Questions tagged [area]

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

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48
votes
2answers
4k 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. ...
110
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8answers
67k 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
116k 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 ...
69
votes
9answers
53k 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 ...
24
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7answers
22k 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 $\...
56
votes
12answers
9k 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: ...
73
votes
9answers
264k 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 ...
274
votes
9answers
40k 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 is 6 ...
8
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3answers
2k 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 you please ...
2
votes
2answers
291 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 $TSA$ ...
88
votes
2answers
33k 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 ...
20
votes
3answers
27k 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?...
39
votes
11answers
245k 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. ...
14
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8answers
14k 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?
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?
20
votes
2answers
22k 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 ...
4
votes
1answer
3k 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-...
58
votes
14answers
34k 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 ...
77
votes
10answers
27k 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?
38
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6answers
12k 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. ...
15
votes
4answers
819 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.
6
votes
2answers
1k views

Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
5
votes
1answer
1k 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-...
4
votes
5answers
6k views

Find all triangles of which perimeter and area are numerically equal

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
4
votes
3answers
54k 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)=\...
4
votes
2answers
18k views

What is the maximum area of a square inscribed in an equilateral triangle?

What is the maximum area of a square inscribed in an equilateral triangle? Please post the approach to solve the above question.
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
2k 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?
14
votes
3answers
9k 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 ...
38
votes
9answers
4k views

Area of parallelogram = Area of square. Shear transform

Below the parallelogram is obtained from square by stretching the top side while fixing the bottom. Since area of parallelogram is base times height, both square and parallelogram have the same area. ...
15
votes
3answers
6k 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 ...
14
votes
1answer
628 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
17k 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 ...
11
votes
3answers
15k views

Check if a point is inside a rectangular shaped area (3D)?

I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example ...
5
votes
3answers
18k views

Why does the magnitude of the cross-product of a and b give the area of a parallelogram spanned by a and b?

I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working ...
9
votes
1answer
513 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 $$\...
5
votes
4answers
4k 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 ...
5
votes
8answers
97k 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
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
5answers
2k 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 ...
7
votes
2answers
1k views

Area of an irregular polygon

I was searching for methods on how to calculate the area of a polygon and stubled across this: http://www.mathopenref.com/coordpolygonarea.html. $$ \mathop{area} = \left\lvert\frac{(x_1y_2 − y_1x_2) +...
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?
1
vote
2answers
75 views

Minimum area of whole quadrilateral given areas of parts

I've attempted to mark segment $AO$ as $a$ and $CO$ as $b$. Now, if draw a line from $D$ that is perpendicular to $AC$, and draw another line from B that is perpendicular to $AC$, then we can use some ...
0
votes
1answer
431 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 ...
63
votes
11answers
16k views

A way to find this shaded area without calculus?

This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600) $ABCD$ is a square with a side of $10$, $APD$ and $CPD$ are ...
46
votes
11answers
10k 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 $d$ ...
10
votes
2answers
514 views

Where is the mass of a hypercube?

Question: Consider a hypercube $C := \{\vec{x}\in\mathbb{R}^n : \forall(i)\: |x_i|\leq 1\}$ and hypersphere $S_r := \{\vec{x}\in\mathbb{R}^n : |\vec{x}| = r\}$. Let $R(n)$ be the radius of the ...
16
votes
4answers
20k 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 ...
16
votes
6answers
2k 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.
7
votes
2answers
6k 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 ...

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