How many rectangles in a 7x7 grid

What is a rectangle?

......... A rectangle is - following the word - a square with right angles ....

......The square is also a rectangle.
It also has sides of the same length and is a special case of the rectangle.

Formulas Top
Basic formulas

......Sizes of the rectangle are
the pages a and b, the diagonal e, the radius of the perimeter R, the scope U and the area A..

In general, a rectangle is given by the sides a and b.
The other quantities can be calculated from this.

Distance from the diagonal
....A rectangle with sides a and b is given.

We are looking for the distance h of a corner point from the diagonal e.


solution
The auxiliary quantities p and q are introduced. According to the catheter principle, a² = pe and b² = qe or p = a² / e and q = b² / e.
According to the height theorem, h² = pq = (a² / e) (b² / e) = (a²b² / e²) = (a²b²) / (a² + b²) or h = (ab) / [sqrt (a² + b²)].
Result: The corner point has a distance of h = (ab) / [sqrt (a² + b²)].

Some special rectangles Top
Golden rectangle

......A rectangle is called a golden rectangle if the aspect ratio is Phi = (1/2) [sqrt (5) +1].

......This is noticeable:
If you draw a square in the rectangle, the rest of the rectangle is again a golden rectangle.
You can find more on my page Golden Ratio.

A4 paper size
......The rectangle made up of the side and the diagonal of a square has the shape of an A-row sheet.

......This is noticeable:
You can halve the rectangle so that the aspect ratio is retained.
You can find more on my A4 paper page.

Here is a compilation of some rectangles.
1) square, 2) 3-4-5 triangle and its reflection, 3) A-row paper format, 4) rectangle around two interlinked squares, 5) golden rectangle, 6) 30-60-90 triangle and its reflection or the equilateral triangle - reassembled, 7) double square

Squaring the rectangle Top
The point is to determine a square with the same area as a rectangle.

......The arithmetic problem is easy. From ab = x² follows x = sqrt (ab).

The classic task relates to the construction of the square.
The construction according to the secant-tangent theorem could look like this.
1 Wear the smaller side of the rectangle on top of the larger one.
2 Draw the vertical line and assume a point on it.
3 Draw a circle.
4 Connect one corner of the rectangle with the center of the circle.
5 Draw the semicircle of the valley.
6 Draw the tangent.
7 Build a square over the tangent section.
The construction according to the string set could look like this.
1 Extend the larger side of the rectangle by the smaller one.
2 Draw the vertical line and assume a point on it.
3 Draw the circle.
4 Connect one corner of the rectangle with the center of the circle.
5 Draw the vertical line through a corner of the rectangle.
6 Build the square over half the string.

The constructions for the cathet set and the height set can be found at Jürgen Ullwer (URL below)

......The "supplementary parallelogram" also contains a square and a rectangle of equal area. But the figure is not suitable for finding the square.

But you can get from a square to a rectangle with the same area.


Squares and rectangles Top
Two center squares

......If you connect the center points of the sides of a rectangle, you get a diamond.

......If you connect the center points of the sides of a diamond, a rectangle is created.

Rectangle within a square
......If you draw the diagonals in a square chord and the incircles in the partial triangles, the centers of these circles form a rectangle.

A representation and a proof can be found on the homepage of Antonio Gutierrez under the name Sangaku Problem (URL below).

Rectangle within rectangle Top
Consider a rectangle with side lengths a and b. A rectangle with a known side length k should be fitted into it as shown in the sketch. Find the second side length d.

......Similar right-angled triangles are created in the corners, the legs of which are x and y or b-x and a-y.
This is used to set up three formulas for the three search variables x, y and d.
Because of the similarity of the triangles, (#) x: d = (a-y): k and (##) y: d = (b-x): k.
According to the Pythagorean theorem, (###) k² = (b-x) ² + (a-y) ².
The calculation of the variable d is extensive and ultimately leads to an equation of the fourth degree:

(d²) ² - (2k² + a² + b²) d² + (4abk) d + [(k²) ²-a²k²-b²k²] = 0

For reasons of writing technique (i.e.2)2 = d4.


invoice
The variables x and y are first determined using the equations (#) and (##).
x: d = (a-y): k and y: d = (b-x): k
<=> kx = d (a-y) and yk = d (b-x)
<=> kx = ad-dy and yk = bd-dx
<=> dy = ad-kx and ky = bd-dx
<=> dky = adk-k²x and dky = bd²-d²x
<=> adk-k²x = bd²-d²x
<=> -k²x + d²x = bd²-adk
<=> x = (bd²-adk) / (d²-k²)
x: d = (a-y): k and y: d = (b-x): k
<=> kx = d (a-y) and yk = d (b-x)
<=> kx = ad-dy and yk = bd-dx
<=> kx = ad-dy and dx = bd-ky
<=> dkx = ad²-d²y and dkx = bdk-k²y
<=> ad²-d²y = bdk-k²y
<=> d²y-k²y = ad²-bdk
<=> y ​​= (ad²-bdk) / (d²-k²)

The equation (###) k² = (b-x) ² + (a-y) ² is converted to k² = b²-2bx + x² + a²-2ay + y².
For the sake of simplicity, one sets x² + y² = d² and instead of x and y the previously determined terms.
k² = b²-2b (bd²-adk) / (d²-k²) + a²-2a (ad²-bdk) / (d²-k²) + d²

Both sides of the equation are multiplied by the term d²-k².
k² (d²-k²) = b² (d²-k²) -2b²d² + 2abdk + a² (d²-k²) -2a²d² + 2abdk + d² (d²-k²)

It is multiplied out.
k²d²- (k²) ² = b²d²-b²k²-2b²d² + 2abdk + a²d²-a²k²-2a²d² + 2abdk + (d²) ²-k²d²

It is ordered and summarized.
(d²) ²-2k²d² + b²d²-2b²d² + a²d²-2a²d² + 4abdk-b²k²-a²k² + (k²) ² = 0
(d²) ²-2k²d²-b²d²-a²d² + 4abdk-b²k²-a²k² + (k²) ² = 0
(d²) ² - (- a²-b²-2k²) d² + 4abdk + [(k²) ²-a²k²-b²k²] = 0

This results in an equation of the fourth degree in d.
(d²) ² - (2k² + a² + b²) d² + (4abk) d + [(k²) ²-a²k²-b²k²] = 0.
It makes little sense and is also frightening to try to specify the variable d you are looking for with a term.
A numerical example can be used to check the usefulness of the equation.


......I found a solution to this rectangle problem on the Internet on the website of David Broughton (URL below) with the integer length specifications as in the drawing.
If you insert a = 118, b = 101 and k = 106 into the general equation of the fourth degree, the equation is called
(d²) ²-46,597 * d² + 5,053,232 * d-144,820,804 = 0.
I found the solutions with the help of Arndt Brünner's applet (URL below)
d1= -260.9, i.e.2= 53 (!), I.e.3 = 85.58 and d4 = 122,4.
The scale drawing holds the solution d2= 53 fixed. So that the drawing works, I calculated x = 45 and y = 28.

......There is a second solution (!), Namely d3 = 85.58. For them x = 84.54 and y = 13.29.

The solutions d1 and d4 are useless for the problem.


Axel Ridtahler encountered this problem when he fitted a rectangular board diagonally into a rectangular box.
It was then that he had the good idea ;-) to point out this problem to me.

Squares in the rectangle Top
Measure area

......If you want to introduce units of area in school, you start with the figure on the left, for example. The area of ​​the yellow square is 1 cm². How big is the area of ​​the rectangle?
The answer 12 cm² is obtained by counting the individual squares. You can also get the result using three bars with four squares each. This then leads to 3 * 4 cm² or the area formula A =.

Number of rectangles in the rectangle
......The question arises, how many rectangles there are using the grid in the 3x4 rectangle.
solution

Result: There are 60 sub-rectangles.


generalization
In an mxn rectangle there are (1/4) m (m + 1) n (n + 1) rectangles.
Source:
> Martin Wohlgemuth Count rectangles in a square grid (Url below)
> Eric W. Weisstein (MathWorld) Rectangle tiling (Url below)

Squares in the rectangle
The drawings above show that it is easy to assemble a rectangle from squares with integer sides.
......
But it is difficult to form a rectangle from different squares.

The smallest rectangle (32x33) for which this is possible is shown on the left.

This is probably the first publication:
R. Brooks, C. Smith, A. Stone, W. Tutte, The Dissection of Rectangles into Squares, Duke Mathematics journal, 1940, Vol. 7, pp. 312-340.


You can learn more about this problem from the links below.

Largest rectangles Top
Dimensions of a sports field

......What dimensions does a soccer field have to have in order to have a career?
of a = 400 m and is as large as possible.

solution
Objective function: A = 2xy
Secondary condition: a = 2y + 2pi * x or y = (1/2) a-pi * x
This implies A (x) = 2x [(1/2) a-pi * x] = ax-2pi * x² and further A '(x) = a-4pi * x.
An extreme value may exist if A '(x) = 0. That means a-4pi * x = 0 or x = (1/4) a / pi.
Since A '' (x) = - 4pi, i.e. smaller than 0, x = (1/4) a / pi is a maximum.
Then y is=(1/2) a-pi * x = (1/2) a-a / 4 = (1/4) a.
Result: The soccer field has the dimensions 2x = (1/2) a / pi and y = (1/4) a.
This leads to the numerical values ​​y = 100m and 2x = 63.7m.

To http://de.wikipedia.org/wiki/Fußballfeld#Spielfeld soccer fields measuring 68 by 105 meters are common.

Square as the largest rectangle
In many extreme value problems, squares are the largest rectangles.
You can find more on my squares page.

Division of a rectangle Top
Task: Divide a rectangle into four equal parts.
Solutions:


Family of rectangles Top
Rectangles of equal area

......The following applies to rectangles with the same area xy is constant or xy = A or y = A / x.

That's the equation of a hyperbola.


Rectangles of the same size
......The following applies to rectangles of the same size 2x + 2y is constant or 2x + 2y = U or y = -x + (1/2) U.

This is the equation of a falling line.


Similar rectangles
......The adjacent rectangles have the same shape.

They are similar.

......If you slide the rectangles down to the left into a perspective position, you can see after the first line of rays that similar rectangles have the same aspect ratio.

Cuboid Top

......
Six suitable rectangles form a body, the cuboid.

This cuboid has the shape of a brick.

The normal format of a brick in Germany is

24 cm x 11.5 cm x 7.1 cm.

......

There are separate websites about other cuboids, the cube and the square prism.

To define the rectangle Top
Above, the rectangle is defined as a quadrangle whose angles are right angles.
By looking, one can easily find the following simple sentences about the rectangle.
Sentence 1: All angles are the same.
Sentence 2: Two opposing sides are parallel.
Sentence 3: Every two opposing sides are of the same length.
Sentence 4: The diagonals are the same length.
Sentence 5: The diagonals are halved.
Theorem 6: There is an axis of symmetry through opposite side centers.


Definition 1
The definition of the rectangle "All angles are right angles" is determined by the name. Strictly speaking, this formulation is a sentence. Because it is enough to only ask for three right angles. From the sentence that says that the sum of the angles in the square is 360 °, the fourth angle is also 90 °. The definition must therefore read:
A square becomes a rectangle when three angles are 90 °.

Definition 2
Theorem 1 (All angles are equal.) Could also be used as a definition. Because again, the sum of the angles results in three or four angles of 90 °.

Definition 3
Sentences 4 and 5 are not sufficient conditions for a rectangle by themselves, but taken together they are.
......
A square becomes a rectangle when the diagonals are the same length and cut in half.

It is not mandatory which property of the rectangle is used to define it. If you stick to definition 1, the other two definitions must be explained and proven as propositions.

Let us take definition 3 for further considerations.
Definition 3 becomes sentence 7: In the rectangle the diagonals are of equal length and halve.

Theorem 7 contains two statements, the clause and the inverse clause:
(I) If three angles are right angles, the diagonals are of equal length and halve.
(II) If the diagonals are the same length and halve, then three angles are right angles.
(The evidence is missing here.)

The two directions are also expressed by the following formulations:
Theorem 7: The diagonals in a square are then and only of equal length and halve if the square is a rectangle.
Or
Theorem 7: The condition "The diagonals are the same length and halve" is necessary and sufficient for a rectangle.

The statements in sentence 2 to 6 are not sufficient, but necessary. They belong to the rectangle.


Is the square a rectangle? Top

......The adjacent square is a square.
If you call it a rectangle, the statement seems wrong.
But the statement is mathematically correct. All squares that have three right angles are called rectangles. And the square has this property. So it is "logical" that the square is a rectangle.

......In doing so, one accepts (see next chapter) that one is opposing a common notion. One expects a rectangle to have sides of different lengths. So I found the wording in a lexicon: In the "actual" rectangle, the sides are of different lengths. - In old textbooks one finds the name oblong for rectangles with sides of different lengths. Then the rectangle captures the square and oblong.

......Of course, one would expect the figure to be called a square rather than a rectangle. Obviously you always have to give a name that captures all the properties of the square.

A mistake made by the editors behind Günther Jauch's “Who Wants to be a Millionaire?” A few years ago fits in here.
The question was: Every rectangle is a ...? A: rhombus B: square C: trapezoid D: parallelogram
The answer was expected Parallelogram. That's right, but Trapezoid is also an answer. The rectangle, like the trapezoid, has a pair of parallel sides.

At de.wikpedia there is a possible explanation for the error: The source was out of date. Since I still own a Brockhaus from 1975, I can quote: (The trapezoid is a) "square with two parallel but unequal sides."

The square is a rectangle Top
It remains to be clarified why it makes sense to define the square as a special case of a rectangle.
Suppose you define the rectangle as follows.
Definition: A square becomes a rectangle when three angles are 90 ° and when the sides are of different lengths. Then you could just take over sentences 1 to 6, but not sentence 7.
Theorem 7 read: The diagonals in a square are then and only of equal length and halve if the square is a rectangle.
The sweeping sentence "If the diagonals are the same length and halve, the square is a rectangle" no longer applies. The square also has this property.
Sentence 7 becomes correct again if it is phrased in this way.
Theorem 7 ': The diagonals in a square are then and only of equal length and bisect each other and the sides are of different lengths if the square is a rectangle.
This example shows how awkward the sentences are. Basically you can of course ask for sides of different lengths for the rectangle. But you have to drag the additional property with you. It's much easier to include the square and think of it as a special case.


In this connection the problem of parallels is famous.
As is well known, two straight lines intersect at one point or, if they are parallel, at no point.
Now there is an approach with the statement: Two straight lines always intersect at one point. So that one can imagine that, one also assigns exactly one intersection point to the parallels, and specifically this is a point that lies "at infinity" and that is assigned to the direction of a straight line. With this approach one can use the so-called Projective geometry develop that elegantly delivers many sentences in connection with intersecting straight lines. Sentences about parallels appear as special cases.

Further pages Top
You can find more about rectangles in other places on my homepage.



Pentominos

Rectangle in the internet Top

German

Arndt Brünner
Determine zeros (solutions) of polynomials

Eckard Specht
MATH4U.DE W.1 Position of the centers of circles

Jürgen Ullwer (mathe-trainer.de)
Area transformation with the help of the height theorem, area transformation with the help of the cathetus set

student circle mathematics (problem of the month June 2004 with solution)
Two circles in a rectangle

Martin Wohlgemuth
Count rectangles in a square grid

Udo Hebisch (Mathematical Café)
rectangle

Wikipedia
Rectangle, squaring the rectangle, aspect ratio (movie)



English

Alexander Bogomolny (cut-the-knot)
Dissection of a Rectangle into Two Chessboards, Equidecomposition of a Triangle and a Rectangle II,
Rectangle in Arbelos, Two Rectangles in a Rectangle, Squaring a Rectangle, Circles and Semicircles in Rectangle,
Orthocentric System From Rectangle

Antonio Gutierrez
Sangaku Problem, The Golden Rectangle and Machu Picchu, The Golden Rectangle and Chichen Itza

David Broughton
RECTANGLE IN A RECTANGLE

Eric W. Weisstein (MathWorld)
Rectangle, Golden Rectangle, Blanches Dissection, Fault-Free Rectangle, Perfect Rectangle, Rectangle Tiling, Rectangle Squaring, Rounded Rectangle, Mrs Perkins's Quilt

Tweedledum (Bill's Graphics)
Rectangles divided into (mostly) unequal squares

Wikipedia
Rectangle, Golden rectangle, Root rectangle, Squaring the square, Oblong, Aspect ratio


credentials Top
Lothar Kusch, Mathematics for School and Work, Part 2, Essen 1971 [ISBN 3 7736 2582 0]


Feedback: Email address on my main page

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2008 Jürgen Köller

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