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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254
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 ...
122
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10answers
19k 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 ...
103
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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 ...
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 ...
75
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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?
67
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9answers
238k 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 ...
63
votes
6answers
5k views

Area covered by a constant length segment rotating around the center of a square.

This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless. I describe my thoughts ...
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 ...
56
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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
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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: ...
50
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8answers
10k 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 ...
48
votes
12answers
6k views

Can area of rectangle be greater than the square of its diagonal?

Q: A wall, rectangular in shape, has a perimeter of 72 m. If the length of its diagonal is 18 m, what is the area of the wall ? The answer given to me is area of 486 m2. This is the explanation given ...
46
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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 ...
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 $d$ ...
39
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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. ...
38
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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. ...
34
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9answers
13k views

Is Area of a circle always irrational

I have learned that $\pi$ is an irrational quantity and a product of an irrational number with a rational number is always irrational. Does This imply that area of a circle with radius $r$, which is $...
34
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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 ...
31
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11answers
188k 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. ...
31
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3answers
3k views

Why does area differentiate to perimeter for circles and not for squares?

I read this question the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle. Why doesn't the same ...
30
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9answers
7k 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 \...
25
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2answers
602 views

$\pi$ in terms of $4$?

I'm trying to define $\pi$ in terms of $4$ by placing a unit circle inside a square, and subtracting the corners of the square. I'm attempting to use summation to define the area of a corner, then ...
25
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1answer
420 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 ...
24
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3answers
3k views

Where does the gap come from? [duplicate]

Can anyone tell me please where does the gap come from? Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.
23
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11answers
6k 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: $...
21
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5answers
2k views

Find the area enclosed by $\sqrt{(x-2)^2+(y-3)^2} + 2\sqrt{(x-3)^2+(y-1)^2} = 4$

Question: What is the area of the interior of the simple closed curve described by the equation $\sqrt{(x-2)^2+(y-3)^2} + 2\sqrt{(x-3)^2+(y-1)^2} = 4$? Comments: I came up with this specific ...
18
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7answers
17k 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 $\...
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 ...
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 ...
16
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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
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7answers
83k 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 ...
16
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4answers
1k views

What is the area of the circle?

In the following diagram, $AB = 4$ and $AC = 3$. What is the area of the circle? I can't find any way to solve this.
15
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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 ...
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?...
15
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6answers
2k views

Unit square inside triangle. [duplicate]

Some time ago I saw this beautiful problem and think it is worth to post it: Let S be the area of triangle that covers unit square. Prove that $S \ge 2$.
15
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1answer
715 views

Bath towel on the rope: minimize the area of self-intersection of a folded rectangle

This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o). There is this rectangle with sides $a<b$. The ...
14
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6answers
4k views

How is the area of a circle calculated using basic mathematics?

Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is $$ A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n $$ which $$ =...
14
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2answers
2k views

Finding the minimal integral area of a circle for which the area is larger than the circumference

I'm sorry maybe it's obvious but English is not my first language. I just want to know what is asked in this question: The area of a circle (in square inches) is numerically larger than its ...
14
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2answers
957 views

Find area of shaded region - is the information insufficient?

This brain teaser turned out to be a brain boggler. As I am the type of math need that dwells on a single problem until it gas been solved ( and understood ). I don't think there is enough ...
14
votes
3answers
1k views

Does the perimeter of a polygon necessarily decrease if more edges are added to it, with the constraint of constant area?

A circle has the lowest perimeter for a 2D shape of a given area. To my understanding, it can also be approximated by a polygon of infinite sides. So, if I take an n-sided polygon and gradually add ...
14
votes
3answers
864 views

Representing the area of a circle as the sum of circumferences

I had this idea of calculating the area of a circle as the sum of circumferences where the radius is being shortened by an infinitesimal amount until it reaches 0. I was told that this was impossible, ...
14
votes
4answers
657 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.
13
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5answers
4k views

Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
13
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5answers
1k views

A flower in a hexagon

The area of the ✽ in a ⬡ This geometry problem comes from a recent math test. The question is the following: We have a regular hexagon with sides equal to $1$ and six circular arcs with radius ...
13
votes
3answers
1k views

Need to find the ellipse of maximum area inscribed in a semicircle.

An ellipse inscribed in a fixed semi circle touches the semi-circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. ...
13
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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?
13
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4answers
471 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: $$...
13
votes
2answers
2k views

Area interpretation of integrals

When integrating under part of a circle, as in $$A=\int_0^a {\sqrt{r^2-x^2}\,\mathrm{d}x}$$ I noted that the simple geometric solution would be to add the areas of the sector and triangle formed by ...
13
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2answers
481 views

Unit diameter pentagons with maximum area

In the euclidean plane, if one considers the set of quadrilaterals having unit diameter (maximum distance between two points in the convex envelope), it is quite easy to give a description of the ...
12
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7answers
1k views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer $$s=\sqrt{p(p-a)(...