What is Rectangle: Definition and 244 Discussions

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

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  1. M

    Maximizing a rectangle in R3

    Homework Statement I've been struggling to figure out how to do the following problem which I came across: Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x+2y+3z=6. Show ALL of your workings.Homework...
  2. G

    Proving a rectangle is connected.

    Homework Statement Let K ={(x,y)\inℝ2:|x|≤1,|y|≤1} Prove that K is a connected subset of ℝ2 The Attempt at a Solution Suppose f:[-2,2]→K and define f(x)={(x,y):|y|≤1} Dist(f(x),f(y))=sup(d(a,b):a\inf(x),b\inf(y))=d(x,y)=|x-y|. Using this equality it is easily shown that f(x) is...
  3. Sudharaka

    MHB Justin's Question about Rectangle in Facebook

    Justin on Facebook writes:
  4. C

    Fitting the 'tightest' rectangle to a set of three dimensional points

    I have a group of 8 3-D points which all sit on an arbitrary plane (The points were generated by projecting the corners of a axis aligned cube, box, onto the plane) I wish to group all the points in a rectangle with the smallest surface area - so a tight fit rectangle. What I am doing at...
  5. Petrus

    MHB The double integral of f over rectangle R and midpoint rule for double integrals

    Hello MHB, I wanted to 'challange' myself with solve a problem with midpoint and rule and the double integral f over the rectangle R. This is a problem from midpoint. "Use the Midpoint Rule m=n=2 to estimate the value of the integrab \int\int_r(x-3y^2)dA, where R= {(x,y)| 0\leq x \leq 2, 1 \leq...
  6. B

    MHB The equation of a hyperbolic paraboloid to derive the corner points of rectangle

    Hi Folks,I have come across some text where f(x,y)=c_1+c_2x+c_3y+c_4xy is used to define the corner pointsf_1=f(0,0)=c_1 f_2=f(a,0)=c_1+c_2a f_3=f(a,b)=c_1+c_2a+c_3b+c_4ab f_4=f(0,b)=c_1+c_3bHow are these equations determined? F_1 to F_4 starts at bottom left hand corner and rotates counter...
  7. B

    Polynomial to represent a linear rectangle element

    Folks, I have attached a picture illustrating the labelling of the linear reactangle element which can be represented by the following equation ##u(x,y)=c_1+c_2 x +c_3 y +c_4 xy## (1) ##u_1=u(0,0)=c_1## ##u_2=u(a,0)=c_1+c_2a## ##u_3=u(a,b)=c_1+c_2a+c_3b+c_4ab##...
  8. L

    Convert a region into a rectangle

    Homework Statement Let R be the region bounded by x^3/2+y^3/2=a^3/2 (x>0, y>0) and the coordinate axes x=0, y=0. Express it in double integral over a rectangle. Homework Equations The Attempt at a Solution How to solve this people please? I tried a couples time but failed to find v u which...
  9. K

    Parametrizing a Self-Intersecting Rectangle

    Homework Statement Let S be the self-intersecting rectangle in ##\mathbb{R}^3## given by the implicit equation ##x^2−y^2z = 0##. Find a parametrization for S.Homework Equations The Attempt at a Solution This is my first encounter with a surface like this. The first thing that came to my mind...
  10. L

    Current in a rectangle on a hinge

    Homework Statement Figure 29-36 shows a rectangular, 15-turn coil of wire, 10 cm by 5.0 cm. It carries a current of 0.90 A and is hinged along one long side. It is mounted in the xy plane, at an angle of 30° to the direction of a uniform magnetic field of 0.50 T. Find the magnitude and...
  11. N

    Differential Equations: Largest rectangle on ty-plane

    Homework Statement Find the largest open rectangle in the ty-plane that contains the initial value point and satisfies the following theorem: Let R be the open rectangle defined by a<t<b, α<y<β. Let f(t,y) be a function of two variables defined on R where f(t,y) and the partial derivative...
  12. N

    Rectangle and set of points

    I found one interesting example... We have a rectangle ABCD with his perimeter o. Where is the set of points when their (for each point) sum of distance lines AB, BC, CD, DA is 2/3o ? I tried it, but geometric problems is quite hard for me and I don't know how do it. So, have you got any...
  13. H

    What forces act on the supports of a rectangle?

    To anyone who saw my previous thread, yes, this is quite similar to it :tongue2:. Homework Statement Given a rectangle, say a painting, with with mass m, height h, and width w with two point supports to a wall at its two upper corners, what force does each support exert? Homework...
  14. R

    Finding dimensions of a rectangle.

    Homework Statement the perimeter of a rectangle is 24 ft. the length is 4 ft longer than the width find the dimensions width x length x+4 however, I should be doing it like this: a first equation should start like: 2x+2y=? and the second should start like x=y+? so what's the...
  15. T

    Second Moment of Inertia of a Rectangle

    Homework Statement Find the second moment of inertia of a rectangle. The Attempt at a Solution Anyone know why my answer is 4 times too big?
  16. P

    Finding the Side-lengths of a Rectangle with Given Area Increase

    Homework Statement A rectangle is 2 metres longer than it is wide. On the other hand, if each side of the rectangle is increased by 2 metres, then the area increases by 24 square metres. Find the side-lengths of the rectangle. Homework Equations The Attempt at a Solution So my...
  17. S

    Laplace's equation on a rectangle with mixed b.c.s

    Homework Statement Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x. Homework Equations The Attempt at a Solution I would be able to solve it by separation of variables if the last...
  18. S

    How many ways to cover this rectangle

    Homework Statement In how many different non-overlapping ways can a (2 x 10) rectangle be covered by (1 x 1) and (1 x 3) rectangles. Homework Equations The Attempt at a Solution I've never done any question like this. The only solution I can think of is a brute force method but I doubt...
  19. N

    Square or Rectangle 30 psi storage tank

    If I want to make a Square or Rectangle storage tank 5 feet deep and 22 feet long and 1 foot wide. The tank will be under 30 psi and also vacuum. That’s not hard to do what I want is the walls not to deflect more then .001 of an inch. The walls can be plastic, plywood with a steel sheet or...
  20. S

    Is the white region a square or rectangle?

    Is the white region a square or rectangle? See my post below for image
  21. Dembadon

    Rectangle Method for approximating an integral has been rediscovered

    "Rectangle Method" for approximating an integral has been rediscovered! http://care.diabetesjournals.org/content/17/2/152.abstract Is this some kind of joke? Has anyone else seen this article before? This is what I felt like after reading it:
  22. Feodalherren

    Maximize area of rectangle inside a circle

    Homework Statement Show that the maximum possible area for a rectangle inscribed in a circle is 2r^2 where r is the radius of the circle.Homework Equations Pre-calc ! NO TRIG ! Doesn't matter if it's easier, it's supposed to be solved with algebra. The Attempt at a Solution I have no clue...
  23. A

    Formula for lengths of a rectangle - why does it give both lengths?

    Why does the formula [P±√(P^2-16A)]/4 give the values of either of two different lengths of a rectangle? (P is perimeter and A is area) I derived it by solving two simultaneous equations, A = xy and P=x+y and then applying the quadratic formula to the resulting second-order equation 2y^2 +...
  24. G

    Probability that a Rectangle lies within a circle

    This is not really a homework problem, I'm just doing it as an exercise puzzle. I think I'm on the right track, but at this point I feel a little exhausted and would love a hint. Homework Statement Let C be a unit circle: x^2+y^2=1 . Let "p" be a point on the circumference and "q" be a point...
  25. S

    What is the Load rating of Aluminum rectangle tubing?

    Greetings, I will describe the problem and my proposed solution which leads me to the load bearing question.. Problem: Pickup camper cabover floor is compromised where it attaches to the driver side sidewall, i.e., separation of floor from sidewall due to some dryrot and a blowout doing...
  26. M

    Rectangle volume using cross sections

    Homework Statement Cross sections are perpendicular to the x-axis and rectangle has the h=1/2b. The region is bounded by the area y=x^2, x-axis and the line x=3 Homework Equations The Attempt at a Solution A=BH A=B(1/2B) A=3/2B V=3/2(x^2-3) just wondering if this is correct...
  27. T

    Limiting the placement of covering rectangle with smaller rectangles algorithm?

    I'm looking for a 'covering rectangle with smaller rectangles' algorithm with the unique feature of being able to exclude some possible center points of rectangles. Basically, limiting the possible areas the smaller rectangles can be placed, while still having the algorithm try to solve for...
  28. C

    Integral of xcos(xy) over a rectangle

    Hi there. I'm having some trouble with some double integrals here. All of them are to be evaluated on the rectangle_{} 1≤x≤2, 0≤y≤1, and the functions are: 1: \int\int_{A}\frac{1}{x+y}dxdy. On this one I made α(y)=\int^{2}_{1}\frac{1}{x+y}dx=ln(\frac{2+y}{1+y}), and finally I should evaluate...
  29. A

    Calculate the work done in Rectangle?

    Homework Statement A force acting on a particle moving in XY plane is given by F= 2y i + x^2 j . particle moves from origin to a final position having coordinates (x,y)=(5,5). calculate the work done by force F along B_____________C | | |...
  30. P

    Mathematica Mathematica: Inertia Tensor w/ 3-d Rectangle

    Hey All, I'm trying to create a 3-D rectangle in Mathematica with the following measurements: Mass M=1.5 kg, and sides of length a=10 cm (parallel to the x-axis), 2a (parallel to the y-axis), and 3a (parallel to the z-axis). Let one corner be at the origin, and let the three adjacent edges...
  31. D

    3D Transformation of Rectangle to a Plane

    Let's say you have four points that define a rectangle in the xy plane centered at the origin (with the x,y axes bisecting the sides). How can you transform these points so that the rectangle lies in an arbitrary plane (defined by a point p and a normal vector n) so it is centered about point p...
  32. L

    Setting area = perimeter of rectangle, does it have any meaning?

    Suppose I have a rectangle with length x and height y:Can this ever be true? That xy = 2x+2y? My guess would be that area of a shape can never equal the perimeter of that shape. And that would be confirmed by the fact that the equation is always false (except if both variables equal zero...
  33. D

    MHB What is the Integral of a Function Around a Rectangle Oriented Clockwise?

    Find the integral $$ \int_C\frac{dz}{z^2 - 3z + 5} = \int_C\frac{dz}{\left(z - \frac{3}{2}-i\frac{\sqrt{11}}{2}\right)\left(z-\frac{3}{2}+i\frac{\sqrt{11}}{2}\right)} $$ Where the path is a rectangle oriented clockwise from (0,0) to (0,4) to (10,4) to (10,0) to (0,0). So $z_1 =...
  34. D

    MHB Drawing a Clockwise Rectangle on the Complex Plane with Tikz

    How can I draw a rectangle oriented clockwise on the complex plane with vertices on (0,0), (0,4), (10,4), and (10,0)? I am guessing the tikz package needs to be used but I am not skilled in making pictures.
  35. K

    Max distance of rectangle that electron beam can pass through

    Homework Statement A beam of electrons is fired into a rectangular region of space that contains a uniform magnetic field in the -z direction. The electrons are moving in the +x direction, as shown. The speed of the electrons in the beam is 6.00 × 106 m/s. The mass of an electron is me =...
  36. T

    I Continious FT of a rectangle waveform is real valued, but the DFT of it is not?

    Continious FT of a rectangle is real valued but DFT of it is not!? Continious Fourier Transform of a rectangle with amplitude of 1 between [-u,u] is a real valued function (u is a positive number). Actually it is a sinc function. However when I use discrete Fourier Transform (fft) I obtain...
  37. R

    The length of a line intersecting a rectangle - not just diagonal

    Hi Given a simple linear equation and the extents of a rectangle, is there a neat way of finding the length of the line section that is contained within the rectangle? The only general method i can think of would involve a bunch of conditional statements to determine whether the line...
  38. C

    Showing Complete Elliptic Integral of First Kind Maps to Rectangle

    Homework Statement Effectively, I'm trying to show the following two integrals are equivalent: \int_1^{1/k}[(x^2-1)(1-k^2x^2)]^{-1/2} dx = \int_0^1[(1-x^2)(1-(k')^2x^2)]^{-1/2}dx where k'^2 = 1-k^2 and 0 < k,k' < 1. Homework Equations One aspect of the problem I showed the following...
  39. L

    Distance to the corner of a rectangle

    Homework Statement This question is taken from 2011 Malaysian Mathematical Olympiad. Mary is standing in a rectangular garden. Her distance to the four corners of the garden are 6 m, 7 m, 9 m and d m, respectively, where d is an integer. Find d. Homework Equations Triangle...
  40. T

    Evaluating A Double Integral over a Rectangle

    Homework Statement Let R be the rectangle bounded by x - y = 0, x - y = 2, x + y = 0, and x + y = 3. Evaluate \int\int(x + y)ex2-y2dA R The Attempt at a Solution First I rewrote the boundaries so that I could graph them more easily. I got y = x, y = x - 2, y= -x, and y = -x + 3. I was going...
  41. L

    Volume of a rectangle through a sphere

    Homework Statement Homework Statement Suppose that a square hole with sides of length 2 is cut symmetrically through the center of a sphere of radius 2. Show that the volume removed is given by where I'm not sure how to approach this, but I figure you can express the sphere in...
  42. T

    Extreme Value Theorem: Maximum Area of Rectangle

    Homework Statement This problem had been previously posted, however i have specific questions about it and don't feel that it was completely answered for it did not explain how you use the extreme value theorem in the problem and specifically, HOW DO YOU GET THE INTERVAL [a,b], The answer has...
  43. G

    Moment of Inertia of a rectangle?

    Homework Statement I was trying to find the moment of inertia of a rectangle with width a and height b were axis of rotation is through it's center of mass Homework Equations The Attempt at a Solution I = integral r^2 dm rho = dm/dA dm = rho dA If I take an infinitely small...
  44. O

    Area between Triangle and Rectangle

    Homework Statement Consider a large triangle, the tip is located at the origo x=0, it is sloped at an angle θ and -θ relative to the x-axis, relative to the x-axis, its dimensions in x can be considered infinite. A large stripe/strip/band is placed on top of the triangle but perpendicular to...
  45. V

    Calculating Electric Flux Through a Rectangle

    Homework Statement Determine the electric flux through a 6m by 4m rectangular area. The area is oriented such that a vector normal to its surface points in the [1;6] direction. The electric field is [7;1] N/C. Homework Equations Φ=E·A=EA*cosθ The Attempt at a Solution...
  46. S

    Fastest way to determine if a circle fully covers a rectangle

    As usual I'm working on a program and I'm having trouble with math/efficiency. Homework Statement I need a way to find out if a circle given as a point and a radius C(x,y,r) fully encloses a rectangle given by the top left corner and the width and height R(x,y,w,h) I only need to know...
  47. T

    2D Circle and rectangle intersection tests

    What I'm looking for is an algorithm to find the details on the intersection of of a circle and rectangle in two dimensional Euclidean space. The information I need to find is straightforward enough; all I need is to know whether the rectangle and circle are not intersecting, partially...
  48. Government$

    Find Rectangle Sides Given Perimeter & Area

    Homework Statement Find the lengths of the sides of rectangle if perimeter is 7,4 m and area is 3m2. Homework Equations S= 2a+2b and P=ab The Attempt at a Solution So i tried getting a form P --> a=P/b and then subsstituting in perimeter formula a with P/b ---> S=2P/b+2b but i am...
  49. A

    A Rectangle Parallel to a Magnetic Field

    Homework Statement file:///Users/LabGuest/Desktop/CH26.tiff Homework Equations FB= qv cross B The Attempt at a Solution If the the rectangle is parallel to the magnetic field and we know that FB is perpendicular to the B field itself than there should be a net Torque but no net...
  50. X

    Mapping points inside one 2D rectangle into another smaller one

    in some work I'm doing i have a 2D rectangle that can be rotated and/or translated in any direction in 2D space. for example it might look like this: (x=30,y=-10) +-----------+ (x=30,y=2) +-- +y | | |...
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