# 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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### Area under the graph of $r\mapsto\binom nr$

The question: Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $r\mapsto\binom nr$, taking the $\Gamma$-function definition of factorial. ...
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### Why is the surface area of a sphere equal to $4\pi r^2$ [duplicate]

I have absolutely no idea where that formula comes from, considering the fact that I am a fifteen year old. According to me, one way to think of it is to arrange $4$ circles having radius equal to ...
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### Area of triangle using double integrals

I have one (rather simple) problem, but I'm stuck and can't figure out what I'm constantly doing wrong. I need to calculate area of triangle with points at $(0,0)$, $(t,0)$, $(t,\frac{t}{2})$. In ...
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### Area of a triangle inside an ellipse

$F_1$, $F_2$ are are foci of the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. $P$ is a point on the ellipse such that $|PF_1|:|PF_2|=2:1\;$, then how could I figure out of the area of $∆PF_1F_2$? As ...
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### Express the distance two points and optimize the area of a triangle.

Consider the parabola $\mathcal{P}$ of equation $\;y = x^2,$ and the line $L$ of equation $y = x+6.$ Let $P(x_P,y_P)$ be a point on the arc of the parabola $\mathcal{P}$ below $L.$ Let $A$ and $B$ be ...
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### $I = \int_{- \infty}^{\infty} \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x}$ should not diverge

I'm having trouble evaluating this integral: $$I = \int_{-\infty}^\infty \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x}$$ where $\delta(x)$ is Dirac delta function, $\mathbf x$ is the real $n$-...
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### Visualizing the area described by the dot product?

Since the dot product of two vectors is an area (if your vectors have units of meters, then the dot product would be in m$^2$), I was wondering if there is a good way to visualize that area. The ...
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### A way to find this shaded area without calculus?

This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600) $ABCD$ is a square with a side of $10$, $APD$ and $CPD$ are ...
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### Area bounded by parabola and line.

The problem is stated as "Find m such that the area of the region bounded by y = mx and y = x^2 - 1 is equal to 36." I tried solving it by systems of equations: ...
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### Calculating square meter area with polygonal geographical coordinates (metric - not DMS system)

I'm working on a program but my problem is not on software side but mathematical. I have the following input : ...
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### Surface Area from a definite integral equation

To find a surface area we need a function, but how to find the surface area if you are given a definite integral question?
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### If $p,q,r$ be lengths of perpendiculars from vertices of triangle $ABC$ on any line, prove $a^2(p-q)(p-r)+b^2(q-r)(q-p)+c^2(r-p)(r-q)=4\Delta^2$

Let : $$A:=(x_1,y_1),$$ $$B:=(x_2,y_2),$$ $$C:=(x_3,y_3)$$ be the vertices of the triangle $ABC$. Consider an arbitrary straight line in perpendicular form $x\cos \theta + y\sin \theta - t = 0$. Then ...
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### Finding area of a triangle with integration

I have a triangle with coordinates (0,0), (1,2) and (1,0). Is the area of this triangle same as finding the integral of the function $y=2x^2$ and substituting the value of x=1 and y=2? Because what i ...
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### How to find area of a triangle by slicing it into squares

First, I tried making a R-angled triangle with base b, height h I want to slice it into squares [not lines] like And as I understand, sum of those square areas = Area of whole triangle How can I ...
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### Calculus Optimization Problem: Wire Triangle and Circle

A wire 5 meters long is cut into two pieces. One piece is bent into a equilateral triangle for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To ...
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### Area between the curves of $2\cos(x)$ and $x/2$

I'm trying to obtain the area between the curve of these two functions (for $x>0$), lets call them $f(x)=2\cos(x)$ and $g(x)=x/2$ and my idea is to get the area under the curve of $f(x)$, then ...
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### How to find the intersection point between 2cosx and x/2

I'm trying to find the solution to this because I need to find the area between the curves, but I need this intersection point to properly subtract the unnecessary parts. I know how to do it with ...
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### What proportion of the quarter circle is shaded?

Interesting yet challenging quiz I found on a website. My answer is a $\frac{1}{ \sqrt{2}}$. After I assumed the semicircle has radius $r\sin{45}$, where $r$ is the radius of the quarter circular ...
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### parametrise $x^2+y^2+(x+y)^2=4$

I am trying to evaluate the double integral $$\iint_D 3 \, dA$$ , where $D=\{(x,y)\in \mathbb{R^2}: x^2+y^2+(x+y)^2\le 4\}$. I need help with a suitable parametrisation for this.
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### How do I find the shaded area?

This is how it looks like: It is given that the area of the shaded region is $35 cm^2$. All of my attempts so far ended up in a two-variable equation in terms of $r_1$ and $r_2$ (the radii of the ...
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### In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC. [duplicate]

In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC, let MD and AN intersect each other at the point Q and let MC and BN intersect each other at the point R. ...
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### Can area of rectangle be greater than the square of its diagonal?

Q: A wall, rectangular in shape, has a perimeter of 72 m. If the length of its diagonal is 18 m, what is the area of the wall ? The answer given to me is area of 486 m2. This is the explanation given ...
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### Shaded Area under square inscribed in a Circle.

Check this Question please I have tried solving this question by first finding the Area of circle and then area of square (via diagonal method). and then subtracted Its value from the total area But ...
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### Given big rectangle of size $x, y$, count sum of areas of smaller rectangles.

Let's say we have two integers $x$ and $y$ that describe one rectangle, if this rectangle is splitten in exactly $x\cdot y$ squares, each of size $1\cdot 1$, count the sum of areas of all rectangles ...
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### Apostol Calculus, Method of Exhaustion

In Apostol's Calculus, he goes through the method of exhaustion to find the area under a parabola from $0 \ to\ b$. Using the fact that, \begin{align} &1^2+2^2+...+(n-1)^2 < \frac{n^3}{3} &...
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### $M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$. [closed]

$M$ is a point in an equilateral triangle $ABC$ with the area $S$. Prove that $MA, MB, MC$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1}{3}S$$
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### Surface Integral of Sphere between 2 Parallel Planes

A circular cylinder radius $r$ is circumscribed about a sphere of radius $r$ so that the cylinder is tangent to the sphere along the equator. Two planes each perpendicular to the axis of the cylinder, ...
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### Find the area bounded by $x^2y^2+y^4-x^2-5y^2+4=0.$

Find the area bounded by $x^2y^2+y^4-x^2-5y^2+4=0.$ I reduced the above equation to $y^2=\frac{x^2-4}{x^2+y^2-5}$ but i am not able to solve further.
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### Find the area of an ellipse inside a circle [closed]

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter. ...
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### Circle segment areas

Is my working out here correct? If not what am I supposed to do? Thanks. 25/2(1.2-sin(1.2)) +72(0.3-sin(0.3)) https://postimg.cc/JyTpchKV
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### How to prove a smooth curve/surface (first derivative continues) has zero area/volume

If a curve is defined as $\begin{cases}x=f_x(t)\\y=f_y(t)\\\end{cases},t\in[a,b]$ Smooth are defined as $f_x'(t)+f_y'(t)\ne0$ and $\lim_{t\to t_0}f_x'(t)=f_x'(t_0),\lim_{t\to t_0}f_y'(t)=f_y'(t_0)$ ...
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### How to find the length of one of the sides of a triangle given the area

The triangles are drawn to scale. The first triangle has side lengths of 17, 17, 16 while the second triangle has side lengths of 17,17,$k$. The triangles have the same area. Find the value of $k$ ...
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### Area of the intersection between a sphere and a cone (located in the center of the sphere)

Please, how do I calculate the area of the intersection between a sphere and a cone, as shown in the image below? The beginning of the cone is located in the center of the sphere, and both geometric ...
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### Why is the area of a $3$-by-$3$ square $3\times 3$ and not $3+3$?

When we need to find the area of a square, we multiply the sides. For example, the area of a square with one side as $3$ cm will be $3\times 3 \text{ cm}^2$. My question is: What is the logic behind ...
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### What's the surface area of a Klein bottle?

I am creating a 3D model of a Klein Bottle based on the Robert Israel formula: Then I need to apply algorithms on the model and I need to know the surface area of this 3D model, then what's the ...
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### Area of a trapezoid with perpendicular diagonals, embedded in a triangle

Let $ABED$ be a trapezoid as in If $AB \parallel DE$, $AE \perp BD$, $AB = 10$, $DE = 4$ and $\angle ACB = 45°$, what's the area of $ABED$? I must also mention that this is for an elementary ...
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### Why is this way of deriving surface area of sphere wrong when a similar method can be used to derive volume?

Suppose a sphere with radius R is centered at the origin, whose cross section is as follows (R is the constant radius while r is variable): then its volume can be easily calculated: \$V=\int_{-R}^{R}\...