${(a+b+c)}^2$ $=$ $a^2+b^2+c^2$ $+$ $2ab+2bc+2ca$

The a plus b plus c whole square formula is derived in algebraic form by geometrical approach as per the areas of square and rectangle.

- Take a square and divide the square vertically into three different parts by drawing two lines. The lengths are $a$, $b$ and $c$ respectively.
- Divide the square horizontally into three parts but the lengths of them should also be $a$, $b$ and $c$ respectively.
- The length of each side is $a+b+c$. Therefore, the area of the square is $(a+b+c) \times (a+b+c)$ $\,=\,$ ${(a+b+c)}^2$

The square whose area is $a$ plus $b$ plus $c$ whole square, is divided as three squares and six rectangles.

- Length of each side of three squares are $a$, $b$ and $c$. So, the areas of them are $a^2$, $b^2$ and $c^2$ respectively.
- The lengths of sides of two rectangles are $a$ and $b$. So, the area of each rectangle is $ab$.
- The lengths of sides of two rectangles are $c$ and $a$. So, the area of each rectangle is $ca$.
- The lengths of sides of two rectangles are $b$ and $c$. So, the area of each rectangle is $bc$.

The area of whole square is ${(a+b+c)}^2$ geometrically.

The whole square is split as three squares and six rectangles. So, the area of whole square is equal to the sum of the areas of three squares and six rectangles.

${(a+b+c)}^2$ $\,=\,$ $a^2+ab+ca$ $+$ $ab+b^2+bc$ $+$ $ca+bc+c^2$

Now, simplify the expansion of the $a+b+c$ whole square formula to obtain its expansion in simplified form.

Thus, the expansion of a plus b plus c whole square is proved in algebraic form by the geometrical approach in mathematics.

$\,\,\, \therefore \,\,\,\,\,\, {(a+b+c)}^2$ $\,=\,$ $a^2+b^2+c^2$ $+$ $2ab+2bc+2ca$

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved