Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional measurement of a shape.

Filter by
Sorted by
Tagged with
6
votes
2answers
3k 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
votes
5answers
2k views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
4
votes
2answers
274 views

How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China The ...
13
votes
4answers
616 views

Closed form for the area of a convex cyclic n-gon, given the set of edge lengths

Let's say we are given a set of positive reals, and we're told that these are the edges of a convex cyclic $n$-gon, and we must compute it's area. For $n = 3$ there is the famous Heron's formula: $$...
6
votes
1answer
234 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 ...
5
votes
2answers
10k 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
votes
3answers
155 views

Triangle tangent to 3 Parabolas, finding the common area

Triangle $ABC$, $AB=4$, $BC=15$, $AC=13$. Two sides are tangents to the respective Parabolas. We have to find the area shaded. My approach- I tried finding the area of the quadratures(Archimedes) ...
3
votes
1answer
1k views

area-preserving iff $|\det |=+1$

Why is a (not necessarily linear) mapping $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ area- and orientation preserving iff the determinant of its jacobian is $\pm 1$ ? (I understand by an area-...
7
votes
1answer
963 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 ...
4
votes
3answers
58k views

Optimization with cylinder

I have no idea how to do this problem at all. A cylindrical can without a top is made to contain V cm^3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can. Since ...
3
votes
1answer
117 views

What is the chance that a PDF with compact support is concave?

Relevant questions and answers, in chronological order: When do equations represent the same curve? Find a smooth function with prescribed moments Does a sequence of moments determine the function? ...
2
votes
3answers
4k views

Why change the sign of the integral when switching the limits of integration?

So I'm refreshing on integration and venture across a property of definite integrals I've been taking for granted for quite some time: \begin{equation}\int_{a}^{b} f(x) \mathrm{d}x =-\int_{b}^{a} f(x)...
2
votes
2answers
109 views

Quadrilateral's area problem

I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
5
votes
2answers
12k 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 ...
3
votes
2answers
2k views

Maximum area enclosure given side lengths

A peer of mine gave me the following problem : Given a sequence of $n$ lengths (i.e.,$L_1, L_2, .., L_n$ ) where each is the length of the side, find a sequence of $n$ points (where $p_k = (x_k, ...
3
votes
3answers
120 views

$M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$. [closed]

$M$ is a point in an equilateral triangle $ABC$ with the area $S$. Prove that $MA, MB, MC$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1}{3}S$$
2
votes
1answer
151 views

Show that the altitude bisects the corresponding angle

$AD, BE$ and $CF$ are concurrent lines drawn from the vertices of $\Delta ABC$ to points $D,E,F$ (respectively) on the opposite sides. If $AD$ is the altitute, show that $AD$ bisects $\angle FDE$ I ...
2
votes
3answers
407 views

What is area of shaded region?

Suppose the side length $a$ of the square is 10mm. A circle is tangent to all four sides of the square. And two quarter-circles with the same radius of 10mm have centers on the opposite vertices. It ...
1
vote
2answers
16k views

Finding the exact area of a circle?

Background: I recently began taking calculus and it has come to alter the way I look at circles, and curves. The equation of a circle is $\pi r^2$, traditionally in school we have always left the ...
0
votes
1answer
51 views

what is the area of the region [ (x,y):0<x , y<1 , 3/4<x+y<3/2 ]? [closed]

can we do this without integration ( breaking the area into triangles & rectangles etc ) ?
8
votes
3answers
14k views

How to find the area of any irregular shape?

how to find the area of any irregular shapes without dividing it into smaller regular shapes ? Example Image:
4
votes
3answers
139 views

Why is $f(4)$ the area under $f'(x)$ specifically from $0$ to $4$ and not for ex from $1$ to $4$ or $2$ to $4$?

I've seen the geometric argument for why any differentiable function $f(x)$ gives the rate of change of the area under its own curve to $x$ for a specific input $x$, and it makes sense to me. It also ...
2
votes
1answer
89 views

Prove area of a quadrilateral is $\frac14[4m^2n^2-(b^2+d^2-a^2-c^2)^2]^{\frac12}$

Someone asked me this question which I am really stuck at, any help is appreciated. If $a,b,c,d$ are the sides of a quadrilateral and $m,n$ are diagonals of the quadrilateral, then prove that area ...
2
votes
1answer
97 views

Prove the equality of circle's areas

Through a random point inside the circle, we draw $4$ lines with $45$ degrees between each other. Prove, that the total area of odd pieces, equals total area of even pieces. I suppose there is a ...
2
votes
3answers
807 views

Dodecagon Area Question

The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon. Is there any way to do this without trigonometry? And could you include a proof also? :O
2
votes
2answers
2k views

When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
2
votes
1answer
2k views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
2
votes
4answers
18k views

What's the difference between width and height?

I've never been to good with Mathematics... I'm wanting to change that as game development has always been a passion of mine. I started studying math on Khan Academy about a month ago and everything ...
2
votes
1answer
682 views

Using overlap area to determine distance between overlapping circles

I'm working on a simple, two-ring Venn Diagram in JavaScript where the area of the sections will be determined by the set size (mapping dataset count to area in pixels). Group 1 area (a1) = 800 ...
2
votes
1answer
4k views

Overlapping area between a circle and a square

I have a circle and a square. They are aligned to their center. The radius of the given circle is less then half the value of diameter of the square. How to find the overlapped area?
1
vote
2answers
1k views

How to Integrate this function $\int(1-x^2)^ndx$

The actual question was to find the area bounded by the curve $y=\int_{-1}^{1}(1-x^2)^ndx$ and coordinate axes. But I haven't came across these type of problems with power $n$.
1
vote
2answers
44 views

To what value does the sum of the areas of the dark triangles converge if the area of ​the square is 1?

To what value does the sum of the areas of the dark triangles converge if the area of ​​the square is 1? I already calculated the first to layers of triangles and got that one is $\frac{1}{2}$ and the ...
1
vote
1answer
106 views

Stretching a Hexagon

Consider the regular hexagon Imagine I want to stretch it uniformly in the horizontal direction ($x$). In order to keep the area constant, I naturally need to compress it in the vertical direction ($...
0
votes
3answers
884 views

Goat tethered in a circular pen

There is a circular pen with a goat in it. The goat is tethered by a rope to the edge of the pen. The rope is the length of the radius of the pen. What area of grass in the pen can the goat graze?
0
votes
0answers
52 views

Complex integral to determine area inside of parameterized closed curve

By combining several things I read in several places, it seems that the area inside a parameterized closed curve in the complex plane ($\;f : [0,1] \to \mathbb C\;$ with $\;f(0)=f(1)\;$ and 'piecewise ...
0
votes
3answers
106 views

Angle subtended at the centre by a segment of a circle

To find the area of segment of a circle, I used the following formula:$\frac{r^2}{2} (\theta$ - $\sin \theta$) But if the area is given and I want to find the angle $\theta$ how can I do that. $\...
0
votes
1answer
754 views

Calculating Centre of area using a method I found but it gives me only approximate answers

As you can see from the title , I am having a problem in calculating centre of areas and specifically the centre of area of a right angle triangle.I know that the centre of area coordinates are $\...
0
votes
1answer
61 views

Is there a geometric intuition for division of area by length?

To give you the context of this question, before I had a question about why an area of a square equals to its side squared, and how to see it geometrically, because multiplying a length by a length to ...
-2
votes
2answers
599 views

Determine the area of a regular octagon with vertices on the unit circle [closed]

How would I determine the area? Help please
135
votes
10answers
20k 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. ...
33
votes
2answers
1k views

Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this? Note that ...
26
votes
1answer
641 views

The generous lazy caterer

It is well-known that $n$ chords divide any convex shape into at most $\frac{n^2+n+2}2=T_n+1$ regions – the lazy caterer's sequence. For example, the pancake below is cut into seven pieces by three ...
31
votes
9answers
9k views

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 \...
23
votes
11answers
9k views

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=...
12
votes
3answers
608 views

How is area defined?

Thinking about area in the context of the Lebesgue measure, I have an intuitive understanding of how area is constructed in $\mathbb{R}^2$: define all rectangles to have the area $length \times width$...
8
votes
4answers
4k views

How to find the shaded area

How to find the shaded area crossed by semi-circle of radius 2 and quarter-circle of radius 4?
6
votes
4answers
2k 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 }\...
9
votes
2answers
128 views

Proof of relationship $S^2−S(a+b+c+d+e)+ab+bc+cd+de+ea=0$ between areas connected to a pentagon

So recently I've been looking around at some other problems to see if they could help me solve an ongoing problem, and I found a theorem that was mentioned that I feel that might be useful to my ...
6
votes
1answer
382 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 ...
3
votes
4answers
501 views

How to show that $\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$

I was reading Advanced Integration Techniques, and found that$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$ The book provides one method using residue theorem and Laurent expansion. ...