To Prove the Volume Relation in Between the Cone, Sphere & the Cylinder of the Same Diameter & the Height. This Volume Proportion Is 3:2:1.

The volume relation in between the cylinder, sphere & the cone of same diameter & height is 3:2:1. If students do this as a hands-on activity they will remember this concept very well.

To prove this concept we have done 3D designs with the help of Tinkercad. The hollow cylinder, sphere & the cone is designed using the Tinkercad. The outer diameter & the height of the cylinder, sphere & the cone is kept as 60mm each. The outer diameter of white colored funnel fixed above the sphere is kept as 65mm. All four designs then were 3D printed using X Max 3D printer. For 3D printing we have used white & orange colored PLA filaments.

We asked students to take 3D printed hollow cone & completely fill it with the sea sand. Then we asked them to empty it in the hollow sphere through funnel fixed above it. Student needed to empty two completely filled cones to fill sphere completely. While it took three completely filled cones to fill cylinder completely. This proves that the volume proportion of the cylinder, sphere & the cone with same diameter & height is 3:2:1.

Students enjoyed this activity when they do it with their hands & may remember forever.

We have given below step by step procedure to 3D design hollow cone, sphere, funnel for the hollow sphere & the cylinder. This way students can be taught to design these shapes.

3D Printer

PLA Filaments Pool - Orange & White colored

Laminated Plywood base

Scissors

Sea Sand

Drag a cone on a work plane. Make it upside down by rotating in 180 degrees. Change its height & diameters to 60mm each. Make it hollow. See the photograph above.

Drag another cone on a work plane. Make it upside down by rotating in 180 degrees. Change its height & diameters to 60mm & 58mm respectively. Make it hollow. 

Drag a 58mm diameter cone into a 60mm cone. Make sure these two cones are concentric. See the top view in above picture.

Elevate inner(58 diameter) cone by 1 to 1.5mm as shown in the photograph above.

Select the outside(60mm) cone & convert it into solid.

Now group these two cones. It will convert into hollow cone as shown above.

Drag a solid sphere from basic shapes to the work plane. Change all three diameters of the sphere to required size. In this case we have changed it to 60mm each. (To change dimension you can click on white square, you will get 20mm as size. Click on the figure & change it to 60mm.) 

Drag a solid sphere on the work plane next to the existing sphere. Change all three diameters of this sphere to 58mm each. Make sure the new sphere is aligned with the earlier sphere as shown in the photograph.

Convert solid sphere of 60mm to hollow sphere by selecting it & clicking on the hole option.

Drag 58 mm hollow sphere into 60mm hollow sphere, see the front view as shown. Are they concentric?

If they are not concentric as shown in the last photograph, adjust the 58mm sphere so that they are concentric as shown in the above photograph. See from all views i.e. front, left, right, top, bottom etc.

Top View

Left View

Convert only the outer (i.e.60mm) sphere into solid sphere by selecting it & clicking on solid option.

Drag a hollow cylinder on the work plane & Change the hollow cylinder’s dimensions as below.

Two diameters=12mm & height 12mm.

Front view sphere & hollow cylinder.

Elevate hollow cylinder to 58mm height as shown above.

Drag hollow cylinder on the top of the sphere as seen in the photograph above.

 Make sure that the cylinder on the top of the sphere should be concentric to the sphere. See the top view of the same in the above photograph.

Group the hollow cylinder & the sphere. See how it looks in the photograph.

Hollow sphere’s drawing is ready for 3D print. One could see the age at the top hole of the sphere. This age indicates that the sphere is hollow. This drawing is to export as .stl file & 3D print it.

Drag a cone on the work plane. Make it upside down by rotating in 180 degrees. Change its diameter to 65mm. Make it hollow.

Drag another cone on the work plane. Make it upside down by rotating in 180 degrees. Change its diameter 63mm. Make it hollow.

Drag a 63mm diameter cone into a 65mm cone. Make sure these two cones are concentric. See the top view.

Elevate inner(63 diameter) cone by 1 to 1.5mm as shown above.

Select the outside(65mm) cone & convert it into solid. 

Now group these two cones. You will have one hollow cone (Can be seen in above photograph)

Drag a hollow cylinder on the work plane. Change its dimensions as Two diameters 12cm & height 10cm. Drag another hollow cylinder on the work plane. Change its dimensions as, two diameters 10cm each & height 10cm.

Drag the smaller (10cm Diameter) hollow cylinder into the bigger (12cm Diameter) hollow cylinder. Make sure they are concentric.

Now make the bigger cylinder (12cm Diameter) solid.

Group these two cylinders, so that you could get one solid hollow cylinder of outer diameter 12mm & height 10mm.

Elevate cone (not cylinder) to 6mm as shown above.

Drag a another hollow cylinder on the work plane. Make its dimensions 12mm height, 12mm two diameters.

Take this hollow cylinder to the center of the cone. See the top view.  

Group the cone & cylinder by clicking on the group command icon. You Could see the hole at the center of the cone.

 Drag a 12mm hollow cylinder (orange colored) under the cone so that they are concentric. See side view & top view

Now group the cone & the cylinder. This is the final design of the 65mm funnel.

Drag a hollow cylinder on the work plane. Change its height & diameters to 60mm each.

Drag another hollow cylinder on the work plane. Change its dimensions as, Height 60mm & two diameters 58mm each.

Drag 58mm diameter cylinder into 60mm cylinder. Make sure these two cylinders are concentric. See the top view in above photograph.

Take the work plane in front view & elevate inner(58diameter) cylinder by 1.3 to 1.8mm as shown in the above photograph.

Select the outside(60mm) cylinder & convert it into solid.

Now group these two cylinders. It will get converted into a hollow cylinder. This is the final drawing of hollow cylinder.

3D Printed Hollow cone (Kept on wooden base)

3D Printed hollow sphere & attached to laminated wooden base.

3D Printed Hollow Sphere with Funnel on top & attached to laminated wooden base.

3D Printed hollow cylinder attached to laminated wooden base.

Cylinder, Sphere & Cone filled with sea sand. There volumes are in the proportion of 3:2:1