Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [rectangles]

Questions about rectangles and their properties.

In Euclidean plane geometry, a rectangle is a quadrilateral with $4$ right angles. A square is a rectangle. Non-square rectangles are called oblongs.

The perimeter of a rectangle with width $w$ and height $h$ is $2(w+h)$, and the area is $wh$.

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A goat tied to a corner of a rectangle

A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze? Well, it seems like the goat can turn a full circle of radius…
space
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Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?

I need help to answer the following question: Is it possible to place 26 points inside a rectangle that is $20\, cm$ by $15\,cm$ so that the distance between every pair of points is greater than $5\, cm$? I haven't learned any mathematical…
Aryan Patel
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How many "prime" rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, the following two tilings are equivalent (some…
RavenclawPrefect
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Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation for the area to be $A=4xy$ but then when trying to…
Gabby
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You have $n$ rectangles of area $1$ (and variable height). Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ (and variable height). Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to pack the rectangles I believe that the most…
Dan
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Largest rectangle not touching any rock in a square field

You want to build a rectangular house with a maximal area. You are offered a square field of area 1, on which you plan to build the house. The problem is, there are $n$ rocks scattered in unknown locations throughout the field. The rocks are…
Erel Segal-Halevi
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How many unit squares can overlap a given unit square without overlapping each other?

How many unit squares can overlap a given unit square without overlapping each other? @calculus has managed to arrange 7 squares (see this GeogebraTube page). This seems like the maximum possible, but how to prove it formally?
Erel Segal-Halevi
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Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under what conditions will the $M_1M_2M_3M_4$ rectangle…
user153012
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Radius of a circle touching a rectangle both of which are inside a square

Given this configuration : We're given that the rectangle is of the dimensions 20 cm by 10 cm, and we have to find the radius of the circle. If we somehow know the distance between the circle and the corner of the square then we can easily find the…
Mooncrater
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cutting a cake without destroying the toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake. I want to divide the cake into 2 non-empty rectangular…
Erel Segal-Halevi
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A goat is tied to the corner of a shed

A goat is tied to the corner of a shed 12 feet long and 10 feet wide. If the rope is 15 feet long, over how many square feet can the goat graze ? I know that this question has already been asked a number of time, but no matter what I do I cannot…
Emperor_Udan
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Bath towel on the rope: minimize the area of self-intersection of a folded rectangle

This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o). There is this rectangle with sides $a
Mikhail G
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Is this a new point on the nine-point-circle of a triangle?

I was trying to get a feel for how to solve another question about the largest triangle that can fit in a unit square, by constructing the smallest enclosing square of a triangle in Geogebra. While doing this I 'discovered' what appears to be a new…
KDP
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Check if a point is inside a rectangular shaped area (3D)?

I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example this on but none for 3D space. I have a cuboid…
Faas
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Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each $R_i$ has at least one integer side length, then…
user2566092
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