Squares are among the first shapes children learn, and there is so much mathematical meaning to unpack in later years. Read on for my shout outs for the geometric, numerical, and algebraic wonders of squares.

A Special Quadrilateral

Geometrically, squares are quadrilaterals with four equal sides and four right angles.  When young students first encounter them, they often think of squares and rectangles as distinct categories, but later on, we define squares as a special category of rectangles that have all sides congruent.  In high school geometry, textbooks might define a square as an equilateral and equiangular quadrilateral, or a type of parallelogram with 4 right angles and 4 congruent sides^1A. 

When I’m teaching the quadrilateral unit, I make a distinction between the DEFINITION of a shape and its PROPERTIES that are consequences of the definition, and my students explore the connections that come from inclusive^1B definitions, even when our textbook uses an exclusive rule.  Here is the typical exclusive quadrilateral hierarchy taught in US geometry courses [Source: Math Monks].

Once students have explored the properties of squares, we discuss what features constitute the “minimal definition”, i.e. what are the fewest set of characteristics that guarantee a square?  I’ve used this one: a square is a parallelogram with ONE right angle (which implies 4 right angles) and a pair of adjacent sides congruent (which guarantees all 4 sides congruent). I also like to have students construct squares using dynamic geometry technology²: what tools can you use?  Must you use the Parallel tool?  How many different ways can you create congruent sides with various construction tools? 

Square Numbers

Thinking numerically, the square of a number is found by multiplying the number by itself.  Let’s also note that any perfect square number can be represented by a square array of dots, a helpful visual model. Even young children who haven’t yet encountered the idea of squaring (or even multiplying) in school can “build” different types of numbers using a dot or tile array as a model; they can distinguish prime vs. composite numbers and notice which composites can make square arrangements, not only rectangular arrays.

Some other fun explorations with square numbers:

• Squares are one type of “figurate numbers”: numbers that can be represented by a regular geometrical arrangement of equally spaced points or dots. Use dots to demonstrate that the sum of two consecutive triangular numbers is a square number.
• Have students look at the digits of square numbers to find some patterns. What numbers never appear in the units’ place?
• There’s a fun trick to quickly square numbers ending in a 5. Howie Hua (@howie_hua) has a short video here demonstrating the trick, and more importantly, explaining why it works.

Square Numbers as Square Areas

Square numbers are not simply special numbers. In fact, the ancient Greeks conceived of the abstract quantity s² as the area of a square of side s. That’s why something raised to the second power is said to be squared (or quadratic, because a square has 4 sides)³.

This brings us to how squares are related to one of the best-known geometry results: The Pythagorean Theorem.  The original statement of the theorem by Euclid (Proposition 47) wasn’t about squaring the lengths of the legs and hypotenuse, but about actual squares constructed upon the sides of a right triangle.

A common diagram to justify the Pythagorean Theorem to students displays the squares on each side, divided up into unit squares: [Source: Wikipedia]

Students can investigate how this diagram might be used for other Pythagorean Triples (set of three integers that satisfy the Pythagorean Theorem a² + b² = c² ).  Also of interest is that one way to generate Pythagorean Triples is to choose two numbers M and N, with M > N.  Then a is the difference of their squares M² – N², b equals double their product 2·M·N, and c equals the sum of their squares M² + N².

The Difference of Two Squares

Which brings us to the algebraic concept of the “difference of two squares” which is a special factoring pattern Algebra 1 students learn when studying quadratic expressions:

When we do the multiplication to check the factoring, we notice that the partial products from the four distributions have two that sum to zero (or cancel out).  This is a consequence of the fact that the two binomials are conjugates, binomials that have the same terms but opposite middle sign⁴.

Often, Algebra 1 class doesn’t mention any geometrical connection here, but let’s take a look.  Create two squares of different sizes (left below), and display them with a common corner as shown in the middle figure below.  The L-shaped uncovered area of the large square is X² – Y² (because it is the small square Y² subtracted from the large square X² (right below).  Can you find a way to express that uncovered L-shape as (X + Y)(X – Y)? Try it first before scrolling down.

First, notice that each short end of the L-shape has length X – Y (left image below).  Then decompose the L-shape into two rectangles as shown (right image below).  The black outlined rectangle has area Y·(X – Y), and the red outlined rectangle has area X·(X – Y). 

So the total area of the L-shape is

I had been teaching this algebraic factoring pattern for years before I heard a teacher share the wonderful geometric interpretation at a conference. Since then, I love to show this geometric connection to my students, to deepen their understanding of the procedure.

Some other algebraic “squares” you can represent geometrically are the Perfect Square Trinomial factoring pattern for X² + 2XY + Y² and the meaning of Completing The Square. For both of these, try out the Algebra Tiles at Mathigon Polypad, use physical algebra tiles or simply sketch with paper and pencil.

The Difference of Two Squares is also a helpful arithmetic procedure. If you are multiplying two numbers that have an even difference, try finding their average and using this idea. For example 97·103 can be rewritten as (100–3)(100+3) which makes the computation much easier; it is 100² – 3² which is 10,000 – 9 or 9,991.

Finally, I found this great problem set from NRICH Maths about writing numbers as differences of two squares, including generalizing some results algebraically.

Other Fun with Squares

Here are some other interesting square items I’ve found recently on the internet⁵:

Area problems: Lots of people post geometry puzzles on the internet, so check the hashtags #geometry #puzzle and also #RecreationalMath. Here are three, the first from James Tanton (@jamestanton) posted here, and the other two were retweeted by Duane Habecker (@dhabecker) but originally posted here and here.

A Two Square Surprise! Tim Brzezinski (@TimBrzezinski) shares many GeoGebra animations all over the internet. Here’s one he animated, originally from Antonio Gutierrez (@gogeometry). Tim’s animation is here, the interactive GeoGebra figure is here, and Antonio’s original post is here.

Take 2 squares (of any size or orientation) and connect corresponding vertices. What do you notice about the midpoints of the connecting segments?

Catriona’s 6-square problem: One of many geometric puzzlers presented by Catriona Agg (@Cshearer41) is this fun one with 6 squares. In this pattern of squares, all six coloured areas are equal. What fraction is shaded? Her original post is here.

Tangrams & Other Manipulatives: Physical and virtual manipulatives are wonderful tools. In addition to Algebra Tiles mentioned above, I absolutely LOVE Tangrams! Paula Krieg (@PaulaKrieg) wrote a recent post about Symmetry with Tangrams; she uses two sets to help young students investigate this idea. Mathigon Polypad has virtual tangrams available in its wonderful playground of manipulatives.

Math Photo 23: For the past several summers, Erick Lee (@TheErickLee) has coordinated a Math Photo sharing with weekly themes. I’m hosting this week’s theme of #Squares (and cubes) which was part of my motivation for writing this post. I’m posting on both Twitter and Mathstodon.

Dividing a Square Into Squares and Rectangles: [Update to the post] There is a fun branch of math concerned with dissecting geometric figures into other figures⁶. Among these investigations are two that I’ve been reading about from John Carlos Baez (@johncarlosbaez@mathstodon.xyz). First, can you divide a square into other squares, either of all different sizes or allowing a size to repeat? Here is a discussion on the different size problem (image on left below); here is an article on which number of component squares are possible, with repeated sizes allowed (image on right below), along with activity from Dan Finkel (@MathForLove).

Then, in December 2022, John Carlos asked a question on Mathstodon about dividing a square into rectangles which are similar to each other. This led to a robust discussion and multi-pronged investigation which John Carlos reported on HERE and Siobhan Roberts wrote about in the NY Times HERE.

A Square Dance: MathArt “Notes on a Square” (Updated 9/2024) Here’s one more bit of square fun: Chris Nho (@nhoskee) has created a joyful animation of transforming (dancing) squares at https://nhoskee.substack.com/p/notes-on-a-square.

Square Surprise! (Updated 6/2025) Tim Brzezinski (@TimBrzezinski) shares his favorite surprising square result… did you know every square contains a 3-4-5 right triangle? Check it out at https://www.youtube.com/shorts/i__d3Ac4WWY.

As you can see, there is so much to shout about for squares. Did you notice that even my numeric and algebraic topics above all can relate back to the GEOMETRY of squares? Enjoy the square and squaring fun!

Notes and Resources

^1AThese are two of nine commonly used definitions of squares found in math textbooks. ^1BInclusive definitions in Geometry consider some categories as subsets of others. For example, if trapezoids have “at least one pair of parallel sides” then they include parallelograms, but if trapezoids have “exactly one pair of parallel sides” then they are distinct categories. Below are two quadrilateral hierarchies: (a) on LEFT is inclusive, (b) on RIGHT is exclusive. [Source: Usiskin & Griffin, 2008, The Classification of Quadrilaterals: A Study in Definition.]

Here is a lovely Venn diagram version using an inclusive thinking of quadrilaterals. [Source: Wikipedia]

²I use all dynamic geometry platforms: GeoGebra, Desmos, Cabri Jr. on TI-84+ calculators, and TI-Nspire calculators. There are at least ten ways I’ve found to construct squares, and no, you do not need to use the parallel tool. To create congruent segments, don’t forget about the circle & compass tools.

³From The Words of Mathematics: An Etymological Dictionary Of Mathematical Terms Used in English by Steven Schwartzman.

⁴I wrote about conjugates in my post Powerful Pairs. Note also that I referred to the multiplication of binomials as “four distributions”; this is often called FOIL (for “first, outside, inside, last”) but I prefer to use the terminology of repeated distribution and partial products since it works universally when multiplying polynomials, not just for binomials.

⁵Math teachers gather on the internet in all sorts of places! We use various hashtags, such as MTBoS (Math Teacher Blog-o-Sphere), ClassroomMath, iTeachMath, and MathEd on a variety of platforms (Twitter, Facebook, Mastodon, etc.). If you are looking for helpful educators, shared resources, and thoughtful discussions, find us wherever you are online.

⁶Dissection proofs abound for the Pythagorean Theorem, which might be a topic for a future blog post. And another wonderful geometry topic is Rep-Tiles, which are dissections of a shape into components that are copies of the same shape (self-similar tilings). Finally, Michaela Epstein (@MathsCirclesOZ) pointed me to the open problem of what is the least number of squares needed to fully tile a rectangle.

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