectangular prisms.
We will have three essential questions that will be guiding our lesson.
Number one, what is the relationship between the volume of a cylinder and the volume of a cone? Number two, Explain the process of determining the formula for the volume of a sphere, volume equals two thirds r squared (times the quantity two r) to volume equals four thirds times r cubed. And number three, Justify and defend the process of developing the formula for the volume of a sphere.
We will begin by completing the warm- up applying the formula of the volume of a rectangular prism and the area of a circle to prepare for volume of cylinders, cones and spheres in this lesson.
SOLVE PROBLEM – INTRODUCTION
We will begin this lesson by completing a SOLVE problem. Katherine is leaving her apartment and moving into a new home. She is packing boxes with books and movies from her bookshelf. Each box has a base that measures two hundred twenty four square inches. Katherine also knows that each box has a height of thirteen inches. If Katherine has five boxes that will all be packed, what is the total volume of all of the boxes that she will use to move?
We will begin by Studying the Problem. First we need to identify where the question is located within the problem and underline the question. The question is what is the total volume of all of the boxes that she will use to move? Now that we have identified the question we want to put this question in our own words in the form of a statement. This problem is asking me to find the total volume of all of the boxes that Katherine uses to move.
In Step O, we will Organize the Facts. First we need to identify the facts. Katherine is leaving her apartment and moving into a new home, fact. She is packing boxes with books and movies from her bookshelf, fact. Each box has a base that measures two hundred twenty four square inches, fact. Katherine also knows that each box has a height of thirteen inches, fact. If Katherine has five boxes that will all be packed, fact, what is the total volume of all of the boxes that she will use to move?
Now that we have identified the facts, we want to eliminate the unnecessary facts. These are the facts that will not help us to find the total volume of all the boxes that Katherine uses to move. Katherine is leaving her apartment and moving into a new home. Knowing where she is moving will not help us to find the volume of all of the boxes together. So we will eliminate this fact. She is packing boxes with books and movies from her bookshelf. Knowing what she is putting in the boxes will not help us to find the total volume of the boxes. So we will eliminate this fact as well. Each box has a base that measures two hundred twenty four square inches. Knowing what the base measurement is of each box will help us to find volume of all of the boxes together. So we will keep this fact. Katherine also knows that each box has a height of thirteen inches. Knowing the height of the boxes will also help us to find the total volume. So we will keep this fact as well. If Katherine has five boxes that will all be packed. In order to find the total volume of all of the boxes, we have to know how many boxes Katherine has. So we will keep this fact as well.
Now that we have eliminated the unnecessary facts, we can list the necessary facts. The area of the base is two hundred twenty four square inches. The height of each box is thirteen inches. And Katherine has five boxes.
Now that we have listed the necessary facts we are ready to move on to Step L, to Line Up a Plan. First we need to write in words what your plan of action will be. We can use the formula Volume equals Base times height and multiply the area of the base times the height to find the volume of one box. Then, multiply the volume of one box by the total number of boxes packed.
What operation or operations are we using in our plan? We will use multiplication.
Now let’s Verify Your Plan with Action. First we can estimate your answer. We can estimate that the total volume will be about fifteen thousand inches cubed.
Now let’s carry out your plan. We said that we were going to use the formula Volume equals Base times height and multiply the area of the base times the height to get the volume of one box. Two hundred twenty four inches squared times thirteen inches equals two thousand nine hundred twelve inches cubed. Then multiply the volume of one box which we just found by the total number of boxes packed. Two thousand nine hundred twelve inches cubed times five boxes gives a total volume of all of the boxes of fourteen thousand five hundred sixty inches cubed.
Now that we have found the total volume of all of the boxes we are ready to Examine Our Results.
First, does your answer make sense? Here compare your answer to the question. Yes, because we found the total volume of all of the boxes.
Is your answer reasonable? Here compare your answer to the estimate. Yes, because our answer is close to the estimate of fifteen thousand inches cubed.
And is your answer accurate? Here check your work. Yes the answer is accurate.
We are now ready to write your answer in a complete sentence. The total volume of all of the boxes that Katherine uses to move is fourteen thousand five hundred sixty inches cubed.
We have now completed the SOLVE problem. We will be referring back to this SOLVE problem as we continue in the lesson.
EXTEND THE SOLVE PROBLEM – DISCOVERING THE VOLUME OF A CYLINDER
Take a look at the SOLVE problem we just completed. What was the problem asking you to find? It was asking you to find the total volume of all of the boxes that were used in the move. What are you finding when you find the volume of a container? You are finding its capacity or the amount a container can hold.
Now take a look at the graphic organizer on your page. What is the figure that is in the second column of the graphic organizer? It is a Rectangular Prism. And what is the figure that is in the third column of the graphic organizer? It is a Cylinder.
What are some items that you see every day that are in the shape of a rectangular prism? Some examples would be, boxes, books, a box of crayons, or a tissue box.
What are some items that you see every day that are in the shape of a cylinder? Some examples would be, a soup can, a drinking glass, or a water tank.
Draw both the rectangular prism and the cylinder in Question five. Now explain the drawing of the rectangular prism. It’s base is a rectangle, and the figure is three dimensional, it has length, width and height. Explain the drawing of the cylinder. The base is a circle, the sides are curved, and it has height.
So how are the two shapes alike? They are both solids, and they are both three-dimensional. Both shapes have a parallel set of congruent bases.
How are the two shapes different? The rectangular prism has a base that is a rectangle. And the cylinder has a base that is a circle. The rectangular prism is made of line segments. It edges are straight. And the cylinder has curved sides.
So taking a look back at the original SOLVE Problem, what formula was used to find the volume of the rectangular prism? We used the formula volume equals base times height. Record this information in Question eight for the rectangular prism.
What is the shape of the base of the rectangular prism? It is a rectangle. Record this information in Question nine and draw a picture of the base.
How can we find the area of the base? The area of the base can be found by multiplying length times width; we are finding the base area. Record this information is Question ten for the Rectangular Prism.
Now looking back at the formula for the volume of a rectangular prism in Question eight, what other formula can we use to find the volume of the rectangular prism? We could use volume equals length times width times height. We know that the area of the base (B) can be found using the formula: Area equals length times width. When we’re looking at the volume formula, volume equals base times height, we know that the base is represented by the length times width. This means that we can rewrite the formula for the volume for our rectangular prism. We can substitute “length times width” in the volume formula for the B. The volume equals the length times width times height.
Now let’s review our answers to Questions six and seven. What is the base of the rectangular prism? It is a rectangle. And what is the base of the cylinder? It is a circle. So would finding volume using the idea of base area times height work for a cylinder? Yes. Explain why. Both of the figures are three dimensional with a parallel set of congruent bases, so we could use the same formula, area of the base times the height, to determine the volume. What was the formula we used in Question eight for volume of a rectangle prism? We used volume equals base times height. What formula can we used for the volume of a cylinder? We can also use volume equals base time height. What is the shape of the base of the cylinder? It is a circle. Record for Question nine in the Cylinder column and draw a picture of a circle. Are there any adjustments we would need to make to the formula for the volume of a cylinder? We are still using the base area times the height, but when working with a cylinder we need to remember that the area of a circle is a different formula than the area of a rectangle for the base. What is the formula for finding the area of a circle? Area equals pi r squared. Record this information for Question ten in the Cylinder column. How can we write the formula for the volume of the cylinder using the area of a circle? Remember that the volume equals the base times the height. For the cylinder our base is pi r squared. So the volume for the cylinder will be pi r squared h. Explain why we can use the two different formulas to find the volume of a cylinder. If we are given the area of the base we can multiply that by the height. If we need to find the area of the base, we use the formula for the area of a circle and then multiply it by the height.
FORMULAS FOR VOLUME OF CYLINDERS
Let’s now take a look at when it would be appropriate to use each of the formulas we know to find the volume of a cylinder. What information do we have from the picture provided here? We know that the radius of the cylinder given is two inches. And the cylinder has a height of seven inches. Based on the information we are given in the drawing, which volume formula should we use? Since we are given the radius of the base, we will use the volume formula, volume equals pi r squared h, because we need to find the area of the base. Explain and defend your answer. We do not know the base area of the cylinder, so we have to find it using the formula for the area of a circle. Then, we multiply by the height.
Note: That we will use the approximate value of three and fourteen hundredths for pi in all of these computations. Now complete the computation and identify the volume of the cylinder. Remember that we are using the formula volume equals pi r squared h. Pi is represented by the value three and fourteen hundredths. The radius in this cylinder is two inches. So we put that two in the formula where the radius is. And the height is seven inches. Our formula is three and fourteen hundredths times two squared times seven. This gives us a volume of proximately eighty seven and ninety six hundredths. Explain why the answer is not exact. The value of pi we are using, three and fourteen hundredths, is an approximation of the irrational value of pi and so our answer is not exact.
What should our units be? The units should be inches cubed. The volume is approximately eighty seven and ninety six hundredths inches cubed. Why is our answer in cubed units? We are working with three different dimensions: length, width and height of a figure. Therefore, the volume has cubed units.
Let’s take a look at another example. What information do we have from the picture provided? We know the base area for this cylinder is twelve and fifty six hundredths inches squared, and the cylinder has a height of seven inches. Explain how the given information for this cylinder is different from the information provided for the last cylinder? Instead of the radius being provided for us to find the base area of the circular base, they are providing us with the base area. So based on the information we are given in the drawing, which volume formula should we use this time? We will use the formula, volume equals base times height. Explain and defend your answer. We know the base area and the height, and the simplest way of finding volume is to multiply the base area times the height. So let’s complete the computation and identify the volume of the cylinder. Remember that we are using the formula volume equals base times height. The base is twelve and fifty six inches and the height is seven. So we will multiply these two values together. The volume equals eighty seven and ninety two hundredths. What should our units be? The units will be inches cubed. The volume equals eighty seven and ninety two hundredths inches cubed. What do you notice about the volumes of both of the cylinders in the top boxes? They are the same. What conclusion can you draw about the two figures? Defend your thinking. They are the same cylinder. The second cylinder must have a radius of two inches, but was already calculated into the base area. Two inches squared times three and fourteen hundredths equals twelve and fifty six hundredths, which was the given area for the base of the second cylinder.
Now let’s talk about a third example. John has a cylindrical container that he uses to store his homemade soup. The base of the container is a circle with an area of twenty eight and twenty six hundredths inches squared. The height of the container is six inches. What information do we have? We know the base area is twenty eight and twenty six hundredths inches squared, and the height is six inches. Based on the information we have, which volume formula should we use? Since we know the base area, we can use the volume formula volume equals base times height. Explain why we would use this formula. We know the base area and the height, and the simplest way of finding volume is to multiply the base area times the height. So let’s identify the volume of the cylinder. Volume equals twenty eight and twenty six hundredths times six. This gives us the volume of one hundred sixty nine and fifty six hundredths inches cubed.
DISCOVER ACTIVITY – VOLUMES OF CYLLINDERS AND CONES
We are now going to complete an activity looking at the volume of cylinders and cones. Take a look at the nets seen here. We are going to cut out each of the nets. Tape these two sides together without overlapping. Fold the circle down to be the “lid,” or base, of the cone.
Now take a look at the other net. Tape these two sides together without overlapping. Fold the circles down to be the “lid,” or “bottom and top” of the cylinder. We’ve created a cone and a cylinder. What do you notice about the base for the cone and the cylinder? The circular base for both figures is the same size. Just to be sure, trace the bases or match them up to prove they are the same size. What do you notice about the height for the cone and the cylinder? The heights are the same for both figures. We could lay a sheet of paper on the top of both and it will be parallel to the desk or surface you are sitting.
Using the beans, fill the cone, leveling the beans off at the top so that they are level with the base. After filling the cone, pour the contents of the cone into the cylinder. Repeat this until the cylinder is full. We can pour the contents of the cone into the cylinder one time and then there is still room left in the cylinder. Continue pouring the contents of the filled cones into the cylinder. We can pour the contents of the cone into the cylinder a second time. And we still have room left in the cylinder. So continue the contents of the filled cone into the cylinder. We can pour the contents of the cones into the cylinder one more time. This time we filled the cylinder. How many full cones were needed in order to fill the cylinder? Three full cones. Explain what this means. This means that the cylinder hold three times the amount of the cone. The cone holds one-third of the amount of the cylinder. So what generalization can you make about the relationship between a cylinder and a cone with the same base and height? The cylinder’s volume will be three times the volume of a cone, or the cone will hold one-third the volume of the cylinder. So what is the formula for finding the volume of a cylinder? We found that the volume of a cylinder can be found with the formula, volume equals pi r squared h or volume equals base times height.