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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2answers
95 views

Area between $y = 5+x-x^2$ and $y = x+4$ [duplicate]

I have the following 2 equations and I have drawn their graphs. $$\begin{cases}y=5+x-x^2\\ y=x+4\end{cases}$$ I have found intercepts as well. The question is asking to find area between these 2 ...
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2answers
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How do I find the area in this question?

Find the area bounded by $$ f(x) = x + 6 $$ $$ g(x) = x^3 $$ $$ h(x) = -\frac{x}{2} $$ Edit Fixed simple error on h(x) I already drew the grew, although it is very hard to really tell where they ...
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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 ...
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4answers
443 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}=\...
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5answers
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Prove, that $\lim_{x \to \infty} 2^x \sin(\pi/(2^k))$ converges to $\pi$.

Prove, that $$\lim_{x \to \infty} 2^x \sin\left(\frac \pi {2^x}\right)=\pi.$$ I've tried calculus to no avail. I found out that every element of this sequence is the area of a regular rectangle with ...
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2answers
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When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
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1answer
954 views

Triangle area division

It is given that in a triangle ABC, a line from A to BC intersects BC at point D. If the ratio in which AD divides BC is given can we say anything about the ratio of areas of triangle ABD and triangle ...
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2answers
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KVPY Scholarship Exam Problem on finding the area of a rectangle

In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the ...
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1answer
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Pre-calculus Rug Area to room area problem.

A rectangular rug is in the middle of a room and there is a uniform width of floor that shows around the rug. The dimensions of the rug are 16ft by 21ft. What would the dimensions of the room have to ...
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9answers
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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 ...
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0answers
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Overlapping Circles Area [duplicate]

I have searched but could not find the exact question. Two circles with radii 5 intersect such that the center of one circle lies on the circumference of another. What is the area of the overlapping ...
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2answers
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Ratio of triangle A and B if the length of the sides are A:25,25, 30 and b:25,25, 40

If $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math
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1answer
752 views

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie $ax+by+c=o$ an equilateral triangle

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie ax+by+c=o an equilateral triangle whose area is $\frac{c^2}{\sqrt{3}(a^2+b^2)}$ is there any other way to solve ...
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4answers
8k views

How do you find the area of a triangle in a 3D graph?

How do you find the area of a triangle in a 3 dimensional graph? Is it any different than a regular 2d graph? How would you solve it, if these were your three points? A(1,-4,-2), B(3,-3,-3), C(5,-1,-2)...
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1answer
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What is the value of $[c,d]$ when $c$ and $d$ be such that $f(x) ∈ [c, d]$ for all $x ∈ [a, b]$?

Let $c$ and $d$ be such that $f(x) \in [c, d]$ for all $x \in [a, b]$. What is the value of $[c,d]$ for the function $f(x)=\sqrt{1-x^2}$ on the interval $[a, b]=[0,1]$? I knew taking the minimum and ...
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1answer
93 views

Finding Area and probability[ hard nut to crack].

Suppose that $X$ and $Y$ are iid uniform distribution with $U(0, 1)$ random variables. (a) What is $\mathbb P((X, Y ) ∈ [a, b]×[c, d])$ for $0 ≤ a ≤ b ≤ 1$ and $0 ≤ c ≤ d ≤ 1$ ? What is $\mathbb P((...
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0answers
514 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
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1answer
216 views

Finding the exact area of a trapizium using similar triangles

IN the trapezium ABCD, the diagonals intercept at M. Let AM= a, BM= b, Cm = c and DM = d, and let Angle AMB be $\theta$. a=6 b=4 c=3 d=2 AB=8 DC=4 $\cos(\theta) = -\frac{1}{4}$ and $\sin (\...
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0answers
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How: Determine area painted by a path with width within a polygon

I have a path that represents the movement of some equipment. The equipment has a width so I'd like to determine the approximate area created by this path within a polygon. If I use the distance ...
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2answers
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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) +...
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2answers
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Converting squared or cubed units

At the moment I am going from square meters to square yards. One meter is $1.0936$ yards. So I figure $1\text{m}^2 = 1.0936 \text{yards}^2$ I know it isn't but I want to learn a good systematic ...
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1answer
785 views

Area ratio in triangle?

Given: $\triangle ABC$. In the side $AB$, we choose point $D$. From this point $D$, we draw a line $DF$ such that intersect side $AC$ and line $DE$ such that intersect side $BC$. If $DF\parallel BC$, $...
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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.
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15answers
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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 ...
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3answers
9k views

Find the rectangle with the maximum area inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ [duplicate]

Find the rectangle with the maximum area inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, whose edges are parallel to the axises. Hint: Find the function we need to maximize. Well in the ...
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3answers
726 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?
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1answer
426 views

Series for $\pi$ which correspond to apollonian gaskets or hyperbolic tilings of the unit disk

Consider the two partitions of the unit disk in $\mathbb{R}^{2}$, the first an Apollonian gasket and the second is the $\{7,3\}$ hyperbolic tiling: Since the unit disk has radius $1$, both of these ...
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0answers
99 views

Area of polygons

I've got a problem that says that I have a triangle with vertices in A, B and C, as the image shows, and the red and the blue surfaces are equal. The question is, how do I say: "The area of the ...
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0answers
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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?
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3answers
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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)=\...
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2answers
176 views

Area of a parallelogram in $ \mathbb{R}3 $

Let $ A = (2, -3, 1), B = (5, -3, -1), C = (-2, -3, 5) $ and $ D = (1, -3, 3) $ I have the above situation. I know this: $$ A = \| \vec{AB} \times \vec{AD} \| $$ gives the area of the parallelogram. ...
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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 $\...
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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 ...
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1answer
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Finding the area of an ellipse using Divergence Theorem

As an example of Divergence Theorem, our textbook mentions finding area of an ellipse, but it isn't clear how it was derived though. Following is an excerpt from the textbook. Suppose there is an ...
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2answers
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Difference between “sq km” and “km sq”

If I have a square with side $2\ \text{km}$, what is its area: $2\ \text{sq km}$ or $4\ \text{km}^2$?
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2answers
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Proving Green's Theorem for Computing Area

I'm having a hard time understanding where exactly the formula for computing area using Green's theorem comes from. It is typically: \begin{equation} \int_C x\,dy = \int_C -y\,dx = Area \end{...
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3answers
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Area of a sector - Inscribed Angle and Central Angle

I know the formula for the area of a sector of an arc made by central angle is $$\text{Area}_\text{Sector}= \frac{\text{Arc Angle} \times \text{Area of Circle} }{360}$$ Now my question is , Is this ...
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2answers
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Why does the cylinder with minimum surface area have a height equal to its diameter?

I'm trying to understand as to why, with a given volume, the diameter and height of a cylinder need to be the same for the minimum surface area. This thread, shows how to derive the minimum surface ...
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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. ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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 ...