A Geometric Interpretation of a Math Olympiad Identity

Introduction

Often it is fun to interpret an algebraic equation or identity in terms of its geometric meaning. It not only helps in working the geometric muscles but also helps understand and remember an equation or an identity better. I’m sure that all math Olympiad enthusiasts have encountered the following identity:

$a(1-b^2)+b(1-a^2)=(a+b)(1-ab)$
Proving the identity is extremely straightforward and is almost a no-brainer:

$a(1-b^2)+b(1-a^2)=a-ab^2+b-a^2b=a+b-(a^2b+ab^2)=(a+b)-ab(a+b)=(a+b)(1-ab)$
However, trying to give a completely visual, geometric interpretation is a pretty interesting exercise to tone up one’s geometric muscles and warm up one’ visualization skills.

In this post, I will be discussing a geometric perspective of the above algebraic identity.

A Generalized Version of the Identity

First, let us note that this identity is a consequence of a more general identity in 3 indeterminates, which is:
$a(c^2-b^2)+b(c^2-a^2)=(a+b)(c^2-ab)$
and again, to prove it algebraically is almost a no-brainer, just expand the brackets and rearrange the terms in a suitable way and factorize, and you’re done, just like we did in the first paragraph.
When we put $c=1$ in the above identity, we get back the identity that we started with. So, if we prove the generalized identity, we are also proving as a particular case, the identity that we started with.

Tune-In to Your Geometric Intuition

How do we come up with a nice geometric proof? Someone with a geometric mind will first note that all the terms (e.g. $ac^2,a^2b,bc^2,\dots\text{etc.}$ ) in the identity have a degree 3. This suggest that we should think of some geometric picture in 3-dimensions. A decent and appropriate 3-dimensional object to start with, in this context, would bea cuboid or in a fancier term – a rectangular parallelopiped. Let’s try that out.

The Geometric Viewpoint

See the picture below. We have a rectangular parallelopiped with a rectangular base with edge lengths $a$ and $b$ units, marked as red and blue in the picture respectively. The height of the rectangular parallelopiped is $(a+b)$ units.

The smaller edges marked with black in the picture are of lengths $c$ units each. Here we have assumed that $a,b>c$ for the sake of simplicity of our picture, but similar proof goes perfectly in other cases also.
Now observe that the small orange cuboid in the left picture has a volume of $ac^2$ cubic units, and the small green cuboid below it, has a volume of $bc^2$ cubic units. Green and orange cuboids of the same size can be found in the right picture, but now at a different place. So, if we remove the orange and green cuboids from our rectangular parallelopiped of volume $ab(a+b)$ cubic units from the right picture, the volume that will remain is $ab(a+b)-ac^2-bc^2$ cubic units, which is the same as the remaining volume when we remove the total orange-green cuboidal column of volume $(a+b)c^2$ cubic units from the left picture, which would give $ab(a+b)-(a+b)c^2$ cubic units. Therefore, we have the equality $ab(a+b)-(a+b)c^2=ab(a+b)-ac^2-bc^2$ . Again, in the left picture, the upper tier is $ab^2$ cubic units and the lower tier is $a^2b$ cubic units. After removing the orange part from the upper tier, we are left with a volume $a^2b-ac^2$ cubic units in the upper tier and similarly, we are left with a volume $ab^2-bc^2$ cubic units in the lower tier. Adding them together, we get the volume of the volume of the parallelopiped minus the thin orange-green column removed, this gives us:
$(a^2b-ac^2)+(ab^2-bc^2)=ab(a+b)-(a+b)c^2$

On the other hand, in the right picture, we are removing the green chunk of volume $bc^2$ from the upper tier, giving remaining volume $a^2b-bc^2$ for the upper tier minus the green chunk. Similarly, the volume of the lower tier minus the orange chunk in the right picture turns out to be $ab^2-ac^2$ . Summing up, we get the total volume of the right side picture, i.e. the parallelopiped minus the orange and green chunks, as $(a^2b-bc^2)+(ab^2-ac^2)$ . But this is the same volume as $ab(a+b)-bc^2-ac^2$ , which is the right side parallelopiped, minus the green and orange chucks. So, we have
$(a^2b-bc^2)+(ab^2-ac^2)$ = $ab(a+b)-bc^2-ac^2$ .
So, we see that each step involved in proving the equality between the left hand side and the right hand side of the identity have a corresponding geometric interpretation in terms of the volumes of the smaller cuboids (orange and green) and the whole rectangular parallelopiped.
So, one could teach this identity even to a kid who is not well versed in algebraic manipulations (e.g. taking terms in common and manipulating the positive and negative signs while expanding brackets etc.) but has an idea of volume and is well-versed in manipulating toy building blocks!

Closing Remark: Algebra You Can See and Touch

This geometric interpretation highlights a deep pedagogical truth: many algebraic identities are shadows of underlying geometric facts. What appears on paper as symbolic manipulation often encodes statements about area, volume, or spatial structure. By lifting the identity into three dimensions and interpreting each term as the volume of a tangible object, the equality ceases to be a formal trick and becomes an intuitive fact.

Such visual arguments do more than build geometric intuition — they also strengthen mathematical memory. An identity understood through volume decomposition is far more likely to be remembered than one obtained solely by mechanical expansion and factorization. More importantly, this perspective emphasizes that mathematics does not live in isolated compartments of algebra and geometry, but flourishes through their continuous interaction.

Seen this way, identities are not merely equations to verify; they are geometric rearrangements waiting to be discovered. When algebra feels opaque, geometry often restores clarity, and when symbols feel abstract, spatial reasoning brings them back to life. In cultivating this interplay, we do not merely solve problems—we learn to see mathematics.

Teaching Note (For Educators)

This identity offers an excellent opportunity to introduce students to the idea that algebraic equalities can be understood through geometric reasoning. Rather than beginning with symbolic expansion, instructors may start with a visual or hands-on activity—using physical blocks, 3D models, or dynamic geometry software—to construct and decompose a rectangular parallelepiped corresponding to the terms of the identity.

For younger students or Olympiad beginners, the focus can remain on volume reasoning and spatial decomposition, allowing learners to verify the equality without formal algebraic manipulation. This approach is particularly effective for students who struggle with symbolic abstraction but respond well to visual or tactile modes of thinking.

At the same time, the example naturally invites deeper discussion for more advanced learners: how polynomial degree relates to dimension, how generalization emerges from geometric insight, and how proof can be guided by structure rather than computation alone. Used thoughtfully, this identity becomes not just an exercise, but a lesson in how mathematical understanding is built.