Questions tagged [area]

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

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Area of a decreasing positive function

Suppose we have a decreasing positive function from 0 to infinity. Now suppose that it starts from somewhere on the y axis, something like f(0)=a, where a is a positive real. Can we say that the area ...
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Find the colored area, tha triangle is a regular one [closed]

The task is to find the colored area, expressed with x (as a function of x). The triangle has a side length named x. We drew its altitudes and we circled from each corner with two different radius. ...
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How to solve the problems in which wire is cut into 2 parts

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=r$ units. If the sum of the areas of the square and the circle ...
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Area bounded by $2 \leq|x+3 y|+|x-y| \leq 4$

Find the area of the region bounded by $$2 \leq|x+3 y|+|x-y| \leq 4$$ I tried taking four cases which are: $$x+3y \geq 0, x-y \geq 0$$ $$x+3y \geq 0, x-y \leq 0$$ $$x+3y \leq 0, x-y \geq 0$$ $$x+3y \...
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Find the area of region between the $x$-axis and the graph of $f(x)=x^3-x^2-2x$, for $-1\le x\le2$

The final answer I have from calculation is $37/12$ sq. units. The final answer was in negative, but because area can't be negative so I solved it making positive. Can someone help me in getting the ...
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5answers
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Find the area enclosed by $\sqrt{(x-2)^2+(y-3)^2} + 2\sqrt{(x-3)^2+(y-1)^2} = 4$

Question: What is the area of the interior of the simple closed curve described by the equation $\sqrt{(x-2)^2+(y-3)^2} + 2\sqrt{(x-3)^2+(y-1)^2} = 4$? Comments: I came up with this specific ...
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Closed Line integral of a scalar field

If $f(\mathbf r)$ is a scalar field, then line integral of this function is a cross-sectional area (the shaded part in the picture) bounded by $f(\mathbf r)$ (in blue) & curve $C$ ( curve is in ...
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1answer
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Finding the Area of a Piecewise Function

I'm learning applications of integration and this is my first experience with this sort of question. How would I get started on a problem like this?
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1answer
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Find the area of this pentagon

Let $BCDK$ be a convex quadrilateral with $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $AB=BC$, $DE=DC$ and such that $ABCDE$ is a convex pentagon. Point $K$ lies in the interior of pentagon $...
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Need to find the ellipse of maximum area inscribed in a semicircle.

An ellipse inscribed in a fixed semi circle touches the semi-circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. ...
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How to show that $A$ is increasing?

Suppose that $f$ is a twice differentiable real function such that $f''(x)>0$ for all $x\in[a,b]$. Find all numbers $c\in[a,b]$ at which the area between the graph $y=f(x)$, the tangent to the ...
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Find $\int_{\left(C\right)}xy{\rm d}x+y^{2}{\rm d}y$ with $\left(C\right)$ bound by $y\geq 0,x^{2}+y^{2}=4\left({\rm clockwise}\right).$

Prob. Find $\int_{\left ( C \right )}xy{\rm d}x+ y^{2}{\rm d}y$ with $\left ( C \right )$ closed by the path $y\geq 0, x^{2}+ y^{2}= 4\left ( {\rm clockwise} \right ).$ My attempt: $\int_{\left ( C \...
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Find the slope of the line go through a point such that the area between the graph and the line is minimum

If $a$ is the slope of $(L)$ that go through point $(-1,2)$ and $f(x)=x^2$ then find $a$ such that the area between the graph and the line is minimum. For what I thought, the area is minimum if the ...
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How to compute the area of a $k$ dimensional sub-manifold in a $n$ dimensional manifold? [closed]

How to compute the area of a $k$ dimensional sub-manifold in a $n$ dimensional manifold? Given a manifold $(M,g)$, where $g$ is the Riemannian metric. If $S$ is a $k$-dimensional sub-manifold of $M$, ...
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Find area of a parallelogram with vertex at the origin and with the given vectors as edges

Find area of a parallelogram with vertex at the origin and with the given vectors as edges: $i +3j - 5k$ and $i +7j - k.$ So matrixes for me take the longest time to put in LaTeX so im just asking if ...
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1answer
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Finding the ellipsoid of maximum area inside a square - and extension to $\mathbb{R}^n$

In an answer here, I read that: Take a max area ellipse. Apply an affine transform to make it a circle; then the problem becomes to show that a minimal area parallelogram containing a circle is a ...
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2answers
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parametrization of surfaces and area

The question is: Paraboloid $z=x^2+y^2 $ divides the sphere $x^2+y^2+z^2=1$ into two parts, calculate the area of each of these surfaces. I know that i need to use $\iint |n| dS$ but i need to first ...
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1answer
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Finding area of quadrilateral inscribed inside of a semicircle

I've been practicing problems recently to study for the upcoming AMC10, and came across one I could not figure out how to solve one of them. The diagram of it is attached below, and the problem goes a ...
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1answer
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Hunt for trapezoid area & arithmetic progression sum formula similarities.

When I was looking into Arithmetic progression sum formula, I found out that it is similar to Trapezoid are formula. Arithmetic ...
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2answers
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Partition the remaining rectangle into equal parts.

There’s a rectangle. A small rectangle is cut from the bigger rectangle ( not necessarily from the center). How will you partition the original rectangle after removing the cut such that the remaining ...
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25 views

Area of a square with an arc passing through it

Assuming the the blacks and blue arcs are quarter circles, how would I go about finding the region A+B? Region A was pretty easy to find, as it's just the area of a circle with radius $r$ divided by $...
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Using jacobian to transform area elements from one system to another.

Please excuse my lack of rigor,I'm just average Physics undergraduate. I have a transformation from $(u, v)$ to $(x, y)$. So an infinitesimal area element from $ (u, v)$ to $(x, y)$ plane will ...
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Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
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3answers
217 views

Area of a region bound by three circular arcs, why doesn't this approach work?

Three circular arcs of radius $5$ units bound the region shown. Arcs AB and AD are quarter-circles, and arc BCD is a semicircle. I tried to find the answer by calculating the total area of the circle ...
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1answer
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A question about the polar equation $r=ln(\theta)$

Take the polar equation r=ln(theta) for theta between $0$ and $2\pi$. 1 In this, we have a loop that connects at a point $(x,y)$ for $2$ unique points $(r_1,\theta_1)$ and ($r_2,\theta_2)$. There are $...
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1answer
44 views

Area of metric ball on n-sphere

Suppose $S_n = \{x \in \mathbb{R}^{n + 1} : ||x||_2 = 1\}$ is the $n$-sphere. Let $d : S_n^2 \to \mathbb{R}$ be the angle metric on $S_n$, i.e. $d(x, y) = \arccos(x \cdot y)$, where $\cdot$ is the dot ...
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Differential area for the lateral surface of frustum of a cone

I am studying Fluid Mechanics and I needed a differential area element of the side or lateral surface of a frustum. This frustum is cut from a cone. In solution manual of the book I study, ...
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Calculate area of a circle

\begin{align} π &= 3.1415\dotsc\\ s&= πr^2 \end{align} Because $π$ infinitely continues, does it means $s$ is not ever a right answer, does it mean we do not know exact $s$ of a circle? Is ...
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The area of the faces of a right rectangular prism are 24, 32, and 48 square centimeters. What is the volume of the prism?

Can someone show me their work, and not just the answer? I need to learn how to do this, and showing work would be greatly appreciated.
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How to calculate the Area of a “Polygon” including Arcs

I have a set of points forming a polygon. However, any 3 points in this polygon can also be represented as an arc (starting at point 1, through point 2, to point 3). I need to find the area of this ...
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5answers
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Area of square inside a triangle [closed]

Considering the attached image, please compute the area of the square. $\\$To elaborate suppose we have a triangle $\Delta ABC$, that its angle $\angle ABC$ is equal to $45^o$. On side $\overline{AB}$ ...
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1answer
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How to find area between curves? ($\frac{1}{x}$ problem)

I need to find the area between those curves: $f(x)=\frac{1}{x}$, $f(x)=6e^x$, $f(x)=1$, $f(x)=6$ I calculated the limits of integration, it is $\ln(\frac{1}{6})$ to $1$. But we cannot integrate ...
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Calculating areas in a square

Enclosed within a square of side A is another smaller square of side B. The center C of side square B is off-centered with coordinates (p,q); OC makes $\alpha$ to x-axis. How to find areas enclosed ...
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1answer
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Analytical approach to find area of parallelogram

I know that to find the area of parallelogram, we have to either find the cross product of its adjacent sides or half the cross product of its diagonals. However I've encountered a question to find ...
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366 views

Area of a circle smaller than one?

If a circle has radius less than one, does it mean that the circumference of the circle is bigger than the area? How can that even be possible? For example, if a circle has radius $0.5$, then the ...
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1answer
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How would I go about solving for the grayed out area?

I was asked to get the grayed out area that you see in this image. All that I'm given to solve this is the three points of the main triangle (A, B, C) and the center of the circle (X) Also the dashed ...
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1answer
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Area between parabolas

I would like to calculate the area between these 2 curves. As you can see the first one is $x^2 = αy$ and the second one is $y^2 = 2αx$ What I tried is to solve the equation when the two curves are ...
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1answer
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How would one approximate the surface area of a curved shape as on these ETFE pillows?

I need to get the surface area of these ETFE pillows:
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1answer
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Is there a formula for calculating the area of 2d shapes on a sphere?

Let's say I have 8 90° triangles on a sphere, like this, where all the angles are 90° when measured: I know that the area of one of those triangles will be (4πr2) * 1/8 as each triangle will take up ...
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Auto Arrangement and Positioning of nested containers

We are trying to solve a problem of auto-layout or auto-arrangements of containers. The containers can be in a hierarchy and can be defined by the rules below: All shapes are containers - Rectangle ...
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1answer
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Did Euclid prove the formula for the area of a triangle?

In Proposition 6.23 of Euclid’s Elements, Euclid proves a result which in modern language says that the area of a parallelogram is equal to base times height. Now Euclid did not have the concept of ...
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Which shapes could have the largest and lowest possible area?

I'm really new to the world of pure mathematics and proofs. I was recently watching a playlist made by Bill Shilito (in YouTube) called Introduction to Higher Mathematics. The first example given is: ...
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Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer $$s=\sqrt{p(p-a)(...
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1answer
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Surface area of quarter of a Sphere

A quarter sphere with a radius of $10 \text{ units}$. Please help, also remember the sides. I used the normal formula of the total surface area of a sphere and divided it by $4$, then added half the ...
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4answers
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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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Illusion in maths?? (Area of sectors and segments) [closed]

I was practising for my maths Olympiad when I stumbled upon this question... This question is more than just a figure. It actually isn't what it looks like. I could've simply seen the solution at the ...
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1answer
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Surface Area of Cap below, Top Part of Sphere

How do I find the Surface Area of an oval cap below (Top of sphere)? Oval Cape Internet is stating: 2πRh However, a simple 8 radius circle is ...
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3answers
392 views

What is area of shaded region?

Suppose the side length $a$ of the square is 10mm. A circle is tangent to all four sides of the square. And two quarter-circles with the same radius of 10mm have centers on the opposite vertices. It ...
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16k 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 ...
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Calculating the area under the curve $e^x$ without calculus

I was thinking about the integral $$\int_{0}^{1}e^xdx$$ This is a problem that can be solved in seconds using calculus: the answer is $e-1$. But, I was wondering, is there a way of solving this ...

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