# Questions tagged [geometry]

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

3,120 questions
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### The staircase paradox, or why $\pi\ne4$

What is wrong with this? Is $\pi=4?$
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### Finding an angle within an 80-80-20 isosceles triangle

The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it. So ...
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### 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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### 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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### Different definitions of trigonometric functions

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two ...
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### 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 ...
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### Proof that Pi is constant (the same for all circles), without using limits

Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
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### If point is zero-dimensional, how can it form a finite one dimensional line?

I have extracted the below passage from the wikipedia webpage - Point (geometry): In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. ...
881 views

### Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
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### Compute center, axes and rotation from equation of ellipse

Suppose I have the equation of an ellipse, in its implicit form $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$$ For example the following: $$4.36\,x^2 + 2.89\,y^2 - 5.04\,xy + 30.8\,x - 0.6\,y + 81 = 0$$ ...
### Integral solutions of hyperboloid $x^2+y^2-z^2=1$
Are there integral solutions to the equation $x^2+y^2-z^2=1$?