Skip to content

Volume Explorer

Run the Volume Explorer MicroSim Fullscreen
Edit in the p5.js Editor

About This MicroSim

Two solids are drawn side by side to the same scale. Both use the same width (the side of a square base or the diameter of a round one) and the same height, so the comparison is always fair.

Below each solid, its volume formula is shown with the current values substituted. The green panel divides the first volume by the second.

Some pairs give a ratio that never changes, no matter how you move the sliders:

  • Square prism and square pyramid: the pyramid holds exactly one third of the prism.
  • Cylinder and cone: the cone holds exactly one third of the cylinder.
  • Cylinder and sphere, when the height equals the diameter: the sphere fills exactly two thirds of the cylinder.
  • Cube and sphere: the sphere fills \(\pi/6\), about 52.4%, of the cube.

How to Use

  1. Choose Solid 1 and Solid 2 from the two menus.
  2. Drag Width or diameter and Height and watch both volumes and their ratio.
  3. Find a pair of solids whose ratio stays the same for every width and height.
  4. Choose a cylinder and a sphere, then set the height equal to the diameter.

Iframe Embed Code

<iframe src="main.html"
        height="587"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

9-12 (High School Geometry)

Duration

15-20 minutes

Prerequisites

Area of squares and circles, and the idea of volume as the amount of space a solid occupies.

Bloom's Taxonomy Level

Analyze (L4)

Learning Objective

Students will be able to apply volume formulas to prisms, cylinders, pyramids, cones, and spheres, and evaluate the volume relationships between related solids.

Activities

  1. Calculate (5 min): Students compute the volume of each of the six solids for a width of 10 and a height of 12 and check against the MicroSim.
  2. Investigate (6 min): Students test all pairs to find which have a ratio that does not depend on the sliders, and record the constant ratios.
  3. Justify (4 min): Students explain the cylinder-to-cone ratio using cross-sections, referring to the Cross-Section Explorer if needed.

Assessment

A cone and a cylinder have the same base and height, and the cylinder holds 540 cubic centimeters. Students state the cone's volume, then explain what happens to both volumes when the radius is doubled.

References

  1. Wikipedia. Volume.
  2. Wikipedia. On the Sphere and Cylinder.
  3. Common Core State Standards Initiative. High School Geometry: Geometric Measurement and Dimension (HSG-GMD.A.1, HSG-GMD.A.3).