Questions tagged [area]

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

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2
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3answers
69 views

Area of a circle passing through two vertices of a parallelogram touching one edge.

For reference: Let $ABCD$ be a parallelogram, $AB = 6, BC= 10$ and $AC = 14$ ; traces a circle passing through the vertices $C$ and $D$ with $AD$ being tangent and this circle is $BC$ a secant. ...
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1answer
56 views

Find the area inside the curve $r^2=2\cos(5\theta)$ and outside the unit circle.

I found the area of one full rose-petal($A_1$) and the area enclosed by the petal and the unit circle($A_2$), subtracted these from one another to get the area enclosed by the curve outside of the ...
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2answers
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Area of the region bounded by $f(x)$ and $g(x)$? ($f(x)=|2-x|$ and $g(x)=2-|2-x|$)

$f(x)=|2-x|$ $g(x)=2-|2-x|$ Find the area of the region bounded by $f(x)$ and $g(x)$. I was told that you actually need two definite integrals to solve this problem, since it involved absolute value, ...
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2answers
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Find the area of ​the shaded region in the figure below

For reference: In the figure $O$ is the center of the circle and its radius measures $a$ and $AQ = QB$. Calculate the area of ​​the shaded region.(Answer: $\frac{a^2}{4}(\pi-2)$) My progress: If $AQ ...
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2answers
47 views

Find the area of ​$​PTHQ$ region in the figure below

For reference: On the side $AC$ of a triangle $ABC$, the points $P$ and $Q$ are marked such that $AQ = PC$ $(QC > CP)$. Then $PT$ and $QR$ were traced in parallel to $AB$ ($T$ and $R$ in $BC$). If ...
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Find the area of ​the shaded region $CEOD$.

For reference: In figure $O$ and $O_1$ are centers, $\overset{\LARGE{\frown}}{AO_1}=\overset{\LARGE{\frown}}{O_1B}$. If $AD = 4\sqrt2$. Calculate the area of ​​the shaded region. (Answer: $2(4-\sqrt2)...
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0answers
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Finding area of a triangle in a quadralateral with only one side known

For a quadrilateral $PQRS$, $PR=40$ and $PS=32$. Suppose for a point $X$ on the line segment $RS$ the triangles $PQX$ and $PRS$ are congruent in that order. If area of triangle $PXS$ is $112$ square ...
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2answers
50 views

Find the area of ​the quadrangular region $EBCD$

For reference: Calculate the area of ​​the quadrangular region $EBCD$, if $BC = 5$, and $AD = AC$. $P$ is a tangent point. (Answer: $64$) My progress: $\triangle OPD \sim \triangle ADC$ $AD=2OD \...
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1answer
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Find the area of ​the triangular region $ABC$ in the figure bellow

For reference: In the figure below, the quadrilaterals $AMNB, AEFC$ and $APQR$ are squares; $S_1=9\ \mathrm{m^2}$, $S_2(BRE)=6\ \mathrm{m^2}$, calculate the area of ​​the triangular region $ABC$. (...
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2answers
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Find the area of triangle $RAQ$ in the figure below

For reference: Through the midpoint $M$ of the side $AC$ of a triangle $ABC$, a line is drawn that intersects in $BC$ in $P$ and extended $BA$ in $Q$. By $B$, a line parallel to $PQ$ is drawn which ...
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1answer
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$f(x)=3x-\sin(\frac{\pi x}{2})$, $\quad$ $g(x)=x^3+2x-\sin(\frac{\pi x}{2})$

$f:\mathbb R \rightarrow\mathbb R$ and $g:\mathbb R \rightarrow\mathbb R$ are two functions such that $f(x)=3x-\sin(\frac{\pi x}{2})$, $\quad$ $g(x)=x^3+2x-\sin(\frac{\pi x}{2})$ then choose the ...
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1answer
44 views

Find the area of ​the non-convex pentagonal region AEIDCA.

For reference: In triangle $ABC$, the interior angle bisectors $AD$ and $CE$ are drwan which intersect at $I$. If $\angle B = 60^\circ$ and the area of triangular region $AIC$ is $A$, calculate the ...
2
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1answer
84 views

Find the area of ​the shaded region: in the figure

For reference: In the figure, calculate the area of ​​the shaded region if $ML=LN, PM=a, NQ=b, AC=c.$ (Answer: $\frac{(a+b)c}{4}$) My progress: Let $ML=LN=BL=x.$ $S_{ALC} = S = S_{APQC}-S_{ALP}-S_{...
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1answer
54 views

Find the area of ​the triangle EGB.

Through the centroid $G$ of a triangle $ABC$, a secant line is drawn, cutting sides $AB$ and $BC$. On the straight line a point $E$ is considered. Calculate the area of ​​the triangle $EBG$ if the ...
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1answer
69 views

Find the area $ S_ {OGBH}$ in the figure below

For reference: In the figure, calculate area $ S_ {OGBH} $ if the triangle area $ABC=9\ \mathrm{m^2}$ and $AB=3\ \mathrm m$ and $AO = 2\ \mathrm m$. (Answer: $5\ \mathrm{m^2}$) My progress: Draw $BO....
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1answer
55 views

Find the ratio of the areas of the triangles: $MPA$ and $PBC$

For reference: The circle inscribed with a triangle ABC is tangent to $AB$ in $M$ ​​and to $BC$ in $N$. The prolongations of $NM$ and $CA$ intersect at $P$. Calculate the ratio of the areas of the ...
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1answer
52 views

Is there a closed form for this definite integral? [duplicate]

It is well-known that $$\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.$$ I noticed that $y = \frac{1}{x^n + 1}$ for even $n$ has a similar shape to the bell curve. Let $$A(n) = \int_{-\infty}^\infty ...
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0answers
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Formula for calculating estimated annual revenue based on month over month growth, total months, and total value.

Background Information I was part of a meeting today where I saw someone build out a table with incorrect data. I spent way too much time coming up with what the correct values should have been and I'...
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1answer
47 views

Find the area of ​the trapezoidal surface $ABCD$

For reference: In the figure shown, calculate the area of ​​the trapezoidal surface $ABCD$, if $\overset{\LARGE{\frown}}{CD} = 150^o$and $BM=MA$. (Answer:$R^2$) Let $R = 1, AM=BM = x:$ $\sin(15°)=\...
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1answer
78 views

Find the area ratio of the $ABNM$ and $ABCD$ trapezoids

For reference: In the trapezoid $ABCD$ ($AB \parallel CD$), $MN$ is the median. $NP$ is traced parallel to $AD$ ($P \in CD$). The area of ​​$MNCD$ is $16\ \mathrm{m^2}$ and that of the triangle $NCP$ ...
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2answers
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Is this formula for the area of a triangle worth being published? $\frac{(a^2+b^2-c^2)^2\sec C\tan C}{8ab}$

I created this trigonometric formula for finding the area of oblique triangles: $$R=\frac{(a^2+b^2-c^2)^2\sec C\tan C}{8ab}$$ where $a, b$ and $c$ are sides of the triangle, angle $C$ is the angle ...
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1answer
57 views

Area of a rectangle inscribed in a semicircle as a function of $x$

The question given is, The figure shows a rectangle with two vertices on a semicircle of radius 2 and two vertices on the x-axis. Let $P(x, y)$ be the vertex that lies in the first quadrant. (a) ...
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0answers
60 views

How to calculate or estimate the area of such implicit region about function $x^{\frac{1}{x+\frac{1}{x}}}$

How to estimate or calculate the area enclosed by the implicit equation. $$x^{\frac{1}{x+\frac{1}{x}}}+y^{\frac{1}{y+\frac{1}{y}}}=\mathrm{e}$$ It is possible to prove that the area is between $13$ ...
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1answer
22 views

The area bounded by the tangential lines to an even-power exponential function

The problem: Consider a function, $f(x) = x^m$, where $m$ is an even, natural number. Then, consider two tangential lines, grazing the points $(-x,y)$ and $(x,y)$, respectively. My question is, what ...
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1answer
70 views

How to find the ratio of area in increasing order for the given figure?

Consider the attached image for the diagram. I'm interested in finding the area of the 4 parts so formed as the ratio of increasing order. Here line segment BD is diagonal for square and AE is so-...
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3answers
125 views

Deriving surface area of a sphere using triangles

I was trying to derive surface area of a sphere myself. I started with a hemi-sphere, sliced it into infinite triangles and then added the area of all the triangles and finally doubled it to get area ...
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0answers
40 views

Question on a similarity I noticed between two unrelated things

I noticed a similarity between two unrelated things that I am trying to look into. To start I will say that I am using my own notation in one part, $A(n | a)$ means the area of a regular, n-sided ...
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0answers
30 views

Legendre's formula for the surface area of scalene ellipsoid

I am looking for a derivation of the formula $$S=2\pi c^2 +\frac{2\pi a b}{\sin \varphi}\left(E(\varphi, k)\sin ^2 \varphi +F(\varphi,k)\cos^2 \varphi\right)$$ for the surface area of the ellipsoid $$\...
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1answer
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Length of EF using area of square.

Please find the length of EF.Given that area of the square is 100 cm².Here is Diagram I tried trigonometry but I can't solve it.
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1answer
42 views

Area expressions in terms of its dimensions

(This question is about area, but the same thing fully applies to volume) Whenever I was finding an area of a shape / geometric figure - triangle, square, circle, ellipse etc - I always had this ...
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1answer
51 views

How does one define area on the complex plane?

I have been thinking, that due to how functions and numbers can go onto the complex plane, and circles can be inscribed over that, such as that shown by ei * a, and it leaves me wondering how exactly ...
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2answers
83 views

How to generalize the area of a kite inscribed within a circle

How exactly would one generalize the area of a kite inscribed within a circle? Through a lot of calculation, which I do think was actually way more complicated then required, using various ...
3
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0answers
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Is the following cartoon mathematically correct?

I saw the following cartoon about probability and statistics the other day: The joke in the above picture being that "the top 1%" (i.e. the integral of the gaussian probability distribution ...
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0answers
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A formula for the area of a triangle defined using 6 distinct points on a parabola

(Note: I have heavily edited this post and I am now asking 1 main question using the same setup.) Let $\gamma: \mathbb{R} \to \mathbb{R}^2$ be the following smooth embedding $$ \gamma(x) = (x, x^2).$$ ...
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1answer
143 views

Is there a geometric proof of this geometric interpretation of the Vandermonde determinant formula?

Let $\Gamma$ be the graph of the parabola $y = x^2$ in the $xy$-plane. Let $\gamma: \mathbb{R} \to \mathbb{R}^2$ be the map $$ x \mapsto (x, x^2). $$ It is actually a diffeomorphism from $\mathbb{R}$ ...
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2answers
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I managed to solve this uniform distribution problem and now i am having difilculties with calculating area on graph

So the task says: There are 2 variables that get random variables from 0 to 1. What is the distribution function for z that represents distance between the two variables. Lets say first variable is x ...
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2answers
122 views

Determine whether a polyomino has a hole

Suppose I know the $(x,y)$ coordinates of the corners of all the unit squares making up a connected polyomino. Is there a simple/elegant way to determine if the polyomino has a hole in it, merely by ...
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2answers
65 views

Derivative of a right triangle's area with respect to its height

I want to find the first derivative of the area of a right triangle as its non-hypotenuse sides change as a function of a third variable. I try it two different ways and get two different answers. ...
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2answers
208 views

What makes one integral converge and a similar integral diverge, e.g., $\int\limits_1^\infty\frac1x dx$ vs $\int\limits_1^\infty \frac1{x^2}dx$? [duplicate]

The area under $\dfrac 1x$ curve within $[1, \infty)$ is considered to be infinite but for $\dfrac 1{x^2}$ curve, it is $1$. Can someone please explain this to me intuitively?
3
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1answer
34 views

Equation to calculate the cap area of an oblate spheroid

I am trying to write a code that calculates the cap area of prolate and oblate spheroids, while avoiding integrals. Through this online calculator I got the equation for a prolate spheroid (i.e. c >...
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1answer
40 views

Finding area between a curve and a line using double integrals?

We are asked to find the area between the circle $x^2 + y^2 = 4$ and the line $x + y = 2$ lying in the first quadrant. One method is to find the area by integrating $2-x$ from $0$ to $2$ and ...
4
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0answers
126 views

Finding the rectangle of minimum area that can be divided into five rectangles such that the lengths of all their sides are different naturals.

Find the rectangle of minimum area that can be divided into five rectangles such that the lengths of all their sides are different natural numbers. I found a 11x11 rectangle such that when we divide ...
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0answers
22 views

Can I compute contour orientation without using polygon area sign?

Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
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0answers
47 views

Volume and surface area of a solid in cylindrical coordinates

Suppose that initially I have a right triangle in the $xy$ plane given by the points $a=(0,0)$; $b=(0,B)$ and $c=(B,B)$, such as shown below: Since from now on, I will work with polar coordinates, I ...
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0answers
20 views

Using iterated integrals to get the area with strict conditions

We were tasked to use iterated integrals over this area shaded in yellow here: I'm confused as to how to approach this as I've only seen ones that encompass the whole space inside the closed shape ...
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0answers
65 views

Finding the area of the largest rectangle bounded by a line and a parabola.

Question: Find the area of the largest rectangle contained within the region defined as $y \geq 2x^2$ and $y \leq 1-x$. Attempts: I attempted to create a line parallel to $y = 1-x$ by taking $y = a-x$ ...
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0answers
63 views

Max Area Under the Curve with $y = \sqrt{6-x}$

I see a lot of examples without radicals for $y$, but I'm not finding the solution to the answer for this one at all. Although I know the answer is supposed to be $4\sqrt 2$. Consider a rectangle in ...
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0answers
29 views

Calculation of the area of the section between a plane and torus

Once I have obtained the curve of the intersection between a horizontal plane and a vertical torus (torus generated by rotation around the $x$-axis), I wish to calculate the area of such intersection, ...
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1answer
68 views

Algorithm for fitting circles into a square

I have the following question. I want to write some code that solves the following problem. Suppose we have a square of 1x1 meters and I want to fill them with circles of 0.22m diameter in a pyramid ...
1
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1answer
65 views

Help with understanding how to find the area of part of sphere that lies inside a cylinder

I really have a hard time understanding how to solve problems where you have two surfaces intersecting. The problem I'm stuck on now is the following: Find the area of the part of the sphere $x^2+y^2+...

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