Geometric proof of the CUBIC FORMULA (a+b)^3

CUBIC FORMULA PROOF

Just like we expressed the geometric proof of expansion of quadratic formula - we prove the expansion of  (a + b)3 

• the expansion is: (a+b)³ = a³ + 3a²b + 3ab² + b³

PROOF:
Lets draw a cube with side length (a+b) , hence we know that the volume of this cube would be equal to (a + b)3 

• Note that in the above diagram, the red part itself is a cube with volume a3  and the blue part is a cube with volume b3
•  Further more, note that the yellow part is a cuboid with height a, width a, and length b ; thus the cuboid has volume a x a x b =  a²b, and as the diagram has 3 yellow cuboids, the total volume of the yellow part would be 3a²b
• Similarly, the green part is a cuboid with height b, length b, and width a ; thus a cuboid with volume ab², and as the diagram has 3 green cuboids, the total volume of the green part is 3ab²

HENCE, THE VOLUME OF THE WHOLE CUBE CAN BE DISSECTED AS FOLLOWS : 

•  We first remove the primary a3  cube and b3 from the image as shown below: 

• Further evaluating the diagram allows us to remove other inscribed combination of shapes :

THEREFORE,

the volume of the cube = a³ + 3a²b + 3ab² + b³

(a+b)³ = a³ + 3a²b + 3ab² + b³