# Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

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### Area of triangle in determinant form

Area of triangle with vertex $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by : $$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1 \end{vmatrix}$$ In this ...
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### The largest equilateral triangle circumscribing a given triangle

Seven years ago, one of my many contributions to the March 2010 edition of Erich Friedman's Math Magic was a packing of eight circles of unit diameter and one equilateral triangle of unit side length ...
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### Maximize area of a quadrilateral given three sides

What is the maximum possible area that a quadrilateral can have, if the lengths of three of its sides are given as 3, 4 and 5, while the fourth side can have arbitrary length? (Thinking of it as three ...
632 views

### How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
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### What is the 'optimal' equal-area partition of a circle?

What is the (an?) n-partition of a circle that meets the following criteria: The boundaries of each partition can be represented as a union of finitely many finite-piecewise-smooth simple closed ...
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### Area enclosed by robot walk

In a previous question I described $n$-robot walks and $(i,j)$-paths: A [$5$-]robot moves in a series of one-fifth circular arcs (72°), with a free choice of a clockwise or an anticlockwise ...
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### Area enclosed by the curve $\lfloor x + y\rfloor + \lfloor x - y\rfloor = 5$

What is the area enclosed by the curve $$\lfloor x + y\rfloor + \lfloor x - y\rfloor = 5$$ $$x\ge y, \forall x, y \ge 0$$ $\lfloor x\rfloor$ stands for the Greatest Integer Function. I think ...
### Area between curves $y=x^3$ and $y=x$
I've tried to done one of my homework problems for several times, but the answer doesn't make sense to me. The question asks to find the area between $y=x^3$ and $y=x$. Those are odd functions, and I'...