What Makes a Square a Special Rectangle? - Down River Resources

What Makes a Square a Special Rectangle?

Kindergarten math and first grade math homework can be confusing and frustration to parents and teachers at times. My favorite messages come from these adults regarding questions on homework pages. Come 4:00 p.m., my cellphone starts blowing up with pictures of homework pages! These parents and teachers know that my love of math runs deep, but my love for helping others understand, especially the students who are depending on them to answers runs deeper! Most recently, I received one of these after-school text messages with a question from first grade geometry. The question read: "I am not a rectangle. I am not a triangle. What shape am I?" Four answer choices were given including a circle, square, rectangle, and triangle. How can you answer this question? More importantly, how you can you explain this answer to your child at home or in your classroom? Read more to learn about why a square is considered a special rectangle.

Math 101: Why is a Square a Special Rectangle?

If you haven't read between the lines yet, the answer to the question is a square.

I am not a rectangle. I am not a triangle. What shape am I?

The child on the other end of the text message instantly crossed out a rectangle and triangle.

I was especially excited to see this because it is evident that the child's parent and/or teacher had taught them the importance of eliminating answer choices!

The circle and square were remaining. The father instantly thought:

 It must be the circle.

Why is that? Why did the parent deduce the circle from those two choices?

The human brain is constantly looking for patterns. The child just eliminated two prominent shapes that have characteristics similar to the square. The rectangle and the triangle have vertices and edges. The brain sends the message that the square fits into a similar pattern; therefore, it must be the circle. 

Right and wrong.

There's more to this question that meets the eye, as with many word problems.

Though "circle" is in fact the correct answer, the justification needs a little work.

This question is testing the knowledge of a first grader, in this instance, if they know that a square is considered a special type of rectangle.

The father instantly proclaimed:

When I was in school, a square was a square and a rectangle was a rectangle!
Though I can't discount this statement {we do know a lot more mathematical explanation in education these days}, this is a common frustration point with many teachers and parents alike.

To learn more about this mysterious "special rectangle," we need to revisit the attributes or properties of shapes, particularly quadrilaterals.

Attributes or Properties of a Quadrilateral 

It's really easy to get lost within all of the academic vocabulary of kindergarten, first grade, and second grade geometry.

Here's the basics:


A quadrilateral has four sides, is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Both rectangles and squares are quadrilaterals. Both shapes are two-dimensional four-sided closed figures with straight sides.

Rectangles, squares, trapezoids, rhombuses, and parallelograms are all part of the quadrilateral family.

What makes rectangles and squares unique within this family?


A rectangle is a quadrilateral. It's sides intersect at 90 degree angles. A rectangle has opposite sides which are congruent, or have the exact same length. . The diagonals, are mutually bisecting, or cut each other in half.


A square is a quadrilateral. It's sides also intersect at 90 degree angle. Similar to a rectangle, its opposite sides are congruent, but ALL of its sides are congruent, or have the exact same length. The diagonals, are mutually bisecting, or cut each other in half. The square also has perpendicular bisecting diagonals.

A rectangle is also classified as a square when both pairs of opposite sides are the same length; thus, a square is a special rectangle. 

Hierarchy of Quadrilaterals

In this illustration above, the rectangle has the properties identified within the red quadrilateral.

The square contains ALL of the properties of the rectangle AND the properties listed solely within the square figure. 

This further depicts that a square is a special type of rectangle.

If you are still following this, you know have passed kindergarten math in Texas! Yeehaw! This just so happens to be first grade math in pretty much the rest of the country.

Content Standards Addressed: TEKS K.6A
The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties. The student is expected to: identify two-dimensional shapes, including circles, triangles, rectangles, and squares as special rectangles

I hope this post inspires you to dust off your college textbook and learn more math lingo, or gives me the privilege to be a bookmarked site for future mathematical assistance.

What is the most confusing homework problem you've seen?

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