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Pyramid’s and Cone’s Volume formulas – Demonstration – pg. 1/3

Posted in Different Math Theory topics, Math Theory by Lia on 04/13/2011

Pyramid’s and Cone’s Volume formulas – Demonstration: 1, 23

 

* To see the Table for Polygons and Circle –  Definitions, Perimeter and Area formulas, click here. 

To see the Table for Prism, Pyramid, Cylinder, Cone, Sphere and other Solid Geometric Shapes – Definitions, Area and Volume formulas, click here.

 

Below is the demonstration for Pyramid’s and Cone’s Volume formulas.

 

We start our demonstration from a known formula, the prism’s volume formula.

The Volume Formula for the right prism (has the lateral faces perpendicular to the base)  is:

[1] tabular{11}{11}{{V = {A_{base}}h}}

Where A_{base} is the base area and h is the height of the prism (Fig. 1). For the right prism, the height h is equal with the lateral edge of the prism. All lateral edges are equal, parallel among themselves and perpendicular on the bases.

We are going to demonstrate now that the volume for the oblique prism ABCDEFGH (Fig. 2)  is equal with the volume for the right prism {A_1}{B_1}{C_1}{D_1}EFGH (Fig.2).

The plane BCGF is parallel with the plane ADHE  and {B_1}{C_1}GF  and {A_1}{D_1}HE are 2 planes {}{ ortho}{} on the parallel planes ABCD  and EFGH.

Then we have:

  • {Delta}B{B_1}F = {Delta}A{A_1}E because:

* BF = AE (given)

* {B_1}F = {A_1}E (equal distances between the two parallel bases)

*B{B_1}F =  ∠A{A_1}E = 90^{circ}

  • {Delta}C{C_1}G = {Delta}D{D_1}H because:

* CG = DH (given)

{C_1}G = {D_1}H (equal distances between the two parallel bases)

* C{C_1}G =  ∠D{D_1}H = 90^{circ}

  • BCGF = ADHE (given)

  • {B_1}{C_1}GF = {A_1}{D_1}HE because:

{B_1}{C_1} = {A_1}{D_1}

{C_1}G = {D_1}H

GF = HE

{B_1}F = {A_1}E

  • B{B_1}{C_1}C = A{A_1}{D_1}D because:

B{B_1} = A{A_1}

{B_1}{C_1} = {A_1}{D_1}

{C_1}C = {D_1}D

BC = AD

[2]  {A_{ABCD}} = {A_{{A_1}{B_1}{C_1}{D_1}}} =  {A_{base}}  because they have equal sides.

But:

[3]  {V_{B{B_1}{C_1}CGF}} ={V_{A{A_1}{D_1}DHE}}   because their faces are equal polygons (quadrilaterals and triangles).

 

Considering [3], [4] will become:

[4]  {V_{ABCDEFGH}} =  {V_{{A_1}{B_1}{C_1}{D_1}EFGH}} - {V_ {A{A_1}{D_1}DHE}}  + {V_ {B{B_1}{C_1}CGF}}

Therefore:

[5]  {V_{ABCDEFGH}} =  {V_{{A_1}{B_1}{C_1}{D_1}EFGH}} = {A_{base}}h

Where h is the distance between the two parallel bases.

 

Continue on page 2/3

 

Pyramid’s and Cone’s Volume formulas – Demonstration: 1, 23

 

 

 

 

 

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