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Volume of Cylinders and Rectangular Prisms

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Volume of Cylinders and Rectangular Prisms

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Use this Animated Math site to practice finding volume of cylinders and rectangular prisms.

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Use this Animated Math site to practice finding volume of prisms.

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Use this board on Discovery Education to learn basics about volume of prisms and cylinders.

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All last week we had Pi on the brain as we learned the formula for calculating the volume of 3D shapes with circles. We got to see Pi in action! In preparing for the week, I really thought about how to help students 1) understand the relationship between the formulas for cones, spheres, and cylinders 2) get really clear on the vocabulary and individual components of the three formulas. Today I want to share with you how I introduced my students to finding the volume of cylinders, cones and spheres.

Introducing Volume of Cylinders, Cones and Spheres with Discovery

First, I did a Volume of Cylinders, Cones, and Spheres Discovery Lab with students. As I've written about before, I really enjoy these Discovery Labs, or guided inquiry lessons, to get students thinking about the concept and making connections and observations before jumping in to the formal instruction- it really just helps students get a deeper understanding of what it is we're studying.

We started by looking at the formulas to find the volume of spheres, cones, and cylinders. I wanted them to really understand what the formulas themselves contained before using them. Students were asked to look at the individual components of each formula, and make observations about the relationship between the three formulas:

Comparing volume formulas

Then, after looking at each of the formulas, students completed this graphic organizer. This chart helped students really understand the individual components that made up each formula. We talked about what each formula has. For example, why do they all have pi? Why doesn't the sphere have height? By going through this process, I wanted to make sure that students really understood what the formula itself means- kind of like translating a foreign language:

To get a free copy of this formula chart, check out the post " How to Teach the Volume of Cylinder, Cones and Spheres Like a Rock Star" or grab the whole discovery lab, including this chart, over at TPT.

Volume formulas in action

Finally, it was time to see the formulas in action. As a class we worked to find the volume of a cone, cylinder, and sphere which all contained the same radius. Before solving, students were asked to make a prediction about which volume would be greatest. We recorded those predictions on the board. My students were super into seeing how their predictions matched up with the reality after solving.

After this introduction to volume formulas, students reflected on their learning. My favorite part was reading their reflections on what surprised them when calculating the volume. For example, the student below said "I was surprised that the cone was so much smaller than the rest."

It was really interesting to hear what students identified after briefly working with the formulas for volume for these three shapes. I sorted students into 3 groups after reviewing their responses: the "Got It" group, the "On the Right Track, but Missing a Piece" group, and the "Pretty Sure That's Not Really What We Were Going For There" group. This process was super helpful to know just what to key in on, and with whom, when doing the formal instruction and first practice problems.

Formal Instruction on Volume Formulas- Interactive Math Notebook

After completing this discovery lab introduction, we did the formal instruction with notes in our Interactive Math Notebooks. I made this foldable for students that helps them look at all three formulas. It also reviewed the key math vocabulary terms of diameter, radius, pi, and height. They need to own these terms in order to talk about what they're doing when finding the volume of these 3D solids.

I love using foldable graphic organizers for notes. They really help students organize what they are learning. And I love the creativity of making shapes that fit the topic. So of course this graphic organizer needed to be a circle itself. Because, obviously, with all of the shapes being related to circles, it would just be silly to have a foldable organizer that is a plain Jane rectangle! Here's what the foldable looks like closed and ready to go into an interactive math notebook:

Getting Proficient with the Formulas to Find the Volume

Last year, I underestimated how much practice students needed to be able to use a formula. I think my teammates and I assumed that it would be pretty straightforward and easy. We just explain the formulas to students and then let them do some practice and voila! students would be able to find the volume of these 3D shapes. But, when we reflected on how they did at the end of the unit, we realized that they still didn't own this concept. And we realized that a major reason why is that they weren't terribly comfortable with the concept of using formulas- it was still relatively new for them. So just saying- here's the formula let's apply it- didn't stick with them the way we had hoped.

To get students a deeper understanding or applying formulas to find the volume of cones, cylinders, and spheres , we spent one day with each shape, really digging in to each formula. During that day we practiced applying the formula to a variety of problems. We followed an "I Do, We Do, You Do" structure to each day, focusing on giving students a gradual release of responsibility.

In addition to more traditional practice questions, students were also asked to grapple with an unusual, or novel problem to apply not only the formula, but also develop their problem solving skills. Here's one example:

Students had gotten pretty automatic using the formula to solve- but they really had to grapple with this. They came to a bit of a halt and had to look at solving this problem differently. It was great to see them work to make a strategy of how they could figure this out, and how they could use the information they knew to find the unknown. My students definitely need more experiences doing this kind of thinking and work. While some of them got it, others didn't, and after struggling with it for a bit wanted to give up and call it impossible.

When we talked through approaches together, it was fun to see the light bulb moment when those who had been struggling realized this was yet another situation where they could set up an equation and solve to find the unknown. I'm hopeful that by the end of the year they'll all make that connection themselves, but they still need more practice with setting up and solving problems.

Getting Practice with Volume- and Having Some Fun

We had fun practicing the formulas to find the volume of cones and cylinders. On our third day we worked with the volume of spheres. After two days of practicing with the volumes of cylinders and cones, students were fairly proficient at using the volume formulas. So, by the time we dug into the formula for spheres, they flew through the practice and we got to do some coloring. 🙂 We enjoyed getting to the fun activity- the Volume of Spheres Color by Answer. It was a blast to watch students work through it and I got to just keep telling them, yep, you got this! My students absolutely rocked this activity.

For even more fun ways to practice, check out the post 11 Engaging Ways to Practice Finding the Volume of Cylinders, Cones, and Spheres.

Final thoughts

What I really enjoyed about this year is that they got much more experience with the formulas themselves. They really got a deeper understanding of what the formulas meant and what the formulas consisted of through the discovery lab. Then, we layered on top of that better understanding more focused practice. Spending a day practicing each of the formulas by themselves really helped them own it. I felt like the whole week just had a really good flow to it and the concepts were chunked out just right for students. Everything we did can be found in this bundle at TPT, including a few games that we didn't quite get to. Do I really have to wait a whole year to teach this topic again?

Thanks so much for reading. Until next time!

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Exploring the Volume of Cylinders, Cones & Spheres - Idea Galaxy

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All last week we had Pi on the brain as we learned the formula for calculating the volume of 3D shapes with circles. We got to see Pi in action! In preparing for the week, I really thought about how to help students 1) understand the relationship between the formulas for cones, spheres, and cylinders 2) get...

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Volume of Cones, Cylinders, and Spheres Review Game

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Volume of Cones, Cylinders, and Spheres Jeopardy Style Review Game

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An interactive math lesson to teach calculating the volume of a cone.

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Volume of Cone, Cylinder, Sphere - Volume of Cylinder - Volume of Sphere - Flocabulary

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Teach or review how to calculate the volume of a cone, cylinder and sphere with Flocabulary's educational rap song and lesson plan.

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Let's take a look at a diagram of an actual prism. A diagram will help us understand what volume actually means and how volume is calculated.

Here is a prism that has a length of 3 units, a width of 6 units, and a height of 4 units.

To understand what volume means, let's start by filling the bottom of the prism with unit cubes. This means the bottom of the prism will act as a box and will hold as many cubes as possible without stacking them on top of each other. This is what it would look like.

The diagram above does not yet represent the total volume. It only represents a partial volume, but we need to count these cubes to arrive at the total volume. We could count them one at a time or we could use a shortcut.

Since the length is 3 units and the width is 6 units, the bottom layer of cubes measure rows and columns that are 3 units by 6 units. So, a quick way of counting the blocks would be to multiply 3 by 6. In general, we would multiply the length of the prism by its width. Multiplying the numbers, we arrive at 18 cubes resting on the prism's base.

If we imagine the prism like a building, we could stack cubes on top of each other until the prism is completely filled. It would be filled so that all cubes are touching each other such that no space existed between cubes. It would look like this.

To count all the cubes above, we could use another shortcut. We already know there are 18 cubes on the bottom level and all levels contain the exact number of cubes. Therefore, we need only take that bottom total of 18 and multiply it by 4 because there are four levels to the prism. 18 x 4 = 72 total cubes to our original prism.

If we review our calculations, we find that the total bottom layer of cubes was found by multiplying the prism's length by its width. Then, we took the result and multiplied it by the prism's height. So the volume of a prism is its length times its width times its height. Sometimes geometers refer to the volume as the area of the bottom times the height. Symbolicly, it is written as V = Bh, where the capitol B stands for the area of the solid's base (bottom).

To understand the units of our answer, we could think in terms of algebra. We know that x times x times x is x ³; similarly, units x units x units = units ³. So if we had to find the volume of our original cube, all we need to do is multiply 3 units x 6 units x 4 units to get 72 units ³ or 72 cubes.

Example 1: If l = 7 ft and w = 5 ft and h = 2 ft, then the volume would be...

Example 2: If l = 12 yds and w = 10 yds and h = 1/2 yd, then the volume would be...

uizmaster: Volume of Prisms

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Let's take a look at a diagram of an actual prism. A diagram will help us understand what volume actually means and how volume is calculated. Here is a prism that has a length of 3 units, a width of 6 units, and a height of 4 units.

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Ice Cream Puddle Activty

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Ice Cream Puddle - National Council of Teachers of Mathematics

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Use volume formulas, and make some decisions about how to use them.

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This just might be my Pi Day activity this year (2013). Story: http://mr-stadel.blogspot.com/2013/03/trashketball-2013-pi-day-task.html

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4. Can a rectangular piece of paper give you the same amount of popcorn no matter which way you make the cylinder? Prove your answer.

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The radius requires the exact center. The circumference is tough because wrapping a string around the glass lumps the glass in with the soda. The diameter can be found by exploiting the fact that the diameter is the longest chord in a circle.

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If the radius of a circle is 17 cm, then the diameter of the circle is 34 cm. If the diameter is 34 cm, then circumference of the circle is approximately 107 cm. Determine if both statements are TRUE or FALSE.

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If the radius of a circle is 17 cm, then the diameter of the circle is 34 cm. If the diameter is 34 cm, then circumference of the circle is approximately 107 cm. Determine if both statements are TRUE or FALSE.

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