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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Finding the area between two regions in the plane

I have two regions, given by $y>\sqrt{2}x - \frac{1}{4x}$ and $y< \sqrt{2}x + \frac{1}{4x}$. How can I find the area of their intersection? If their is no easy analytical way, could someone ...
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34 views

Ratio between the areas of two rectangles inscribed in a triangle and the area of the triangle

Let's denote $A(X)$ the area of the polygon $X$. Let $S$,$R$ be rectangles inside a triangle $T$. Find the maximum value of: $$\frac{A(R)+A(S)}{A(T)}$$ My try The only property i know (from a ...
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2answers
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Find the area bounded by a quartic curve

Find the area bounded by the curve defined by: $x^4+y^4=x^2+y^2$ . The final solution should be $\sqrt{2}\pi$. I tried to change it into polar coordinates, but I couldn't calculate the integral.
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What values of $\rho$ can give the answer? $V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$

Find the Volume of the region bounded above by sphere $x^2 + y^2 + z^2 = 2a^2$, and below by the paraboloid $az = x^2 + y^2$? $$V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$...
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Why is area under some symmetric curves zero and others not?

When finding the area under a curve, there appears to be a contradiction Like area under integral sinx from 0 to 2pi is zero because, the areas above and below cancel each other. But when finding ...
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Calculating density of individuals within an area (tree stand density)

Any help greatly appreciated! I want to find the density of trees surrounding a sample point (measured in trees/m$^2$). The distance from the sample point to the three nearest trees has been ...
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1answer
23 views

Max possible area, of a rectangle shape where one side is a half circle. circumference of 100m

A picture of the shape! I recently took a maths test where one of the questions was just unsolvable for me. I'm going to try to make it as clear as possible, to not create confusion. The question ...
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2answers
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Area of Parametric Curves

Compute, in terms of $A, B, h$, and $k$, the area enclosed by the curve defined by the parametric equations: $x(θ)=Acosθ+h$ $y(θ)=Bsinθ+k \quad \quad \quad \quad \quad \quad $ for $0 ≤ \theta ≤ 2π$....
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1answer
44 views

Maximizing Area of a quadrilateral inside of a square

The square ABCD has point M located on side AB and point N on side CD. Lines CM and BN intersect at point U. Lines DM and AN intersect at point V. Determine where points M and N should be placed to ...
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2answers
78 views

Area under curve $\frac{1}{x}$ is infinite, volume of revolution $\frac{1}{x}$ is $\pi$?

Stumbled across this weird phenomenon using the equation $y = \frac{1}{x} $. Surface Area: When you calculate the surface area under the curve from 1 to $\infty$ $$\int_1^\infty \frac{1}{x}dx = \...
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3answers
90 views

Area Inside A Loop Formed By Parametric Equations

We are given: $x=49-t^2$ $y=t^3-16t$ The curve apparently makes a loop which lies along the x-axis. I need help finding total area inside the loop. I don't know where to even start. If it helps, ...
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1answer
46 views

Area between curves.

I have to calculate area bounded by curves : $(x^3+y^3)^2=x^2+y^2 $ for $ x,y \ge 0 $. I tried to use polar coordinates, but I have : $r^4(\cos^6\alpha +2\sin^3\alpha\cos^3\alpha + \sin^6\alpha)=1$
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1answer
54 views

Help me to calculate polyhedron area or let me know the references about it.

I'm a student who is studying the Management Science in the Republic of Korea. I'm looking for a reference to calculate the area of the system of inequalities. I can easily calculate when the number ...
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1answer
153 views

Why does $\int_0^R 2 \pi r \,\mathrm d r$ give the area of a circle?

There's a method of computing the area of a circle by dividing it in concentric rings with infinitesimal width. Let $R$ be the radius of the circle and $r$ be the radius of the rings. The area of the ...
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1answer
64 views

Prove the equality of circle's areas

Through a random point inside the circle, we draw $4$ lines with $45$ degrees between each other. Prove, that the total area of odd pieces, equals total area of even pieces. I suppose there is a ...
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3answers
45 views

Area enclosed by general equation of second degree

What is the area of the figure enclosed by the curve: $5x^2+6xy+2y^2+7x+6y+6=0$. My attempt: $2y^2+6(x+1)x+5x^2+7x+6=0$. $\displaystyle y=\frac{-6(x+1)\pm\sqrt{36(x+1)^2-4(5x^2+7x+6)}}{4}$ $\...
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2answers
58 views

If the sides of a quadrilateral are $a,b,c,d$, prove that the area cannot exceed $(ac+bd)/2$.

MOP 1997: Let $Q$ be a quadrilateral whose side lengths are $a,b,c,d$ in that order. Show that the area of $Q$ does not exceed $(ac+bd)/2$. My solution: Without loss of generality, let $a$ be the ...
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67 views

Area of shaded portion inside a circle. [closed]

How do you find the area of orange shaded region given the inner diameter of the green circle is $14\sqrt{2}$ units?
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The coordinate of a circle [closed]

Suppose two circles intersect and form three regions A, B, and C. The center of circle A is (2,2) and the center of circle B is (x,y). The three regions formed by the two circles are equal in area. ...
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37 views

Surface area of a sphere over a disc

What's the surface area of the sphere $x^2 + y^2 + z^2 = 1$ over the disc $(x-1/2)^2 + y^2 \le 1/4$ ? I've tried something, but I don't think it's right, as it's not a "nice answer" So here is what ...
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1answer
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Calculate Area of 4 points polygon from distance between poins

I have 4 points on ground, something like what is depicted in this sample: https://www.mathopenref.com/coordpolygonarea.html The problem is that I don't know how to determine the coordinates of my ...
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2answers
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Finding the area of shaded region using co-ordinate system

I am trying to solve the problem above by co-ordinate geometry. Here is what I have done: I flipped the figure (just for simplicity) such that the point B is to the right of AC. Then I took C as the ...
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Finding the area of inner triangle constructed by three cevian lines of a large triangle

QUESTION: In a triangle $ABC$, $AD, BE$ and $CF$ are three cevian lines such that $BD:DC = CE:AC = AF:FB = 3:1$. The area of $\triangle ABC$ is $100$ unit$^2$. Find the area of $\triangle HIG$ ...
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1answer
61 views

How to find the radius of the circle after adding the area of segment?

I am making a game where a circle can hit a wall an interact with it to simulate a ball hitting a wall. When the circle is 1/4 the radius through the wall the area of the part in the wall will be ...
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Surface area of revolution query

I've been struggling to grasp this concept. As you see, I was able to achieve the arc length asked for in the first example. Now, however, I seem to be in a pinch regarding the surface area. I ...
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21 views

Finding the width of a smaller rectangle

We have a large rectangle, with a width of 14m and a length of 9m. In the middle of this rectangle is a smaller one, with a length given of 5m. The text prompts me to use the width of the larger ...
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46 views

Using Mid-Points for Area of Triangle does not support answer.

I was preparing for my end-sem exams and I ran into this problem that asks me to figure out the area of the triangle given its mid-points. Either I could've computed the vertices of the triangle which ...
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Area of a quatrefoil inside a circle

What is the maximum area of a quatrefoil that is inscribed in a circle of radius 6? My first guess was to cut the circle into small regions but that doesn't seem to work. The solution does not have ...
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Computing hyper area of a contrained simplex

Let $D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] = \{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid \sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \}$, where $r \geq b_i \geq a_i \geq 0$...
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Easy derivation for area of obtuse triangle

I am looking for an easy derivation for the area of an obtuse triangle. As everybody knows the formula for any triangle is the same: half the product of the base and its height. The derivation for an ...
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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 ...
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1answer
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Volume and surface area of region by a curve and line

Let $g(x) = (1-2x)(x-3)$, B be the region enclosed by $g(x)$, $x = 1$, $x = 2$ and $y = -1$ after revolve the region B about x-axis, we have a solid. The volume of solid and the total surface area is?...
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1answer
57 views

Area between trigo function and a line

How can I compute the area of region enclosed by $\cos^2x$ and $y=1-\frac{2}{\pi}x$? I am just stuck on the first step, finding their intersection point, I know I can plot the graph using online ...
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1answer
33 views

area under parametric curve

I have difficulty on how to eliminate parameter especially the equation involved trigonometry equation. The question is asking for the area bounded by the curve , the 2 axes and the line $y=1$. $x=4 ...
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Triangle problem, $AC=3$ , $AB=5$ ,…, then $PH=?$

In $\triangle ABC$ : $AC=3$ , $AB=5$ , $\angle ACB= 90 ^ \circ$, $P$ is a point inside $\triangle ABC$ such that $PA+BC=PB+AC=PC+AB$, $H$ is a point on the line $AB$ such that $\angle PHB=90^\circ$ ...
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4answers
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unable to understand tan/arctan-related integral transformation

I'm currently studying area between curve and x axis on Khan Academy. I understand that the But I am puzzled by how did we get to this step from the previous step?
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2answers
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Geometrical meaning of the formula for the area of a triangle using a $3\times3$ matrix.

I know the geometrical meaning of the determinant of a matrix, and I know that, for example, the area of a parallelogram defined by two vectors $$ v=\begin{bmatrix}a \\b \end{bmatrix},\quad w=\begin{...
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1answer
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Intuitive Explanation to Pappus Theorem

Pappus's theorem is as follows: First theorem: The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on ...
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1answer
61 views

Perimeter of a parallelogram just with the area [closed]

The area of a parallelogram is $2019^{2019}$ square unit. What is the perimeter of this parallelogram? I know the basic formula but could not break down $2019^{2019}$, and connect that to the ...
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1answer
41 views

Area of touching part of Sphere to the wall.

I believe that it has a very simple explanation but one question stuck in my mind. What is the area between sphere and wall when it touches to it. If it is zero, why it is not occurring in real life?...
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How to compare centrality of points in different shaped and sized polygons

Suppose I have several polygons of varying shape and size, each containing a single point. I want to compare the centrality of each point, and determine which point is more central to its polygon. So ...
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1answer
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What is the area of $OEAF$ in $ABC$ triangle in the following diagram?

The area of $ABC$ and $OBC$ triangle is $120$ and $24$ respectively. $BC=16$, $EF=8$. Find out the area of $OEAF$ quadrilateral. Source: Bangladesh Math Olympiad 2014 Junior Category. I can find ...
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Find the area of ​the figure bounded by the curves using the replacement

I need to find the area of ​​the figure bounded by the curves $y^2 = 2x , x+y = 4, x + y = 12 ,$ using the replacement $u = x + y, v = y^2 -2x$. Going to the new coordinates $u,v$, I get an unlimited ...
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6answers
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What is the area of $\triangle ABC$ if area of $ABEF$ and $BCFD$ is 15 and 23 in the following diagram?

$ABEF$ and $BCFD$ parallelograms have areas respectively $15$ and $23$. Find the area of $\triangle ABC$. Source: Bangladesh Math Olympiad 2014 Junior Category Is only areas of two parallelograms ...
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Showing that the area of a parabolic sector is half the area of a corresponding region bounded by the directrix (without Calculus)

Given parabola: It is necessary to prove that the area of the parabolic sector (green) is equal to half the area of the parabolic rectangle (orange) without calculus. ("Something like" ...
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2answers
106 views

How to find the area of the rectangular region 2 ≤ x ≤ 5, -1 ≤ y ≤ 3

How to find the area of the rectangular region $2 ≤ x ≤ 5$, $-1 ≤ y ≤ 3$. I tried to plot the graph in $xy$-plane, but I'm not sure how to find the area.
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How do I calculate?

How do I calculate cross-sectional area of this: ${r}=\sqrt{sin\theta} \, $, when $ 0 \le \theta \le \pi,$ Don't know what is the right answer but I have get that the area is 1. Is that right answer?...
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1answer
33 views

How do I calculate area?

How do I calculate the area of this: $D=\{ (x,y)\mid 0 \le x \le 1, x^2 \le y \le x^2+2 \}$ ${A}=\iint_D \, \textrm{d}A.$ Don't know what is the right answer but I have get that the area is 6. Is ...
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1answer
48 views

minimum area occur when $c=\frac{a}{2}$

Let $f(x)=\sqrt{\tan x}$ . Then show that the area bounded by $y=f(x),y=f(c),x=0$ and $x=a,0<c<a<90^\circ$ is minimum when $\displaystyle c=\frac{a}{2}.$ what i try enclosed area $$A=\int^{...
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111 views

Given that $[ABC]$ : Area of small circle = $\frac{3\sqrt3}{4}$ : $\pi$. How many parts of area of small circle is inscribed in large circle? [closed]

In the common region of two circle, $\triangle ABC$ has been drawn with its maximum area such that the proportion of the maximum area of $\triangle ABC$ and the area of small circle is equal to $\frac{...