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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103
votes
8answers
59k 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 ...
16
votes
7answers
81k views

Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation for ...
18
votes
7answers
16k views

Show that the area of a triangle is given by this determinant

I'm not sure how to solve this problem. Can you guys provide some input/hints? Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$. Show that the area of $\...
39
votes
2answers
3k views

Areas versus volumes of revolution: why does the area require approximation by a cone?

Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. ...
56
votes
9answers
41k views

Why determinant of a 2 by 2 matrix is the area of a parallelogram?

Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$. How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$? Are the areas of the ...
55
votes
12answers
7k views

Any smart ideas on finding the area of this shaded region?

Don't let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can't think of an easy/smart way of finding it. Any ideas? For reference, the Area is: ...
67
votes
8answers
236k 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 ...
252
votes
8answers
35k 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 ...
8
votes
3answers
1k views

Proof of Heron's Formula for the area of a triangle

Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula: $$A = \sqrt{p(p-a)(p-b)(p-c)},$$ where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$. Could ...
1
vote
2answers
195 views

Consider a right angled $\triangle PQR$ right angled at $P$ i.e ($\angle QPR=90°$) with side $PR=4$ and area$=6$.

Consider a right angled $\triangle PQR$ right angled at $P$ i.e ($\angle QPR=90°$) with side $PR=4$ and area$=6$. If $\triangle PQR$ is rotated through $360°$ about the side $PR$ , what is the $TSA$ ...
85
votes
4answers
24k 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 ...
57
votes
14answers
12k views

Area of a square inside a square created by connecting point-opposite midpoint

Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each. $H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively. What will the area of the square formed in the middle be? I know that ...
8
votes
4answers
2k views

Area of intersection between 4 circles centered at the vertices of a square

The centers of four circles are at the vertices of a square of sidelength 100m. Each circle has the radius of 100m. Which is the area of their intersection?
30
votes
11answers
186k views

How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
18
votes
2answers
18k views

Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides a, b, c. The "square" which I will refer to, from now on, has all its four vertices on the sides of the ...
13
votes
8answers
13k views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
2
votes
1answer
2k views

Given a polygon of n-sides, why does the regular one (i.e. all sides equal) enclose the greatest area given a constant perimeter?

This doesn't require much more than the title. I just need an explanation, but an algebraic proof would be a bonus. We can demonstrate this for quadrilaterals, a square is best as shown by this graph-...
75
votes
9answers
24k 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?
15
votes
3answers
13k views

Rigorous proof that $dx dy=r\ dr\ d\theta$

I get the graphic explanation, i.e. that the area $dA$ of the sector's increment can be looked upon as a polar "rectangle" as $dr$ and $d\theta$ are infinitesimal, but how do you prove this rigorously?...
38
votes
6answers
11k views

Why square units?

I was recently asked "Why is the area of a circle irrational?", to which I replied that it was not necessarily irrational—there are of course certain values for $r$ that would make $\pi r^2$ rational. ...
14
votes
4answers
646 views

Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.
3
votes
3answers
47k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\...
2
votes
2answers
2k views

Circle areas on squared grid

There is a circle. On 9 equal squares. Every square has some value assigned to it. Every square gets weight, depending of what percentage of it is circle (area-wise). I need to find circle radius, ...
2
votes
3answers
993 views

How calculate the shaded area in this picture?

Let the centers of four circles with the radius $R=a$ be on 4 vertexs a square with edge size $a$. How calculate the shaded area in this picture?
15
votes
3answers
4k views

Which area is larger, the blue area, or the white area?

In the square below, two semicircles are overlapping in a symmetrical pattern. Which is greater: the area shaded blue or the area shaded white? My Solution Let the length of each side of the square ...
12
votes
1answer
562 views

Find the area where dog can roam [duplicate]

A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam? I tried to find the area of all semicircles and then to ...
46
votes
15answers
14k views

Do two right triangles with the same length hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
9
votes
1answer
442 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\...
4
votes
4answers
3k views

Area of the field that the cow can graze.

How do we find the area that the cow can graze? The question goes as follows-- There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as ...
4
votes
7answers
76k views

maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: Let ...
3
votes
1answer
444 views

what is the surface area of a cap on a hypersphere?

According to mathworld, let the sphere have radius $R$, then the surface area a spherical cap of height $h$ and base radius $a$ is given by $$S=2\pi Rh=2\pi(a^2+h^2).$$ What is this value for an n-...
3
votes
5answers
899 views

Calculate the area of center square in the following figure [duplicate]

Calculate the area of center square in the following figure:(the big square has a side length of 1 and each vertex of big square has been connected to the midpoint of opposite side) answer ...
4
votes
0answers
3k views

Finding the area of the shaded square inside a square created by connecting point-opposite midpoint [closed]

If the lines intersect the vertices of the square. The area of the square is $1$ and the lines also intersect the midpoints of the square lines. How to find the area of shaded region?
4
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, ...
0
votes
1answer
383 views

Calculate the area of an irregular cyclic convex polygon

I want to write a program in C++ to calculate the area of irregular cyclic convex polygons. However, the inputs are in the form corner point angles. I am just not sure what the inputs mean and what ...
43
votes
11answers
8k views

How was the area formula for a circle ($A = \pi r^2$) derived before the introduction of calculus?

How did mathematicians prior to the coming of calculus derive the area of the circle from scratch, without the use of calculus? The area, $A$, of a circle is $\pi r^2$. Given radius $r$, diameter $d$ ...
11
votes
3answers
7k views

Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
16
votes
6answers
1k views

Why is the area of the circle $πr^2$? [duplicate]

I searched many times about the cause of the circle area formula but I did not know anything so ... Why is the area of the circle $\pi r^2$? Thanks for all here.
16
votes
4answers
14k views

How to maximize the area of a triangle, given two sides?

I am having a question that is it possible to find what will be the maximum area of a triangle if suppose we know any two of the sides and none of the angles? We don't know whether one of these sides ...
7
votes
2answers
3k views

Show that the Area of image = Area of object $\cdot |\det(T)|$? Where $T$ is a linear transformation from $R^2 \rightarrow R^2$

Prove that the area of an image in $2d$ cartesian coordinates is equal to the determinant of the linear transformation times the area of the initial shape. I've tried to formulate general expression ...
3
votes
2answers
219 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
461 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
210 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 ...
3
votes
5answers
683 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 ...
7
votes
1answer
612 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 ...
5
votes
2answers
6k 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 ...
4
votes
3answers
50k 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
93 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
2answers
89 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 ...
2
votes
1answer
641 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-...