Students will engage directly with solids in a guided activity meant to deepen their understanding of the formulas for surface area and volume. From creating a set of cylinders with the same volume but different surface areas, to exploring the variables in each formula, students will gain further insight into what goes in and comes out of these formulas.

*A deep dive into surface area and volume formulas for basic geometric solids.*

**Formula Exploration** (20-40 minutes)

*Formula Exploration*student sheets.- iPad with Volumize app.
- Wifi access to send work to other iPads or to the online project space.

**Intro:** You may have learned about surface area and volume before, but what do the formulas for basic solids really tell us? Can five different cylinders share the same volume but have different surface areas? What would six distinctive shapes with the same volume look like? Could those same six shapes keep the same volume but look different than the first set? When companies need to package their products, they want to use materials in the most efficient way possible. Why do you think it’s important for companies to think about shape, volume and surface area when considering which packaging to go with?

This activity will dive into these questions and you will come out of it with a better understanding of geometric solids, their formulas for surface area and volume, and how this can help you investigate and design three-dimensional objects in the real world.

**Formula Exploration:** There are a few steps to understanding surface area and volume formulas in this activity: 1) exploring each solid individually, 2) exploring across different geometric solids, and 3) reflecting on how the formulas incorporate linear (1D) measurements, area (2D) measurements and volume (3D) information about the solids.

**Formula Exploration** (20-40 minutes)

- Draw the six two-dimensional solids on the board or interactive white board:
- 1) rectangular prism, 2) cylinder, 3) cone, 4) sphere, 5) triangular prism and 6) square-based pyramid.

- Have students draw the 2D shapes on their paper and take pictures of their drawings.

Once each group has drawn and photographed the six shapes, students will build on top of those flat images with 3D representations in Volumize by following along with the student sheet.

- The first step is to set a scale so that the height of their shape is 1m. This will be helpful to manage data as the lesson unfolds.
- Students will be prompted to create a rectangular prism with a volume of 1m-cubed. Then, they will be prompted to create two more that look different but also have a volume of 1m-cubed.
- Have students note in the table the varying surface areas for each prism that has the same volume. They will be prompted to reflect on these differences as well.
- For each new solid, have students start a new project in Volumize and repeat the process. If working in groups, have each student take responsibility for one of the shapes.
- Once groups have worked through each geometric solid, have them create another new Volumize project in which they create six different solids, all with the same surface area, but taking note of their different volumes.
**Tip:**Start by creating a base (a rectangular prism works well) that each of the six solids is built off of. You will not need this for calculation or data purposes but it allows for all the shapes to be seen in the same frame, yet not connected to each other (i.e., each individual shape will be connected to the base.) - Leave time for students to explore and reflect on the data they have gathered. Use student sheet questions to guide the conversation and ask them to deconstruct the formulas, shedding light on how each one works.

Extensions and Inquiring Further

This activity would extend well to explorations of geometrically *similar* figures and comparing their surface areas and volumes. This activity would likely incorporate ratios and proportions as well.

or:

Conversion between metric units and feet and inches could work well with the data created by this activity. Choose one or more data sets and explore the conversions. Have students convert from meters-cubed to liters, for example, and feet-cubed to gallons. Spend some time with the histories of these different units. Discuss various situations that might be meaningful to students, like mph vs. km/h, or gallons of gas vs. liters.

In 2D, begin by drawing the six geometric solids in the provided table. You will build off of these drawings using the Volumize app.

- Open Volumize and tap Get Building. You will be prompted to take a picture of something that you would like to model. Take a picture of the sketch you made of the first shape, a rectangular prism.
- Set the scale so that the height of your prism is 1 meter (1m).
- Add a 3D rectangular prism and align it closely with the sketch you made. Now edit one of the handles — length, height or width, so that the volume is 1m-cubed. (Use the info panel on the right to see the volume of the prism as you make adjustments)
- Add a second rectangular prism to one of the sides, and manipulate it to look different from your first shape, but maintain a volume of 1 meter cubed. Experiment with making it narrow or wide to get to 1 meter-cubed.
- Add one more rectangular prism, and change it so that it looks different from the first two, but also has a volume of 1 meter-cubed.
- Now tap on Surface Area, notice the varying values, and fill in the data table in your handout.
- Go back to the Volumize start page, and start a new project, this time use the cylinder, and repeat steps 2-6. Then do this for the other solids that you made sketches of.
- Answer the reflections questions in your student sheet.
- Create one more Volumize project. Start with a picture of the blank wall or tabletop. Set the scale again to be around 1m for roughly half the screen.
- Add a rectangular prism, and modify the size of the sides until the surface area is 3 meters squared. Then add each of the other solids, making them all arrive at the same surface area. Open the info panels for each solid and note their volumes in the data table.
- Go back and edit each solid so that they look different but maintain a 3 meter-squared surface area. Note the new volumes in the Volume 2 column in the data table.
- Be prepared for a class discussion about the things you noticed were at play.

- Why are linear measurements defined as meters, surface area as meters squared and volume as meters cubed?

- In the space below, draw the following shapes:

Rectangular prism | |

Cylinder | |

Cone | |

Sphere | |

Triangular Prism | |

Square-based Pyramid |

- As you work through the activity, save each project and fill in the table with the data from the side-panel of Volumize.

Shape | Volume | Surface area |

Rectangular prism 1 | 1 m cubed | |

Rectangular prism 2 | 1 m cubed | |

Rectangular prism 3 | 1 m cubed | |

Shape | Volume | Surface area |

Cylinder 1 |
1 m cubed | |

Cylinder 2 |
1 m cubed | |

Cylinder 3 |
1 m cubed |

Shape | Volume | Surface area |

Cone 1 | 1 m cubed | |

Cone 2 | 1 m cubed | |

Cone 3 | 1 m cubed |

Shape | Volume | Surface area |

Sphere 1 | 1 m cubed | |

Sphere 2 | 1 m cubed | |

Sphere 3 | 1 m cubed |

Shape | Volume | Surface area |

Triangular prism 1 | 1 m cubed | |

Triangular prism 2 | 1 m cubed | |

Triangular prism 3 | 1 m cubed |

Shape | Volume | Surface area |

Pyramid 1 | 1 m cubed | |

Pyramid 2 | 1 m cubed | |

Pyramid 3 | 1 m cubed |

- Why is it that shapes with the same volume can have different surface areas?

- Create six different shapes in Volumize that have the same surface area of 3 meters-squared, and fill in the table below. Note their volumes in Volume 1. Then, edit each shape by changing one dimension (height, radius, etc.) but keeping the surface area at 3 m-squared. Then take note of the new volumes as Volume 2.

Shape | Volume 1 | Volume 2 | Surface area |

Rectangular prism | |
3 m-squared | |

Cylinder | |
3 m-squared | |

Cone | |
3 m-squared | |

Sphere | |
3 m-squared | |

Triangular prism | |
3 m-squared | |

Pyramid | |
3 m-squared |