Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

Filter by
Sorted by
Tagged with
2
votes
1answer
34 views

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. ...
3
votes
1answer
77 views

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 ...
2
votes
2answers
83 views

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 ...
4
votes
3answers
645 views

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 ...
0
votes
0answers
20 views

Green’s theorem double into line integral

How I can convert this double integral into line $\int \int \frac{dx dy}{y^2} $ we have $\frac{dQ}{dx}-\frac{dP}{dy}=\frac{1}{y^2}$ how to find $Q, P$ functions? Any two functions works? $P=1, Q=\frac{...
3
votes
3answers
505 views

Is it possible to integrate something that isn't a function?

How would you "find the area under the curve" of something that isn't quite a curve? The graph may be curved at places, but if it's not a function (the same x value has more than one corresponding y ...
0
votes
1answer
40 views

Area and Perimeter GCSE Exam Question. Help Please..

I need help on this Area and Perimeter Question: What I did: 11 * 7 - 10* 6 - 77-60 = 17 Thank You and Help is Appreciated
0
votes
0answers
45 views

Finding the Polar Area of two circles intersecting each other

The equations for the two circles are: $r=18\cos{\theta}$ $r=3$ $r_f= 9$ and $r_g=3$ I can see that I need to subtract the $r=3$ circle, however im not sure on how to get the boundaries of ...
0
votes
1answer
63 views

Error in cross sectional area of a cylinder, given circumference is $8$ feet $\pm 1$ inch

Given a cylinder with the circumference of $8$ feet $\pm 1$ inch, how could the error in the cross sectional area be found? As per request: Given that the area of the cross section is a circle, $A=\...
0
votes
1answer
290 views

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 ...
4
votes
2answers
129 views

$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 $n$-...
1
vote
1answer
90 views

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 ...
52
votes
8answers
10k views

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 ...
1
vote
1answer
32 views

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: ...
1
vote
1answer
32 views

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 : ...
0
votes
0answers
33 views

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?
5
votes
2answers
106 views

In the square $ABCD$, prove that $BF+DE=AE$.

Consider the Square $ABCD$. The point $E$ is on the side $CD$. If $F$ is on the side $BC$ such that $AF$ is the bisector of the angle $BAE$. Prove that $$BF+DE=AE.$$ By Pythagorean theorem we have $(...
0
votes
0answers
79 views

Finding the area of a polygon with complex numbers.

If we consider a regular polygon defined by the $n$-tuple $Z=(z_0, z_1, \dots, z_{n-1})$, and set $$A(Z)=\frac{1}{2} \Im \left ( \sum_{k=0}^{n-1} \bar{z_k} z_{k+1} \right )$$ Given that the number $...
2
votes
1answer
80 views

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 ...
0
votes
1answer
184 views

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 ...
0
votes
0answers
42 views

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 ...
0
votes
2answers
314 views

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 ...
1
vote
2answers
55 views

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 ...
0
votes
1answer
58 views

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 ...
1
vote
2answers
97 views

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 ...
0
votes
1answer
55 views

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.
0
votes
1answer
68 views

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 ...
0
votes
0answers
17 views

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. ...
48
votes
12answers
6k views

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 ...
2
votes
1answer
121 views

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 ...
2
votes
2answers
37 views

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 ...
0
votes
0answers
65 views

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} &...
3
votes
3answers
113 views

$M$ 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 $$S'\leq \frac{1}{3}S$$
0
votes
1answer
95 views

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, ...
0
votes
2answers
94 views

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.
1
vote
1answer
60 views

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. ...
0
votes
0answers
17 views

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
0
votes
1answer
32 views

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 $\lim_{t\to t_0}f_x'(t)=f_x'(t_0),\lim_{t\to t_0}f_y'(t)=f_y'(t_0)$ ...
4
votes
2answers
100 views

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 $k$ ...
0
votes
0answers
26 views

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 ...
1
vote
2answers
94 views

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 ...
2
votes
4answers
156 views

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC|= |\triangle PAC|$.

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC| = |\triangle PAC|$. Here, $|\triangle XYZ|$ denotes the area of $\triangle XYZ$. I've tried drawing it up, and $...
0
votes
1answer
51 views

Finding the surface area of a solid of revolution

I'm given the function $x=\frac{1}{15}(y^2+10)^{3/2}$ and I need to find the area of the solid of revolution obtained by rotating the function from $y=2$ to $y=4$ about the $x-axis$. I've tried ...
0
votes
2answers
150 views

Find the length of x for an arrow head

I am struggling to get some traction on this and need someone to show me how to calculate this. I have a GCSE question that asks: In the attached image I'm being asked the following: The arrowhead ...
0
votes
0answers
21 views

Functions of perimeter and area of parts from two circles

We have a circle with radius $1$. Next we draw a circle with radius $r$, whose center lay on the first circle, so $0\leqslant r\leqslant2$. When $r=0$ there is only $1$ part with $$C(0)=2\pi, S(0)=\pi$...
2
votes
0answers
174 views

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 ...
2
votes
3answers
129 views

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 ...
2
votes
2answers
52 views

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}\...
-3
votes
1answer
32 views

Pizza Problem, percents

A medium size pizza at Ristorante Porcupine is 10 inches in diameter. A large pizza is 21 inches in diameter. What percent larger is the area of a large pizza? Express your answer to the nearest ...
0
votes
1answer
45 views

How to bisect a quarter of a cheese to get minimal surface on the two halves?

The cheese is kind of "trappista cheese" ( https://en.wikipedia.org/wiki/Trappista_cheese ), which has a shape of "circle sect based prism". For simplicity it has a height equal to the radius of the ...