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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1answer
596 views

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 ...
7
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1answer
143 views

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 ...
7
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2answers
102 views

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 ...
7
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1answer
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 ...
7
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1answer
281 views

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 ...
7
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2answers
92 views

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 ...
7
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1answer
121 views

Conjecture: Every convex set of area $1$ is contained in a triangle of area $2$.

Consider the following: Conjecture. Every convex set of area $1$ is contained in a triangle of area $A = 2$. I can prove it if $A$ is changed to $4$. The convex set being a square shows that $A = ...
7
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2answers
92 views

Problem involving the square root of a trigonometric term

I was trying to find the shaded area in this figure: And no, it isn't homework. I just chanced upon it on Facebook and had a go at it. I managed to find it using a very simple method. I now want to ...
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6answers
12k views

How do you find the area of a parallelogram with the vertices? [closed]

How do you find the area of a parallelogram with the following vertices; $A(4,2)$, $B(8,4)$, $C(9,6)$ and $D(13,8)$.
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3answers
801 views

The area of circle

The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$ but I don't know what to do next.
6
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3answers
717 views

The area of a right spherical triangle

Is there a compact formula for the area (excess angle – assuming a unit sphere) of a right spherical triangle given its side lengths $a$ and $b$? As explained in an answer to an earlier question ...
6
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4answers
560 views

Heron's formula when side lengths include radicals

I am helping a 7th grade student prepare for a math contest. I have a copy of a previous year's test, and one of the questions has me perplexed! I think perhaps I'm making this problem harder than it ...
6
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2answers
608 views

Finding the area remaining after flipping a rectangle inside a rectangle

Let $r$ be the inside rectangle of base $b$ and height $h$. Let $R$ be the outside rectangle of base $B$ and height $H$ The dimensions of $r$ and $R$ are related in the following way: I want to ...
6
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4answers
444 views

How find this maximum $S_{\Delta ABC}$

in $\Delta ABC$,and $\angle ABC=60$,such that $PA=10,PB=6,PC=7$, find the maximum $S_{\Delta ABC}$. My try:let $AB=c,BC=a,AC=b$, then $$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$ then $$S_{ABC}=\...
6
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1answer
111 views

Primary school competition problem find the area of a square

Someone posted online a primary school competition problem. I solved it using the Cayley-Menger determinant (finding the answer is 169/2 after getting the length of AD being 13) but it probably isn't ...
6
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2answers
1k views

Area of triangle

A triangle is inscribed in a circle. The vertices of triangle divide the circle into three arcs of length 3, 4 and 5 units, then find the area of triangle.
6
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2answers
454 views

Area of greatest integer function

Question: Find the area enclosed by the function: $$\left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3$$ where $\lfloor\cdot\...
6
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1answer
2k views

Finding the area of a implicit relation

Let's say we have the function: $$x^2+y^2+\sin(4x)+\sin(4y)=4$$ I haven't taken Calculus III, in fact I'm just taking Calculus I. Since I learned how to find the derivative of implicit relations I ...
6
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1answer
303 views

Area of Shaded Region Inside Quadrilateral

I recently saw a similar problem online but found the area in a completely different way. Problem: There is a unit square, i.e. a square with side length equal to $\text 1$. Two lines are placed ...
6
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4answers
984 views

Layman's proof that the area of a circle of radius $r$ equals $\pi r^2$.

There are many, many clear and simple proofs of basic but nontrivial facts in high-school mathematics, such as Pythagoras' theorem or the identities $$\sum_{k=1}^nk=\binom{n}{2}\qquad\text{ and }\...
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2answers
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Area under parabola using geometry

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without ...
6
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1answer
352 views

To find area of the curves that are extension of ellipse

I like to draw an ellipse via 2 fixed points and a rope between the fixed points (2 focuses). I wanted to extend the idea. Point A,B,C,D are fixed points and Point E can move freely. Point E,B,C have ...
6
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1answer
211 views

Area bounded by $\cos x+\cos y=1$

What is the area of the region $\cos x+\cos y > 1$, where $|x|,|y|<\pi$? In other words, is there a "closed" form -- using functions that are well-known and nice to work with -- for this ...
6
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4answers
331 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. We ...
6
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3answers
520 views

Surface optimization for a volume

One of my children received this homework and we are a bit disoriented by the way the questions are asked (more than the calculation actually). This is exactly the wording and layout of the homework: ...
6
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2answers
347 views

Area and Polar Coordinates

Would anyone be able to help me with this problem? I think I know the area formula in polar coordinates that should be used: the antiderivative of ((1/2)r^2 dtheta) from alpha to beta but I'm not ...
6
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1answer
779 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
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0answers
68 views

Expected area of a random $n$-gon

Choose $n$ points $\{z_1, \ldots, z_n\}$ from the unit circle $\partial \mathbb{D} = \{z \in \mathbb{C}: |z| = 1\}$ uniformly at random, and let $P_n$ be the convex hull of the $z_i$'s. Let $X_n = ...
6
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1answer
150 views

Definite integral over a semicircular area $\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$.

change the order of integration in $$\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$$ I was able to change the order of integration here to $$\int_0^a\...
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5answers
477 views

What is the least number of (fixed) parameters I can ask for, when calculating area of a triangle of unknown type?

I need to calculate the area of a triangle, but I don't know, whether it is right angled, isoscele or equilateral. What parameters do I need to calculate the area of a triangle of unknown type?
5
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4answers
274 views

Area of a triangle with sides $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$,$\sqrt{z^2+x^2}$

Sides of a triangle ABC are $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$ and $\sqrt{z^2+x^2}$ where x,y,z are non-zero real numbers,then area of triangle ABC is (A)$\frac{1}{2}\sqrt{x^2y^2+y^2z^2+z^2x^2}$ (B)$\...
5
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2answers
7k views

Finding number of “Pixels” in a Circle using diameter

I'm trying to figure out how to calculate the number of whole pixels in a pixel circle using the diameter of the circle. I understand how to find the area of a circle using diameter. But I'm ...
5
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4answers
250 views

Integrating with respect to an angle [duplicate]

Hello maths community! One day I was solving a geometry problem and I thought I had found a way of solving it. When I was solving the problem, I kind of invented a new way of finding an area of a ...
5
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1answer
9k views

Area of a five pointed star

A 5 pointed star is inscribed in a circle of radius $r$. Prove that the area of the star is $$ \frac{10 \tan\left(\tfrac{\pi}{10}\right)}{3-\tan^2\left(\tfrac{\pi}{10}\right)} r^2 $$
5
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2answers
106 views

In the square $ABCD$, prove that $BF+DE=AE$.

Consider the Square $ABCD$. The point $E$ is on the side $CD$. If $F$ is on the side $BC$ such that $AF$ is the bisector of the angle $BAE$. Prove that $$BF+DE=AE.$$ By Pythagorean theorem we have $(...
5
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5answers
122 views

The ellipse of the maximum area contained between the curves $\frac{\pm 1}{x^2+c} $

The ellipse of the maximum area contained between the curves $\frac{\pm 1}{x^2+c}, c > 0 $ It seems to me that the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ of the maximum area when $ a ...
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3answers
3k views

How do I find the area of a general closed curve? (And then generalisation to multiple dimensions)

Let us say I have a general equation where $f$ is a general function: $$f(x,y)=0$$ If the function in the Cartesian plane forms a closed curve, is there a general way to find the area bounded by the ...
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1answer
9k views

Area of a circle inside a quarter circle

I'm trying to figure out a couple things. The main question I have is how to find the area of a circle inscribed inside a quarter circle with a radius of x. The secondary question to that is if the ...
5
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2answers
251 views

Grazing area for a goat around a circle.

I am doing this math question and i am really confused on how to approach it. This is the question: A retired mathematics professor has decided to raise a goat. He owns a silo and a barn. The barns ...
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2answers
1k views

Find the area of the shaded region in the figure below:

Find the area of the shaded region in the figure below: I am completely stuck on how to start off this question. Please help on some guidance on how to start it off.
5
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2answers
1k views

How to create a two circle Venn diagram with 3 equal sections?

I had a student ask if I could draw a Venn diagram in which each region was of equal area. I have played around with this a little but have not landed on an answer I'm satisfied with. I was able to ...
5
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1answer
93 views

Area of ascending regions on implicit plot - $\cos(y^y) = \sin(x)$

Take the equation $$ \cos \left( y^y \right) = \sin(x) $$ The plot forms a kind of skew checkerboard, where each "tile" shrinks as $y$ increases. I was attempting to find the sum of the areas (the ...
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2answers
150 views

1987 AIME Problem #4

The Question: Find the area of the region enclosed by the graph of $|x-60|+|y|=|\frac x4|$. Answer: What I know: Because of all the absolute values I only need to find one side of the graph of this ...
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2answers
10k views

Why does the integral of the surface area of a sphere equal its volume? [duplicate]

Why does the integral of the surface area of a sphere equal it's volume? $$\int{4\pi r^2 \ \mathrm{d}r =\frac{4}{3}\pi r^3}$$ I don't quite understand why the relationship between surface area and ...
5
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1answer
256 views

Prove that the maximum volume of a triangular-base prism is $\sqrt{\frac{K^3}{54}}$ where $K$ is the area of three triangles containing a vertex $A$

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and ...
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votes
2answers
67 views

Area below the curve $y=\left[\sqrt{2+2\cos2x}\right]$

Find the area below the curve $y=[\sqrt{2+2\cos2x}]$ and above the $x$-axis , $x\in [-3\pi,6\pi]$, (where $[.]$ denotes the greatest integer function) . My approach: $$y=[\sqrt{2+2\cos2x}]$$ $$=[\...
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2answers
129 views

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 ...
5
votes
1answer
4k views

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'...
5
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2answers
524 views

Surface area of cone

thanks for any help. I'm trying to find the surface area of a cone via integration. I know that the parametric equation of a cone is $$x=u\cos(p) \\ y=u\sin(p) \\ z=u$$ So as a vector, $\vec{R} = \...
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1answer
73 views

Minimal surfaces

Among the definitions of minimal suraface I found these two: (1) A surface $M\subset\mathbb{R}^3$ is minimal if for any point $p\in M$ there is a neighborhood $U$ of $p$ in $M$ that minimizes the ...