1. A Class Where Every Student Said a Square Is Not a Rectangle
In one reported lesson, when a teacher asked, "Is a square a rectangle?", every student answered "No, it is not a rectangle" [6]. The reason given for this answer is the visual image of a "long rectangle." This response is less of a mistake and more of a natural feeling. Because the word for rectangle in Japanese contains the character for "long," it feels natural to give a different name to a shape where the vertical and horizontal lengths are the same.
So, how does mathematics define a rectangle? It is likely defined by conditions rather than visual images. The next section looks at those conditions.
2. Why the Definition of a Rectangle Does Not Mention Side Lengths
The key factor is that "all four angles are right angles." A second-grade arithmetic practice problem explains that a quadrilateral with four right angles is called a rectangle [1]. A square is taught as a quadrilateral with four right angles and all sides of equal length [1]. In the Course of Study, squares, rectangles, and right triangles are included in the geometry content for the second grade [3].
It is important to note that the condition for a rectangle does not include "opposite sides are equal." The fact that opposite sides become equal is a result of the angles being right angles; it is not a condition itself. If a rectangle is defined only by its angles, the difference between it and a square is just one thing. A square is a shape that adds "all side lengths are equal" to the rectangle's conditions. Because one condition is added, a square can be said to be a more special shape.
3. Why a Square Is Both a Rectangle and Counted Separately in School Problems
A shape that adds conditions does not lose the original conditions. A square has four right angles and all sides equal. It satisfies all the conditions for being a rectangle. Therefore, Britannica describes a square as a "rectangle with equal sides," meaning a special type of rectangle [4]. A square also fits the dictionary definition of a rectangle (two pairs of parallel sides and four right angles) [5].
On the other hand, some second-grade practice problems include instructions that say, "Do not include the number of squares in the number of rectangles" when counting rectangles in a pattern [2]. This is an example of an instruction in this specific teaching material; it does not mean all textbooks or problems do this. There is a paper explaining that in the second year of junior high school, students learn the inclusion relationship, where parallelograms are the general shape and rectangles, rhombuses, and squares are special cases of them [7]. It seems the flow is to first develop the eye to distinguish shapes in elementary school, and then consider the relationships between shapes in junior high school.
4. What Is Left When You Cut a Square Off a Rectangle?
There is a rectangle where the remainder is the same shape as the original. If you cut a square with a side equal to the shorter side from a rectangle, the remainder is a thin, long rectangle. In an ordinary rectangle, the remainder is a different shape from the original. However, in a rectangle with a height of 1 and a width of about 1.618 (the golden rectangle), the remainder is a rectangle similar to the original [9]. If you cut a square from the remaining rectangle, another similar shape remains. This simple game of cutting squares from rectangles connects two shapes.
What happens if you try this with familiar A4 paper (210 × 297 mm)? If you cut a square with a side of 210 mm (the same as the short side), the remainder is 210 × 87 mm. Dividing the long side by the short side gives 2.41, which is a different shape from the original ratio of about 1.414. This means A4 is not a golden rectangle. So, what is special about A4 paper?
5. Why Folding A4 Paper in Half Gives the Same Shape
A4 paper is 210 × 297 mm, and the ratio of height to width is about 1:1.414 (1:√2). For a rectangle with this ratio, if you fold the long side in half to make two parts, the resulting rectangle is also the same shape (similar) as the original. This ratio is called the silver ratio [8].
Let's check this. 297 divided by 210 is about 1.414. When folded in half, the size is 148.5 × 210 mm, and 210 divided by 148.5 is also about 1.414. The ratio does not change after folding. Folding A4 in half makes A5, and folding A5 in half makes A6; all have the same shape.
The silver ratio (about 1.414) and the golden ratio (about 1.618) are different numbers [8]. The golden rectangle has the property "similar when a square is cut off," while the silver rectangle has the property "similar when folded in half." Paper sizes in Japan's A series are the same dimensions as the international standard ISO 216. The B series dimensions differ between Japan's JIS and ISO standards; for example, B5 is 176 × 250 mm in ISO, but 182 × 257 mm in JIS [10].
6. How to Check for the Same Shape with Paper and a Ruler
You can make a square from copy paper without scissors. Fold the top corner of the paper into a triangle so that the horizontal side aligns perfectly with the vertical side. The area above the point where the tip of the triangle reaches the vertical side is a square, and the part remaining below is a rectangle. You can verify with your hands what remains when you take a square from a rectangle.
To check if the shape is the same when folded in half, fold an A4 sheet in half, orient it vertically, and place it on the bottom-left corner of the original paper. If you connect the bottom-left and top-right corners of the original paper with a ruler, the top-right corner of the smaller paper should fall almost exactly on that line. In the case of A4 paper cut in half, the diagonal lines align almost perfectly. If you have a B5 notebook, you can try the same overlay method. See if a B5 sheet folded in half also results in roughly the same shape.