Relationship between Young's modulus, bulk modulus and Poisson’s ratio

In isotropic materials, there is a relationship between Young's modulus, the bulk modulus and Poisson’s ratio. This calculator determines Young's modulus based on the relationship, and provides the derivation of the relationship.
CalculatorInput﻿﻿K
:200.00GPa​
﻿
﻿﻿v
:-0.20​
﻿
﻿
Output﻿﻿E
:840.00GPa​
﻿
﻿
E=3K(1−2ν)E = 3K(1 - 2\nu)E=3K(1−2ν)
Where:
﻿﻿ν
 is Poisson's ratio, a measure of how much a material expands or contracts in the direction perpendicular to its loading direction (unitless)
﻿﻿K
 is bulk modulus, a material property that is a measure of the resistance to longitudinal deformation ﻿(GPa)
﻿
﻿﻿E
 is Young's modulus, a material property that is a measure of the resistance to longitudinal deformation ﻿(GPa)
﻿
ExplanationImagine a segment of rubber, in the usual shape of a unit cuboid. Let us consider, a tensile force ﻿P
 (such as you pulling on it) acting normally outward on each surface of the unit cube. 
﻿
Tensile forces, P acting on each surface of a unit cube
The force ﻿P
 acting along the direction of x-axis produces an extension ﻿ε
 along that direction. Since the cube is a unit cube, ﻿P
 and ﻿ε
 indicate the tensile stress and tensile strain along that axis, respectively. By definition, Young’s modulus, ﻿E
 is therefore given by ﻿E=εP​
 or ﻿ε=EP​
.
Also by definition, the contraction of the cube in the other two directions (y-axis and z-axis) is given by ﻿νε=YνP​
 where ﻿ν
 is Poisson’s Ratio. 
The force ﻿P
 acting in the x-direction produces an extension ﻿EP​
 in the x-axis and a contraction ﻿−EνP​
 in the y-axis and z-axis.
Similarly, the tensile force ﻿P
 acting in the y-direction produces an extension ﻿EP​
 in the y-axis and a contraction ﻿−EνP​
 in the x-axis and z-axis. The tensile force ﻿P
 acting in the z-direction produces an extension ﻿EP​
 in the z-axis and a contraction ﻿−EνP​
 in the x-axis and y-axis.
The total (longitudinal) extension, ﻿εlong​
 along each of the three axes is given by:
﻿
εlong=PE−2νPE=PE(1−2ν)→E=Pεlong(1−2ν)\varepsilon_{long}=\frac{P}{E} - \frac{2\nu P}{E} = \frac{P}{E} (1 - 2\nu)\\\rightarrow E =\frac{P}{\varepsilon_{long}} (1 - 2\nu)εlong​=EP​−E2νP​=EP​(1−2ν)→E=εlong​P​(1−2ν)
All the tensile forces acting together produce a volume stress of magnitude ﻿P
. This stress produces a volume strain, ﻿εvol​
 of magnitude ﻿KP​
 where ﻿K
 is known as the bulk modulus. 
Volume strain, ﻿εvol​
 is equal to three times the longitudinal strain, ﻿εlong​
 along each direction. And therefore we get:
﻿
E=Pεvol/3(1−2ν)→E=3K(1−2ν)E =\frac{P}{\varepsilon_{vol}/3} (1 - 2\nu)\\\rightarrow E = 3K(1 - 2\nu)E=εvol​/3P​(1−2ν)→E=3K(1−2ν)
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