Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

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842 votes
23 answers
102k views

The staircase paradox, or why $\pi\ne4$

What is wrong with this proof? Is $\pi=4?$
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465 votes
10 answers
506k views

Is this Batman equation for real? [closed]

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-...
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  • 5,499
293 votes
9 answers
45k views

V.I. Arnold says Russian students can't solve this problem, but American students can -- why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it is 6 ...
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  • 7,871
257 votes
4 answers
17k views

What is the maximum volume that can be contained by a sheet of paper?

I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty nontrivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet of ...
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248 votes
26 answers
61k views

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 ...
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  • 4,211
223 votes
13 answers
56k views

How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on ...
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210 votes
4 answers
77k views

Why can a Venn diagram for $4+$ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $...
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208 votes
15 answers
16k views

Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to ...
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191 votes
15 answers
32k views

What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 \text{ ...
189 votes
10 answers
472k views

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
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  • 11.2k
166 votes
26 answers
184k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
162 votes
6 answers
8k views

Why is the Penrose triangle "impossible"?

I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. ...
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158 votes
20 answers
20k views

How to distinguish between walking on a sphere and walking on a torus?

Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view. If ...
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154 votes
16 answers
22k views

What's new in higher dimensions?

This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in ...
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152 votes
3 answers
18k views

Slice of pizza with no crust

The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza has ...
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146 votes
5 answers
19k views

Can someone explain the math behind tessellation?

Tessellation is fascinating to me, and I've always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would've gone about calculating tessellating shapes. In my ...
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  • 1,747
143 votes
6 answers
3k views

Studying Euclidean geometry using hyperbolic criteria

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
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  • 4,753
136 votes
10 answers
21k views

Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red. ...
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  • 9,125
128 votes
4 answers
21k views

What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw ...
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  • 1,887
125 votes
10 answers
40k views

How can I find the surface area of a normal chicken egg?

This morning, I had eggs for breakfast, and I was looking at the pieces of broken shells and thought "What is the surface area of this egg?" The problem is that I have no real idea about how to find ...
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  • 7,126
118 votes
9 answers
6k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here. Randomly break a stick in five places. ...
115 votes
8 answers
74k views

Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
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  • 8,461
109 votes
18 answers
11k views

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and ...
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109 votes
5 answers
7k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
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  • 19.4k
108 votes
18 answers
83k views

Why is the volume of a sphere $\frac{4}{3}\pi r^3$?

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! ...
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  • 8,913
107 votes
11 answers
212k views

What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...
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  • 6,463
106 votes
11 answers
181k views

Why is the volume of a cone one third of the volume of a cylinder?

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by ...
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  • 8,461
104 votes
8 answers
7k views

Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the ...
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  • 1,562
98 votes
7 answers
10k views

Mathematicians' Tensors vs. Physicists' Tensors

It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector ...
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  • 2,564
94 votes
17 answers
62k views

What is the most elegant proof of the Pythagorean theorem? [closed]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
93 votes
6 answers
7k views

Can you make a sphere out of a plane?

I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could ...
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  • 1,368
91 votes
2 answers
4k views

Modelling the "Moving Sofa"

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
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  • 4,685
90 votes
2 answers
43k views

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 ...
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  • 4,433
87 votes
8 answers
17k views

Is there a size of rectangle that retains its ratio when it's folded in half?

A hypothetical (and maybe practical) question has been nagging at me. If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches ...
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  • 973
87 votes
8 answers
17k views

Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?

Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there. Circular manhole:...
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87 votes
4 answers
3k views

Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I've now also asked this question on mathoverflow. Notes: Obviously I'm ignoring the triangle of side $0$. ...
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86 votes
5 answers
6k views

Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

Having a bit of a problem calculating the volume of a take-away box: I originally wanted to use integration to measure it by rotating around the x-axiz, but realised that when folded the top becomes ...
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  • 847
85 votes
5 answers
22k views

Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
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  • 2,260
82 votes
6 answers
29k views

Why is a full turn of the circle 360°? Why not any other number?

I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number? What is the significance of 90° for the perpendicular or 360° for a circle? I didn't ever ...
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  • 1,091
82 votes
3 answers
16k views

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
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  • 3,675
81 votes
6 answers
7k views

20 circles in the plane, all passing through the origin

Suppose I draw $20$ circles in the plane, all passing through the origin, but no two tangent at the origin. Also, except for the origin, no three circles pass through a common point. How many regions ...
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  • 1,140
80 votes
10 answers
25k views

What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not ...
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  • 6,539
80 votes
9 answers
63k views

What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the ...
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  • 1,073
80 votes
4 answers
6k views

Volumes of n-balls: what is so special about n=5?

The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$ For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ $$...
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80 votes
3 answers
3k views

Escaping infinitely many pursuers

The fugitive is at the origin. They move at a speed of 1. There's a guard at every integer coordinate except the origin. A guard's speed is 1/100. The fugitive and the guards move simultaneously and ...
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  • 1,659
79 votes
10 answers
29k views

Is there a shape with infinite area but finite perimeter?

Is this really possible? Is there any other example of this other than the Koch Snowflake? If so can you prove that example to be true?
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79 votes
6 answers
177k views

Finding a point along a line a certain distance away from another point!

Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$. The gradient of the line between them is: $$m = (y_1 - y_0)/(x_1 - x_0)$$ And therefore the equation of the line between them is: $$y =...
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  • 959
78 votes
6 answers
160k views

What is the equation for a 3D line?

Just like we have the equation $y=mx+b$ for $\mathbb{R}^{2}$, what would be a equation for $\mathbb{R}^{3}$? Thanks.
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  • 22.4k
77 votes
1 answer
5k views

Gerrymandering on a high-genus surface/can I use my powers for evil?

Somewhat in contrast to this question. Let's say the Supreme Court has just issued a ruling that the upper and lower roads of an overpass need not be in the same congressional district. This makes ...
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  • 58.3k
75 votes
9 answers
282k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
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