A square is a special case of a rectangle. What is a single-word term for a rectangle that is not a square? I am looking for a word that excludes squares. I am also looking for a word that is not "rectangle".
[Math] the name for a rectangle that is not a square
definitiongeometryterminology
Related Solutions
Among the rectangles with the same perimeter, the one with the maximal area is the square.
So you have to ask yourself how close to a square you can get with those segments.
First of all, the sides need to be pairwise equal. Note that the average segment is $9.75$.
So let's try adding numbers whose sum is close to $2\cdot 9.75$, say it is between $18$ and $20$.
$$16+3 = 6+13 = 19$$
$$15+5 = 11+9 = 20$$
is the only working combination ($15+3 = 13+5 = 18$ but $6,9,11,16$ do not add up to two equal numbers)
So that is your answer.
For definiteness, let's assume explicitly we're working in Euclidean plane geometry. Particularly, we have concepts of points and lines, subsidiary concepts of polygons, and numerical measures of length and angle.
The area of a rectangle (four-sided polygon bounded by two pairs of parallel segments meeting at right angles, including the boundary segments) is defined to be the product of the side lengths. Congruent rectangles (i.e., differing by a Euclidean motion) obviously have the same area.
Less obviously but pretty clearly, if a rectangle is subdivided into two rectangles sharing one side, the area of the large rectangle is the sum of the areas of the subrectangles. Inductively, if a rectangle is subdivided into finitely many non-overlapping subrectangles, the area of the large rectangle is the sum of the areas of the subrectangles.
It's reasonable to try extending this concept of area to non-rectangular subsets of the plane. Here's a naive approach: Let $P$ be a subset of the plane. To invent a name, let's say a "tile covering" of $P$ is a finite collection of rectangles such that
- The set $P$ is contained in the union of the rectangles;
- Distinct rectangles share no interior points. (Our rectangles must generally all be aligned on the same Cartesian axes.)
Each tile covering of $P$ has an area, the sum of the areas of the selected rectangles. The area of a tile covering is an upper bound for what we mean by the area of $P$. Now consider all tile coverings of $P$, or rather, consider the set of real numbers that are the area of some tile covering of $P$. The greatest lower bound of this set is a "candidate" for the area of $P$; we might call this greatest lower bound the outer area of $P$ and denote it $OA(P)$.
Dually, we might similarly look at the set of all "tile inclusions," finite collections of non-overlapping rectangles whose union is contained in $P$. We could define the area of a tile inclusion to be the sum of the areas of its rectangles. Finally, we could consider the least upper bound of the areas of all tile inclusions of $P$. This number $IA(P)$, the inner area of $P$, is another candidate for the area of $P$.
Finally, we might say that the set $P$ has area if $IA(P) = OA(P)$, and define the common value to be the area of $P$, acknowledging that many subsets of the plane do not have area in this sense. On the bright side, polygons do have area in this sense. Triangles, particularly, have area equal to one-half the base times the height (regardless which of three choices we make of base-and-height!). This should be believable, but it's probably also easy to see that calculating the area of a general region, such as a disk or other set whose boundary is not a finite collection of line segments, is generally a non-trivial undertaking.
Now to the question, What if we used sets other than rectangles as the basis of area? Any choice for an "area primitive" (such as squares, triangles, L-shaped hexagons, or what) assigns a numerical measure to some plane sets and not to others using the strategy outlined above. The question, as I see it, amounts to How do we know our "new" measure of area coincides with the rectangle measure of area? Unsurprisingly, the issue comes down to whether or not the "new" area of rectangles themselves coincides with length times width.
For squares as primitives (with the area of a square defined to be the square of the side length), it turns out rectangles have the same area as before. This is related to other answers that speak of counting how many squares of given size fit into a rectangle, or cover a rectangle.
For triangles as primitives (with the area defined to be one-half the base times height), the new area of rectangles again coincides: A rectangle can be diagonally subdivided into two congruent right triangles. For isosceles right triangles...you get the idea. Consequently, any polygonal primitive for which tile coverings exist leads to the same definition of area, since every polygon can be "tile covered" by triangles.
As others have commented and answered, area is not some Universal, True Concept, but a mathematical definition made in the context of a mathematical model. Even speaking mathematically, the preceding does not adequately define area of (e.g.) subsets of a sphere or other smooth surfaces in Euclidean three-space. In "reality" matters are even more complicated. Mandelbrot, at the very latest, articulated that length (e.g., the coast of Britain) is a scale-dependent quantity. Area has the same character when we consider real objects: tree leaves, lungs, the surface of the earth.
Searching for Jordan content, Lebesgue measure, and geometric measure theory will unearth as many additional details as desired.
Best Answer
It's called oblong. The following picture is from wikipedia.