Questions tagged [area]

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

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Area bounded by $f(x) = -2(x-3)(x^2+1)$ and $g(x) = ax$

I have the following equations and trying to determine the value of a $g(x) = ax$ $f(x) = -2(x-3)(x^2+1)$ Hello, I am trying to identify the unknown value of a > 0 such that the following holds: ...
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Computing the area between 2 curves

$x+y = 5$ and $x+7 = y^2$. $$$$ It is possible to calculate the area between two curves as follows: $\int_a^b f(x) dx$ + $\int_b^c g(x) dx$. Where a = ___ , b____, c_____ and $f(x) = $________ and $g(...
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Finding an area under the curve with no area

Hello I am trying to explain why the function h(x) $h(x)= x^{99} + x$ will always have the area without determining an antiderivative of h(x) that we must have First I thought of drawing the graph ...
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What is the surface area of a plane on a sphere like Earth?

If I have a $1 \text{km} $ x $1 \text{km} $ plain of grass, then the surface area is $1 \text{km}{^2} $. Well, no, because the Earth isn't a flat plane but rather a sphere that's curved. The surface ...
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How to prove that the area of a shape is independent of the choice of axes?

Suppose we have some shape in a plane, and we want to find its area using calculus. I set my $x$ and $y$ axes arbitrarily. They are perpendicular to each other. I can calculate the area of the region ...
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Equation Relating Surface Area of Higher Dimensional Spheres

Let $\omega_d$ denote the surface area of the $d$ dimensional unit sphere in $\mathbf{R}^{d+1}$. I want to show that $$ \frac{\omega_d}{2} = \int_0^{\pi/2} \omega_{d-1} \sin(t)^{d-1}\; dt. $$ My ...
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A doubt regarding change of variables in Double Integrals.

So I have recently learnt to calculate "Double Integrals" and I have a doubt on change of variables. Lets say we are evaluating a double integral of the form I = $\int\int _R F(x,y) dx dy $ ,...
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surface area in ${\mathbb{R}}^n$

Let the surface $E_2 = \{(x_1,x_2,x_3)\in{\mathbb{R}}^3 : x_1^2+x_2^2+x_3^2=1, x_1^2+x_2^2\leq \frac{2}{3}, x_1^2\leq \frac{1}{3} \}.$ We want to compute the surface area $a(E_2)$ of $E_2.$ That is, $$...
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Area of sub-triangle inside a triangle

Let $ABC$ be a triangle of area $a$. The segment $\overline{AB}$ is divided in $n$ equidistant points and segment $\overline{AC}$ is divided in $m$ equidistant points. Find the area $b$ of triangle $...
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What is the difference between meters squared and square meters?

So I'm in year 11 and I just had a question about area. What is the difference between meters squared and square meters? Because all my teachers told me that they are the same thing but when I googled ...
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Finding the maximum area of a rectangle

I'm having trouble with this problem and I need help to solve it... A person has $800$ ft of fencing. He wishes to form a rectangular enclosure and then divide it into three sections by running two ...
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Calculate ${S_n} = \sum\limits_{k = 1}^n {\frac{n}{{{n^2} + kn + {k^2}}}} ;{T_n} = \sum\limits_{k = 0}^{n - 1} {\frac{n}{{{n^2} + kn + {k^2}}}} $

Let ${S_n} = \sum\limits_{k = 1}^n {\frac{n}{{{n^2} + kn + {k^2}}}} ;{T_n} = \sum\limits_{k = 0}^{n - 1} {\frac{n}{{{n^2} + kn + {k^2}}}} $, for n=1,2,3,.....Then (A) ${S_n} < \frac{\pi }{{3\sqrt 3 ...
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25 views

Why doesn't the proving work if the vectors change sides?

Instead of calculating the area of F4 as shown in the solution I could also calculate it by doing 1/2 (B-A)x(C-A) since for area the direction doesn't matter but if i do it this way the proving doesn'...
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Roots and point of inflections

Let $b$ and $c$ be the roots of a four degree polynomial. Also $x=b$ and $x=c$ are the real points of inflection of this four degree polynomial. If the other two roots of the polynomial be $a$ and $d$ ...
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Area , average value , energy and power of notable signals

Can someone please link me a table where I can find all the values of area, $ \langle x(t) \rangle $ , $ E_x $ and $ P_x $ of signals as constant, $ u(t) $, $\operatorname{rect}(t) $, $\operatorname{...
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Find the area bounded by the curve $x^4+y^4=x^2+y^2$ [closed]

I am stuck with this problem which deals with evaluating an Area The problem reads : Find the area bounded by the curve $x^4+y^4=x^2+y^2$. I tried factorizing the expression and expressing $y$ in ...
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337 views

Find the area of the garden planted with flowers.

A garden is shaped in the form of a regular heptagon (seven-sided), $MNSRQPO$. A circle with centre $T$ and radius $25\ \text{m}$ circumscribes the heptagon as shown in the diagram below. The area of $...
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To Prove $\frac{1}{b}+\frac{1}{c}+\frac{1}{a} > \sqrt{a}+\sqrt{b}+\sqrt{c}$

The three sides of a triangle are $a,b,c$, the area of the triangle is $0.25$, the radius of the circumcircle is $1$. Prove that $1/b+1/c+1/a > \sqrt{a}+\sqrt{b}+\sqrt{c}$. what I've tried: $$\frac{...
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Areas and volume ambiguity

For my question, I came up with this very simple analogy for the original question I have in mind. Case 1 let's say there is a rectangle with the length $X$ and width $Y$, so the area will be $XY$. If ...
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Area of a constrained and bounded strictly non-decreasing function - geometric solution

I have boiled a problem I have down to the following: Assume a non-decreasing function $pb=f(pt)$, with domain [0,1] and range [0,1]. Require that $f(0)=0$, $f(1)=1$ Require that from point $A(0,0)$ ...
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Area of $A:= \lbrace (x,y) \in \mathbb{R}^2 : x^2+(e^x +y)^2\leq1 \rbrace$

I am having trouble calculating the area of $$A:= {\lbrace(x,y) \in \mathbb{R}^2 : x^2+(e^x +y)^2\leq1 \rbrace}.$$ I hope someone can help me. I have tried using Fubini with the following boundaries ...
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Proof of relationship $S^2βˆ’S(a+b+c+d+e)+ab+bc+cd+de+ea=0$ 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 ...
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65 views

Point $B$ lies on line segment $\overline{AC}$ with $AB = 16$ , $BC = 4$ .

Point $B$ lies on line segment $\overline{AC}$ with $AB = 16$ , $BC = 4$ . Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\Delta ABD$ and $\Delta BCE$ . Let $M$ be ...
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Prove that $\frac{PQ}{MN} = \frac{|[BCE] - [ADE]|}{[ABCD]}$ in a quadrilateral ABCD where P and Q are related to the diagonals

I've recently been given a few challenge problems that I really want to find out. But for the most part, I just can't figure out how to completely prove the problems. Now one of the problems goes ...
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Proving the concurrence of the three lines formed by connecting midpoints of opposite sides of a hexagon

Let's consider a convex hexagon where we know the midpoints of each side. I'm currently trying to show that if we connect the midpoints of the opposite sides into lines that the three lines will all ...
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1answer
30 views

Is quadrilateral with two equal opposite sides and joining mid points of other sides divide equally?

Let $ABCD$ a convex quadrilateral such that $AB=CD$. Let $P$ and $Q$ are the mid points of the sides $BC$ and $AD$ respectively. Now if we joint $PQ$, is it divide the quadrilateral in equal area? To ...
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64 views

Area of Triangle inside a Rectangle

Rectangle $WXYZ$ has an area of $25$. Point $U\ \&\ V$ lie at the sides $XY\ \&\ YZ$,respectively$. $$\triangle WXU$ has an area of $6$ & $\triangle WZV$ has an area of $5$. Find the area ...
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1answer
38 views

In triangle ABC a point X is taken on AC and a point Y is taken on BC if AY and BX meet at O

This question is from pre collage mathematics. The question goes on like this: In $\triangle ABC$ a point $X$ is taken on $\overline{AC}$ and a point $Y$ is taken on $\overline{BC}$. If $\overline{AY}...
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Deriving Area of Circle

I wanted to find the area of a circle with radius $r$ described by $x^2+y^2=r^2$ I decided to describe the whole circle by multiplying $2$ to the semicircle, $y=\sqrt{r^2-x^2}$. I integrated this from ...
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How do I find the lateral area of a cuboid given its height, base area and the area of the diagonal cross-section?

H, M and B are given and I need to find the lateral area (area of all the sides): Sketch of the cuboid Since it's a cuboid, I know that the lateral area is $$S = 2(aH + bH) = 2H (a+b)$$ I found the ...
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Least triangular convex polygon

(This question is based on a question posed in a math riddle post on Reddit.) Let $P$ be a convex polygon. Let the non-triangularity of $P$ be the minimum area of the symmetric difference (shown with ...
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Area of irregular quadrilateral with diagonals proof

I have an irregular convex quadrilateral with diagonals d and D. These diagonals form an acute angle $\alpha$. I know that I can find the area of this quadrilateral by using this formula: $A = \frac{D\...
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A random result for area of plane

In cartesian coordinate system we take the x-y plane a d draw the line x=y in 1st quadrant now we can say that this line divides the plane of 1st quadrant into two equal halves which means area under ...
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The image of a function defined as a regular polygon with $n$ > 0 sides inside a circumference of radius 1 and center 0.

The following problem has been passed to me trough a friend studying for a local math olympics, but I can't get my head around it: Consider C a circle with center (0,0) and radius 1, that is: $$ C = \{...
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Finding the area under the inequality $\sin^2 \pi x + \sin^2 \pi y \le 1$ for $x,y \in [-1,1]$

Find the area under the inequality $$\sin^2 \pi x + \sin^2 \pi y \le 1 \text{ for } x,y \in [-1,1]$$ I coudn't do this problem without using a graphing calculator: It's easy to see now that in each ...
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Calculate the dual basis and tangent basis vectors

A coordinate system with the coordinates s and t in $R^2$ is defined by the coordinate transformations: $s = y/y_0$ and $t=y/y_0 - tan(x/x_0)$ , where $x_0$ and $y_0$ are constants. a) Determine the ...
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Area of triangle under tan conditions.

For an acute triangle $ABC$, the following conditions hold. $$\frac{1}{\tan A} + \frac{1}{\tan B} + \frac{1}{\tan C } =2$$ $$ a^2 + b^2 + c^2 =50 $$ Compute the area of such a triangle. I used a tan ...
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$A,B,C$ and $D$ are concyclic.$AC$ is the diameter of the circle and $AD=DC$.The area of quadrilateral $ABCD$ is $20cm^2$.

$A,B,C$ and $D$ are concyclic . $AC$ is the diameter of the circle and $AD=DC$ . The area of quadrilateral $ABCD$ is $20c$m$^2$. Draw a line $DE$ such that $E$ is a point on $AB$, and $DE$ $\bot$ $AB$....
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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 ...
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The plane $𝑥/4+𝑦/4+𝑧/7=1$ intersects the $𝑥-$ , $ 𝑦-$ , and $𝑧$- axes in points $𝑃, 𝑄, 𝑅$. Find the area of the triangle $Ξ”𝑃𝑄𝑅$.

The plane $π‘₯/4+𝑦/4+𝑧/7=1$ intersects the $π‘₯-$ , $ 𝑦-$ , and $𝑧-$ axes in points $𝑃, 𝑄, 𝑅$. Find the area of the triangle $Δ𝑃𝑄𝑅$. So here's my attempt. First I find the normal vector:...
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Finding the length between the two vertices of two different triangles having the same base side and same perimeter by a shorter method.

My approach:- Using distance formula I found the lengths of AC,CB and AB. Using them I found out the perimeter of triangle ABC and it came out to be 18 square units. Hence, $$AB+BC+CA=18 units$$ ...
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289 views

Integral Calculus, Infinitesimal

To integrate $y=f(x)$ from $a$ to $b$ we break the function into small rectangles of width $dx$. So the $n$-th rectangle will be at a distance of $n\,dx$ from $a$ on the $x$-axis. Let there be $t$ ...
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How do I use cross products to find the area of the quadrilateral in the $𝑥𝑦$-plane defined by $(0,0), (1,βˆ’1), (3,1)$ and $(2,8)$?

How do I use cross products to find the area of the quadrilateral in the $π‘₯𝑦$-plane defined by $(0,0), (1,βˆ’1), (3,1)$ and $(2,8)$? So what I first do is find two vectors. Gonna use (0,0) as ...
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1answer
126 views

Diagonals dividing an hexagon into regions of the same area.

Let's say that a diagonal of a convex polygon with an even number of vertices is a line segment connecting opposite vertices. I found the following question in an old exercise list that a teacher gave ...
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36 views

Compare the area of two different catenoids spanned by the same two circles

A catenoid is obtained by rotating the graph of the function $f(x)=a \cosh (x/a)$ around the $x$-axis. Consider catenoids that satisfy the boundary condition $f(c)=f(-c)=r>0$. We have $a\cosh(c/a)=...
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106 views

How to calculate the area of $\triangle ABC$ when the distance from $BC$ to the circumcircle at $G$ is 10?

$\triangle ABC$ is right angle triangle and its circumcenter is $O$. $G$ is a point where $BC$ is tangent to the incircle. The perpendicular distance from $BC$ to circumcircle at $G$ is 10. How to ...
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2answers
57 views

Finding the area of the region bounded by the graphs of $y=|x|$, $y=|x|+3$, and $y=5-|x|$

Find the area of the region bounded by the graphs of $y = |x|$, $y = |x| + 3$, and $y = 5 - |x|$. I got $\left(\dfrac{5\sqrt2}2\right)^2$, but this is incorrect. I don't really understand what the ...
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3answers
56 views

Let $ABC$ be triangle with sides that are not equal. Find point $X$ on $BC$ From following conditions.

Let $ABC$ be triangle with sides that are not equal. Find point $X$ on $BC$ such that $$\frac{\text{area}\ \triangle{ABX}}{\text{area}\ \triangle{ACX}}=\frac{\text{perimeter}{\ \triangle{ABX}}}{\text{...
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21 views

Determining the areas above, below, and between two cumulative distributions

I have two cumulative distributions, which reflect two different beta distributions (one with mean = $.5$, precision/phi =$ 3$, the other mean is $.35$, and the precision/phi is $20$). The cumulative ...
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3answers
217 views

Area of a region bound by three circular arcs, why doesn't this approach work?

Three circular arcs of radius $5$ units bound the region shown. Arcs AB and AD are quarter-circles, and arc BCD is a semicircle. I tried to find the answer by calculating the total area of the circle ...

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