While most of the geometry we have tackled in this course thus far has been presented on a flat screen, the world around you is not flat. For this activity, your task is to collect two 3-D items from your environment in order to calculate their volumes and surface areas.

Question

Because the 3-D objects studied in this module had either all straight surfaces or at least one curved surface, you will need to gather two items. One item will be either a prism or a pyramid, the other item will be either a cone or a cylinder. Some items that will work for this activity are canned goods, food storage containers, recyclable cardboard items, etc.

To show your work, you will create a page with two columns, one column for the prism/pyramid calculations and the other column for the cone/cylinder calculations. You may use the Quickstart Template for your work.

You will need to decide on the units for your measurements. This will depend on your measuring tool. Once you have decided what measuring tool and units you will be using, begin by studying the base of your objects. Measure the base of each item. If the base is a polygon, you will need to measure the length and width. If the base is circular, you will need to measure the diameter or radius. Record your measurements and include the units. Using these measurements, calculate the base area of your items. Record these area calculations, along with proper units. Use 3.14 for π and round your calculations to the nearest tenth of a unit.

To calculate the volume of your 3-D objects, you need two things, the area of the base of the object and the height of the object. Using your measuring tool, measure the heights of your items. Use the same units you used to measure the length, width, and diameter or radius in step 1. Record your measurements. Using the area of the base from step 1 and the height you just found, calculate the volume of your items. Show all your work and be sure to include the proper units with your final volume calculation. Use 3.14 for π and round your calculations to the nearest tenth of a unit.

Look at your items again. Notice the surfaces that make up your 3-D items. You will now calculate the area of all these surfaces in order to find the total surface area of your items. Calculate the areas of all the surfaces that make up your items, and record your area calculations, including proper units. Add all these areas up to find the total surface area of your items, and record the final total surface area for each item. Make sure to include proper units. Use 3.14 for π and round your calculations to the nearest tenth of a unit.

Question 1: What should your units on your base area calculations be, and why? How is this different from the units on your volume calculations? Explain in two to three sentences.

Question 2: If you were to take a cross-section parallel to the base for one of your items, what shape would you see? Can a cross-section be a sphere? Explain in two to three sentences.

Submit the following to your instructor:

Your table of work, including base, volume, and surface area calculations.
Your answers to the two reflection questions.

Answer 1

To complete your activity on calculating the volume and surface area of 3-D objects, here's how you could structure your work based on the guidelines provided. Below is an example using a rectangular prism (a box) and a cylinder (a can).

Example Table of Work

| Prism/Pyramid (Rectangular Prism) | Cone/Cylinder (Cylinder) | |-------------------------------------|--------------------------| | Measurements: | Measurements: | | Length: 10 cm | Diameter: 6 cm | | Width: 5 cm | Radius: 3 cm | | Height: 8 cm | Height: 10 cm | | Base Area Calculation: | Base Area Calculation:| | Area = Length × Width | Area = π × r² | | Area = 10 cm × 5 cm = 50 cm² | Area = 3.14 × (3 cm)² | | | Area = 3.14 × 9 cm² | | | Area = 28.26 cm² | | | | | Volume Calculation: | Volume Calculation: | | Volume = Base Area × Height | Volume = Base Area × Height| | Volume = 50 cm² × 8 cm | Volume = 28.26 cm² × 10 cm| | Volume = 400 cm³ | Volume = 282.6 cm³ | | | | | Surface Area Calculation: | Surface Area Calculation: | | SA = 2lw + 2lh + 2wh | SA = 2πrh + 2πr² | | SA = 2(10 cm)(5 cm) + 2(10 cm)(8 cm) + 2(5 cm)(8 cm)| SA = 2(3.14)(3 cm)(10 cm) + 2(3.14)(3 cm)²| | SA = 100 cm² + 160 cm² + 80 cm² | SA = 188.4 cm² + 56.52 cm² | | SA = 340 cm² | SA = 244.92 cm² |

Total Measurements and Calculations:

• Rectangular Prism:
  • Base Area: 50 cm²
  • Volume: 400 cm³
  • Surface Area: 340 cm²
• Cylinder:
  • Base Area: 28.26 cm²
  • Volume: 282.6 cm³
  • Surface Area: 244.92 cm²

Reflection Questions

Question 1: The units for base area calculations should be in square centimeters (cm²) because area is a two-dimensional measurement expressed in square units. In contrast, volume calculations are in cubic centimeters (cm³) because volume is a three-dimensional measurement that accounts for length, width, and height, thus requiring cubic units.

Question 2: If you were to take a cross-section parallel to the base of the rectangular prism, you would see a rectangle with the same dimensions as the base (10 cm x 5 cm). A cross-section cannot be a sphere because a sphere is a three-dimensional shape, and its cross-section will always result in a circle, not a complete sphere. A cross-section represents a two-dimensional slice of a three-dimensional object.

Once you complete your table and reflections, you can submit them as required. Make sure that your measurements and calculations are accurate and clearly presented.