Questions tagged [area]

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

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Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. Actually there isn't So, here are the lengths of sides ...
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1answer
595 views

Proof that for any n-sided polygon P, and any integer m greater than n, there is an m-sided polygon with the same area and perimeter as P?

The answers to a recent question established that it is possible to construct families of polygons all with the same area and perimeter. Some comments on some of the answers inspired this very ...
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110 views

Software to approximate area of curve

Does any know of a math program where I can measure the area of a closed parametric curve ? I know that I can measure the area between 2 curves with the TI-Nspire, but not for one curve in ...
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1answer
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Parameterization of a triangle in $\mathbb R^3$

I'm asked to find the area of a triangular region $T$ with vertices at $(1,1,0)$, $(2,1,2)$, and $(2,3,3)$. With the help of software, I was able to conjure up the following parameterization for $T$, ...
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How to find the area of region $R$?

Region $R$ contains all the points $(x,y)$ such that $x^2+y^2\leq100\;$ and $\:\sin(x+y)\geq0$. Find the area of region $R$. $\:\:\:\:\:\:\:\:\:$$\sin(x+y)\geq0$ $\:\:\:\:\:\:\:\:\:$$\implies2n\pi\...
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127 views

Area enclosed by curve $f(x,y)$ defined implicitly

Find area bounded by the curve $$5x^2+2y^2+6xy+7x+6y+6=0$$ I can find the area using integration for curves defined explicitly in $x$ or $y$. I have no idea how to do this.
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Calculate $3$ sides of any triangles from $S,P,R$ and $r$.

Is there any way to calculate $3$ sides of any triangles $(a,b,c)$, If we know Area $S$, Perimeter $P$, Circumradius $R$ and inradius $r$. I took a deep look in to Wikipedia page, but no clue: https:/...
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Length of sides of a triangle and area

Let $T$ be a triangle with sides of length $a,b,c$ and $T'$ a triangle with sides of length $a',b',c'$. If $a<a'$ and $b<b'$ and $c<c'$, is it true $Area(T)<Area(T')$? I've tried to use ...
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2k views

Find the area of the largest hexagon that can be inscribed in a unit square

How so I find the area of the largest regular hexagon that can be inscribed in a unit square? I am using this http://www.drking.org.uk/hexagons/misc/deriv4.html to figure it out but not sure if it is ...
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214 views

How to find the projected area in the x-z plane of an ellipsoidal cap rotated by angle β in x-y plane?

I have ellipsoidal cap rotated in the x-y plane by an angle $\beta$; where the axis size in x coordinate is 'a', the axis size in y-coordinate is 'b' and axis size in z coordinate is 'c'. I am trying ...
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find a certain ratio of areas in a triangle where other areas are given [closed]

Given : [BDE]=8, [BDC]=12, [CDF]=9 where [.] is the area of the respective triangles. Find ratio [AFD]/[AED] from the figure. NOTE:-Question may have errors please just edit that or comment the ...
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941 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
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Proof that each of the three cevians is divided in the ratio $1:3:3$

Points $D,E,F$ are the first trisection points of $BC,CA,AB$ respectively. Let $[ABC]$ denotes the area of triangle $ABC$. If $[ABC]=1$, find $[GHI]$, the area of the shaded triangle. (the two images ...
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472 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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1answer
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What is the expected area of a cyclic quadrilateral inscribed in a unit circle?

Choose four points randomly on the circumference of a circle with radius $1$. Connect them to form a quadrilateral. What is the expected area of this quadrilateral? I have attempted to simulate to ...
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2answers
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Surface area of cone

thanks for any help. I'm trying to find the surface area of a cone via integration. I know that the parametric equation of a cone is $$x=u\cos(p) \\ y=u\sin(p) \\ z=u$$ So as a vector, $\vec{R} = \...
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Function diverges but area under curve is finite?

I have a curve given by $f(x)=\frac{x}{b^2}(1-\frac{x^2}{b^2})^{-1/2}$ for $x \in (0,b)$. Upon integration, I get that the area under the curve on the interval $(t, b)$ is given by $F(t) =\sqrt{1-\...
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216 views

Triangle Quadrilateral and pentagon whose areas form a set of consecutive positive integers.

Find a triangle, quadrilateral and pentagon with integer side lengths whose areas form a set of three consecutive positive integers. Make the areas as small as possible subject to these constraints. ...
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332 views

Proof the area of a given triangle with coordinates is half determinant

I was given a problem, tried to solved it but couldn't get to a solution. It goes like that: There's a triangle ABC with area S. $$ \vec{AB} = (a,b) $$ $$ \vec{AC} = (c,d) $$ Prove that $$ S = \...
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1answer
301 views

Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem. Suppose $T$ is the set of all possible triangles made from the vertices of a ...
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3answers
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Calculate the area of the ellipsoid obtained from ellipse $\frac {x^{2}}{2}+y^{2} = 1$ rotated around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
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1answer
203 views

Does the symmetry of a parabola in finding the maximum area of a rectangle under said parabola matter?

Apologies for my English, I'm a not a native speaker... So I've got this homework, about finding the maximum area of a rectangle under a parabola. I'm using this as a reference to do my work: How to ...
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2answers
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Integration of x^(1/2) sinx

My book say that integration of $x^{1/2} \sin x$ is not possible, why is it so? Which functions do not have an anti derivative? Does it mean that they do not have any area under the curve? But that'...
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2answers
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Calculating torus surface area

I was trying to calculate the surface area of a torus, whose tube radius is r, and distance from "singularity" to the center of the torus tube is R. Here's what I've tried to do (The reason ...
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1answer
83 views

Are closed simple curves with this property necessarily circles?

Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple curve and $\Gamma$ be the region enclosed by $\gamma$. Let $O$ be the center of mass of $\Gamma$. Suppose that any line that goes through $O$ ...
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1answer
160 views

Finding the area enclosed by the locus of the vertex of the rectangle at which the normals meet.

Let a and b be the lengths of the semimajor and semiminor axes of an ellipse respectively. Draw a rectangle whose two sides are tangent to the ellipse and the other two are normal to the ellipse. I ...
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4answers
292 views

Finding the maximum area of a quadrilateral when three points are given

I am working on problems in the chapter "Applications of Derivatives". I encountered the following problem: Question: Four points A,B,C, and D lie in that order on the parabola $y=ax^2+bx+c$...
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1answer
138 views

Surface area of ellipsoid given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis

Calculate the surface area of the ellipsoid that is given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis. My idea is that if $f(x)=\sqrt{1-\frac{x^2}{2}}$ rotates around the x-axis we will end ...
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5answers
196 views

Area of a critical triangle ABC if PA,PB known and PC unknown

help me to solve this this problem please: In a triangle $ABC$, $\angle BAC$ = $60\,^{\circ}$,$AB=2AC$.Point P is inside the triangle such that $PA=\sqrt{3}$,$PB=5$. What is the area of triangle $ABC ?...
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1answer
107 views

Contract expression of circle segment area contingent on height

I want to determine a function for the area of the segment's height. I have made it this far, but I would like to contract the equation further - sadly, I do not know how to do this while still ...
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1answer
448 views

Prove surface area of a sphere using solid of revolution surface area formula.

I have to prove the surface area of a sphere with $r=1$ using the solids of revolution through revolution abouth both the $x$ and the $y$ axis. The formulas are easy. From top to bottom, surface area ...
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1answer
64 views

Prove that the cube roots of areas are equal.

In $\triangle ABC$, $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively. If $Z$ is on the segment $XY$ such that $\frac{AX}{XC}=\frac{CY}{YB}=\frac{XZ}{ZY}$. Prove that the area of $\...
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1answer
52 views

Is convolution area-preserving?

Consider two functions f(x) and g(x) with indefinite integrals ("area under the curve") Af and Ag. Does the convolution f * g preserve the area, i.e. is Af *g = Af * Ag i.e. ∫ ( ∫f(x)g(t−x)...
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1answer
680 views

Linear Algebra, meaning of 0 determinant in linear transformations

Lets say the area of a figure in $\Bbb R^2$ was $10$. Then after a noninvertible linear transformation from $\Bbb R^2$ to $\Bbb R^2$, is there enough info to determine the new area? Since its ...
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0answers
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Calculating the Area of a Circle Occupied by a Rectangle

This is a question regarding how to calculate the area of a circle occupied by a rectangle when that rectangle is larger than the circle (see this link for a example image http://i57.tinypic.com/...
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2answers
304 views

Why is the area of a rectangle with sides a and b defined as axb?

Imagine $a \times b$ was not defined and we need to come up with something. Here is the justification I could come up with: Suppose, I have to measure this thing called area, I have to come with a ...
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1answer
605 views

Proving area equal to zero of a continuous function.

I'm kinda stuck with this exercise: Prove that the graph of a (uniformly) continuous function $f\:[a,b] \to \mathbb{R}$ has area zero. I was thinking that maybe I should use uniform continuity ...
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1answer
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Determining the average area as the point $X$ varies

We consider a fixed triangle $ABC$ with side lengths $a = BC$, $b = CA$, $c = AB$, and a variable point $X$ in the interior. The lines through $X$ parallel to $AB$ and $AC$, together with line $BC$, ...
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227 views

Proof of bizarre form of Cavalieri's Principle?

In a book I'm really enjoying, The Irrationals by Julian Havil, in Chapter 2 he mentions that John Wallis often used Cavalieri's Principle, which he explains. However, he then says that "in modern ...
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In triangle $ABC$, $AA_1$, $BB_1$, $CC_1$ divide sides in ratio of $1: 2$ and meet at $M$, $K$, $L$. Find area relation of $KLM$ and $ABC$

Points $A_1$, $B_1$, $C_1$ divide sides $BC$, $CA$, $AB$ equilateral triangle $ABC$ in a ratio of $1: 2$. The line segments $AA_1$, $BB_1$, $CC_1$ determine the triangle $KLM$. Is the triangle $KLM$ ...
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559 views

Alternative proof of lateral surface area of a conical frustum

I am trying to come up with an alternative proof of the lateral surface area of a conical frustum with parallel bases by making use of the linear increase in perimeter $P$ of the base with respect to ...
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2answers
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How would I calculate the area of a rectangle on a sphere using vertical and horizontal angles?

Imagine a sphere being one's eyeball and the rectangular area being the picture of one's view. Like putting a name tag sticker on a balloon. How can I find the area of the rectangle on the sphere?
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129 views

Making the area of the quadrilateral and the area of a triangle the same

$ABCD$ is a quadrilateral and $X$ is a given point on AD. Find a point Y in AB such that the area of the $\triangle AXY$ is equal to that of $ABCD$. Hence show how to divide the quadrilateral $ABCD$ ...
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Area of Circle Overlapped by Rectangle

I'm trying to determine 'how much' (as a percentage) a 2D rectangle fills a 2D circle. Actual Application: I was comparing the accuracy of some computer game weapons by calculating the max possible ...
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2answers
142 views

Indefinite integral with sector of ellipse

An ellipse is given by the following equation: $$ 152 x^2 - 300 x y + 150 y^2 - 42 x + 40 y + 3 = 0 $$ After solving for the midpoint we have: $$ 152 (x-1/2)^2 - 300 (x-1/2) (y-11/30) + 150 (y-11/30)^...
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3answers
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Maximal possible area of a rectangle inscribed in the given right triangle

A rectangle is to be inscribed in a right triangle having sides of length $6$ in, $8$ in, and $10$ in. Find the dimensions of the rectangle with greatest area assuming the rectangle is positioned as ...
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1answer
669 views

2D Gaussian integration over arbitrary eccentric circle. Analytical solution?

How can I find the solution for the integral of an axisymmetric Gaussian distribution over a circular surface? (A circular surface eccentric to the centre of the Gaussian distribution). I am trying ...
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1answer
394 views

Decide if a point is inside quadrangle

Let's say computer randomly generate 4 points on canvas. points are A[x1,y1] B[x2,y2] C[x3,y3] D[x4,y4] I need to decide if a point P [x,y] is insided the area ...
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How to find the area of intersection of two circles using axiomatic geometry?

Problem: square(ABCD) is a regular square, and a circle touches internally in the square. Also, arc(BD) divides the square. Then calculate the area of the colored region. This question is easily ...
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2answers
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Find the dimensions of the largest rectangle that can be inscribed in a semicircle of radius r.

I know that If I were to make a loose coordinate plane graph than the radius and (X,Y) of the rectangle would have to mixed into make an equation out of the whole thing. What exactly that equation is ...