Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [quadrilateral]

For questions about general quadrilaterals (including parallelograms, trapezoids, rhombi) and their properties.

In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Squares, rectangles, rhombi, parallelograms and trapezoids are special kinds of quadrilaterals.

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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the…
Sid
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Area of a square inside a square created by connecting point-opposite midpoint

Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each. $H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively. What will the area of the square formed in the middle be? I know that this problem can be solved by trigonometry by using…
Agile_Eagle
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Area of parallelogram = Area of square. Shear transform

Below the parallelogram is obtained from square by stretching the top side while fixing the bottom. Since area of parallelogram is base times height, both square and parallelogram have the same area. This is true no matter how far I stretch the top…
AgentS
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Is every parallelogram a rectangle ??

Let's say we have a parallelogram $\text{ABCD}$. $\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \text{ADC}=ar\triangle \text{BCD}$$ Now the things…
Harsh Kumar
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A conjecture involving prime numbers and parallelograms

As already introduced in this post, given the series of prime numbers greater than $9$, let organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are displayed is the ten to which they belong, as…
user559615
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Is there a formula to calculate the area of a trapezoid knowing the length of all its sides?

If all sides: $a, b, c, d$ are known, is there a formula that can calculate the area of a trapezoid? I know this formula for calculating the area of a trapezoid from its two bases and its height: $$S=\frac {a+b}{2}×h$$ And I know a well-known…
Newuser
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Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides $a, b, c$ . The "square" which I will refer to, from now on, has all its four vertices on the sides of the triangle, and of course is completely…
Sawarnik
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Japanese theorem for cyclic quadrilaterals Proof Inversion

Two weeks ago our professor taught us the Japanese theorem for cyclic quadrilaterals. It states that the inscribed centres of the four triangles formed by two sides and a diagonal of a cyclic quadrilateral always form a rectangle. After the proof…
Tobi
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Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.
Amir Kazemi
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Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90, C=96, D=78$ and $BC=2*AB$, then the measure of the angle $ABD$ is?

The problem Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90°, C=96°, D=78°$ and $BC=2*AB$, then the measure of the angle $ABD$ is...? The idea As you can see I calculated above $B$ as $96$.Let point $X$ be the midpoint of…
IONELA BUCIU
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Angles of triangle $\triangle XYZ$ do not depend on the position of point $P$ (proof needed)

Let $ABCD$ be a fixed convex quadrilateral and $P$ be an arbitrary point. Let $S,T,U,V,K,L$ be the projections of $P$ on $AB,CD,AD,BC,AC,BD$ respectively. Let $X,Y,Z$ be the midpoints of $ST,UV,KL$. Is it true that the angles of triangle $\triangle…
user613853
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How to calculate the area of a quadrilateral given the (x,y) coordinates of its vertices

This problem is from the 2017 Gauss Contest (Grade 7). Four vertices of a quadrilateral are located at (7,6), (−5,1), (−2,−3), and (10,2). What is the area of the quadrilateral in square units? I solved this by graphing the quadrilateral, then…
Will.Octagon.Gibson
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Can a quadrilateral polygon have 3 obtuse angles?

I was messing around with quadrilaterals trying to draw one that has three obtuse angles. I couldn't create one because with 3 obtuse angles the shape would "open up too much". I have finished high school math with a perfect score but hadn't messed…
Dor Goldreer
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Point that divides a quadrilateral into four quadrilaterals of equal area

Consider an irregular quadrilateral $ABCD$. Let $E,F,G,H$ be the midpoints of its edges. It seems that there is a point $K$ such that $$ S_{AHKE} = S_{EKFB} = S_{KHDG} = S_{KGCF} \left(= \frac{1}{4} S_{ABCD}\right) $$ I'm curious whether the point…
uranix
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Volume of an Irregular Octahedron from edge lengths?

Does anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. Alternatively, how might I calculate the length of a line…
Hoffi.D
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