Explaining Volume Formulas

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will give an informal argument for the formulas for the circumference of a circle, area of a circle, and the volumes of a cylinder, pyramid, and cone.
  • Mathematical Practices—You will use appropriate tools strategically.
  • Inquiry—You will perform an investigationin which you will make observations anddraw conclusions.
  • STEM—You will apply mathematical and technology tools and knowledge to grow in your understanding of mathematics as a creative human activity.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-CheckedActivities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Area and Circumference of Circles
  1. You will use the GeoGebra geometry tool tolook for a relationship between the circumference and the diameter of a circle. Go tocircumference ofa circle,[1] and complete each step below.If you need help, follow these instructions for using GeoGebra.
  1. Follow theseinstructions, and fill in the table for different values of the diameter. Round your answers to the hundredths place.
  • Move the slider to roll the circle. As you roll the circle, the circumference of the circle unfolds along the x-axisand the length of the circumference appears below.
  • Once you have completely rolled out the circumference, fill in the table for the specific diameter you used.
  • Roll the circle back to its original position.
  • Change the diameter by moving the blue dot on the y-axis.
  • Repeat these steps for each new diameterthat you choose.

Type your response here:

Diameter / Circumference / Ratio of Circumference to Diameter
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  1. What relationship between the circumference and the diameter is suggested by your observations?

Type your response here:

  1. The number 3.14 is a rounded value of the mathematical quantity ?.If you replace 3.14 with?, what is the exact formula for the circumference in terms of the diameter,d, and in terms of the radius,r?

Type your response here:

  1. Now you will use GeoGebra to explore how the value of ?is related to the area of the circle. Go to area ofacircle,[2] and follow the directions below.

Follow these directions, and answer the questions that follow:

  • Move the slider to straighten the circumference of the circle.
  • Click the circumference checkbox to see the length of the circumference.
  • Dissect the circle by clicking the dissect checkbox.
  • Move the rearrange slider to rearrange the dissected pieces of the circle. First you’ll see the partsline up on the straightened circumference,and then the parts move into an arrangement with the curved segments on alternate sides.
  • Use the lowest slider to increase the number of dissected parts.
  1. As you increase the number of dissected parts, the shape they form begins to resemble a geometric figure. Which geometric figure does the shape resemble?

Type your response here:

  1. What are the dimensions of this shape in terms of the dimensions of the circle?

Type your response here:

  1. What is the general formula for the area of the shape that you see?Once you decide, rewrite the formula in terms of the dimensions of the circle.

Type your response here:

  1. Based on the formula that you developed in part iii, what is the area of the circle? How do you know your answer is correct? Explain.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Volume Formulas: Volume of an Oblique Cylinder

You will perform a simple experiment to examine how the volume of a three-dimensional object changes if you slant its vertical axis. You can perform this activity using a stack of identical coins, CDs, or any available set of identical disk-shaped objects. Ifyou cannot locate a set of disk-shaped objects, go tooblique cylindersto simulate the experiment online. Scroll down to the Volume section.

Begin by stacking the disks one on top of the other so they create an upright stack. That is,the disks should form a right cylinder.

  1. Think about the volume of the stack. By definition, whatis the volume of the stackequal to? Express you answer in words instead of numbers.

Type your response here:

  1. Now slowly slide the disks so the stack leans in one direction, forming an oblique stack. Be sure the stack does not fall over. Does the volume of the stack changeas you slant the stack of disks? Explain.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Volume Formulas: Volume of a Cone

In this activity, you will compare the volume of a cone with the volume of a cylinder with the same base and height as the cone to find how their volumes are related.Go to volumeof a cylinder and volume of a cone, and complete each step below.

  1. Drag the orange points on the cylinder and the cone to change their radii and heights. (Make sure theFreeze height checkboxesare not checked.) Set equal radii and heights for the cylinder and the cone, andnote their respective volumes. Record the volumes for a few sets of heights and radii,and calculate the ratio of the volumes in each case. Remember, make sure the height and radius of the cone and the cylinder are the same in each pair.(You might see some discrepancies in the tool due to rounding of decimals.) Round your calculations for ratio to the hundredthsplace. The first one has been done for you.

Type your response here:

Radius / Height / Volume of Cone / Volume of Cylinder / Ratio of Volumes

1 / 12 / 16 / 2,412.7 / 7,238.2 / 0.33
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  1. Based on your results in part a, how is the volume of the cone related to the volume of the cylinder, given that their bases and heights are the same?(Keep in mind that the volumes you recorded were rounded, not exact values.)

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Volume Formulas: Volume of a Pyramid

In this activity, you will compare the volume of a cube with the volume of a pyramid that has the same base and height as the cube to find how the volumes are related. Go to volume of a cube and volume of a pyramid, and complete each step below.

  1. Drag the orange pointon the cube to change its side length. Then set the base of the pyramid to the same dimensions as a face on the cube. Also set the height of the pyramid to the same side length as the cube. Note the volumes of the cube and the pyramid. Record the volumes for a few sets of heights and bases, and then calculate the ratio of the volumes. Be sure the dimensionsfor the cube and the pyramid are the same in each set.(You might see some discrepancies in the tool due to rounding of decimals.) Round your calculations for the ratio to the hundredths place. The first one has been done for you.

Type your response here:

Side of Pyramid Base / Height of Pyramid / Side of Cube / Volume of Pyramid / Volume of Cube / Ratio of Volumes

1 / 5 / 5 / 5 / 41.6 / 125.0 / 0.33
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  1. Based on your results in part a, how is the volume of the cube related to the volume of the pyramid, given that their bases and heights are the same? (Keep in mind that the volumes you recorded were rounded, not exact values.)

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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[1]“Rolling a circle to find pi,” created with GeoGebra by dhabecker, July 14, 2012, is used under a Creative Commons Attribution-ShareAlike license.

[2]This work is adapted from “Area of Circles,” created with GeoGebra by Anthony C.M. OR (orchiming), Sept. 7, 2011, and used under a Creative Commons Attribution-ShareAlike license.