volume of sphere and volume of cone by simulation method

Cone & Sphere Volume Lab

Interactive 3D Proofs of Cylinder, Cone, and Sphere volumes

| Created by: Ghanashyam Adhikari

Part 1: Volume of Cone

Select Lab Stage:

Part 1: Pouring Progress

Ready for Pour #1

Drag on screen to rotate viewpoint in 3D space.

Cone Fill 100%

Cylinder Fill 0%

Sphere Fill 0%

Loading Geometric Equations...

Cone Cup ($V_{\text{cone}}$)

Cylinder Jar ($V_{\text{cyl}}$)

Sphere ($V_{\text{sphere}}$)

Adjustable Parameters

Radius ($r$) 1.20 cm

Height ($h$) LOCKED $h=2r$ 2.40 cm

Pour Simulation

Interactive variables auto-scale the 3D models while retaining Archimedes' proportions.

Proof & Derivation Steps

Discover the exact relationships linking the Cone, Cylinder, and Sphere.

1

First Cone Pour (1/3 Filled)

Pouring one full cone into the empty cylinder fills exactly one-third (1/3) of its total volume.

\(V_{\text{filled}} = \frac{1}{3} V_{\text{cylinder}}\)

2

Second Cone Pour (2/3 Filled)

Pouring a second full cone into the cylinder fills exactly two-thirds (2/3) of its total volume.

\(V_{\text{filled}} = \frac{2}{3} V_{\text{cylinder}}\)

3

Third Cone Pour (3/3 Complete)

Pouring the third full cone fills the cylinder exactly to the top brim, proving the ratio is exactly 3 to 1.

\(V_{\text{cylinder}} = 3 \times V_{\text{cone}} \implies V_{\text{cone}} = \frac{1}{3}\pi r^2 h\)

1

First Cone Pour (1/2 Filled)

Pouring one full cone into the empty sphere fills it up to exactly half (50%) of its total capacity.

\(V_{\text{filled}} = \frac{1}{2} V_{\text{sphere}}\)

2

Second Cone Pour (Sphere Full)

Pouring a second full cone tops the sphere off exactly to the brim, establishing the base relationship: Volume of Sphere = 2 × Volume of Cone.

\(V_{\text{sphere}} = 2 \times V_{\text{cone}}\)

3

Substituting Sphere Height (h = 2r)

Since a sphere's effective bounding height is equal to its diameter (h = 2r):

\(V_{\text{sphere}} = 2 \times \left(\frac{1}{3} \pi r^2 h\right) = \frac{2}{3} \pi r^2 h\)

Substitute \(2r\) for \(h\):

\(V_{\text{sphere}} = \frac{2}{3} \pi r^2 (2r)\)

Simplify:

\(V_{\text{sphere}} = \frac{4}{3} \pi r^3\)

Mathematical Formulation Summary:

Volume of Cylinder:

\(V_{\text{cylinder}} = \pi r^2 h\)

Volume of Cone:

\(V_{\text{cone}} = \frac{1}{3} \pi r^2 h\)

Proof Check Ratio:

\(\frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 3\)

Volume of Sphere (Relative to Cone):

\(V_{\text{sphere}} = 2 \times V_{\text{cone}} = \frac{2}{3} \pi r^2 h\)

Deriving via Radius Substitution (h = 2r):

\(V_{\text{sphere}} = \frac{2}{3} \pi r^2(2r) = \frac{4}{3}\pi r^3\)

Geometric Integration

Geometric relationship established among cylinder, cone, and sphere volumes is a foundational concept in 3D geometry proofs.

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