Questions tagged [area]

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

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22 views

How to calculate area of a curved rectangle. [closed]

I need help with this task. It's a rectangle with two quarter-circles cut out. Nothing is known about the curves. a=30 b=20 r1=5 r2=10 Is it possible, to calculate ...
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Find the value of constant a such that triangle ABC has the smallest area [closed]

I'm not sure how to go about solving this problem here. Any help would be appreciated. Thank you! $$\text{Let A(4, a, −1), B(2, 2, a) and C(1, −2, a) be three points in } \mathbb{R}^3\text{.}\\ \...
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Minimum area of whole quadrilateral given areas of parts

I've attempted to mark segment $AO$ as $a$ and $CO$ as $b$. Now, if draw a line from $D$ that is perpendicular to $AC$, and draw another line from B that is perpendicular to $AC$, then we can use some ...
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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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Average number of holes supporting a puck on an air hockey table

How would I go about finding the average number of holes supporting a puck on an air hockey table? I'm trying to find a generalized equation using the distance between holes, the area of the puck, and ...
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Is this a valid way of deriving the area of a circle?

On the Wikipedia article about deriving the area of a circle, it mentions that the formula $$ \text{area} = \pi r^2 $$ can be derived by evaluating the integral $$ 2 \int_{-r}^{r} \sqrt{r^2-x^2} \, dx ...
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Need help understanding computation of area and volume using Matrices and linear algebra?

I am currently studying linear algebra from Gilbert R Strang's book. I have a few questions regarding computing area using matrices. In the book, for a triangle with coordinates: $(x_1, y_1)$, $(x_2, ...
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Find the volume of the solid generated by revolving the triangular region enclosed by y = |x| and y = 1 about the line x = −2.

Question: Find the volume of the solid generated by revolving the triangular region enclosed by y = |x| and y = 1 about the line x = −2. My solution: pi91 - pi11 - 1/3 pi 1 - 1/3 pi 1 = 22/3 pi Use ...
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Find the area of the region enclosed by the curves $y = \tan(x)$, $y = -3\tan(x)$ and $x = \pi/4$

Find the area enclosed by the curves $y = \tan(x)$, $y = -3\tan(x)$ and $x = \pi/4$. What can possibly be the area, $\tan(x)$ and $-3\tan(x)$ never intersect unless it is $\frac{\pi}{2}k$. There is no ...
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Area of a cyclic quadrilateral.

Question : The distance $SR$ from $PQ$ is 7cm and arc $SR$ is 48cm and arc $SP \cong$ arc $QR$. Then find the area of quadrilateral $SRQP$($PQRS$ are taken in order and $O$ is centre). What we(me ...
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What is the cross-sectional area of a cupcake liner? [closed]

The shape is shown here including the dimensions.
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5-set scaled Venn diagram

Let's assume that we have $5$ sets, each having up to $100$ elements. Some elements belong to multiple sets. Even more, there is at least one element for each of the $2^5=32$ configurations (by ...
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In $\triangle ABC$ where $AB = AC$, $D$ and $E$ are points on $AB$ and $AC$ respectively, such that $AB = 4BD$ and $AC = 4AE$.

In $\triangle ABC$ where $AB = AC$, $D$ and $E$ are points on $AB$ and $AC$ respectively, such that $AB = 4BD$ and $AC = 4AE$. If area of quadrilateral $BCED$ is $52$ $cm^2$ and $[\triangle ADE] = x$ $...
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Area bounded by the curve y=xcosx and the x-axis

Find the area of the closed region bounded by the curve $y=xcosx$ and the x-axis (in other words the total area) for $((2n-1)pi))/2 \leq x \leq $(2n+1)pi))/2 where n is an arbitrary positive integer. ...
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What function describes an area of projection of a circle that is tilting so that we start with the area of a circle and end up with a straight line?

My question is simple. I would like to know the function that describes the area of a projection of a circle on a plane (meaning something like an area of a shadow of the circle on a paper) based on ...
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Area of a region in first quadrant

Find the area of the region bounded by paraboloid $z = x^2 + y^2 $ lies below $z = 4$ and in the first octant. Where I am going wrong? What is the correct area? My work: $A = \int \int_D \sqrt{ \...
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The volume and the surface area of a cylinder having base radius $r$ and height $h$ inscribed in a fixed sphere of radius $a$ as functions of …

We are given a sphere of radius $a$, and a cylinder of base radius $r$ and height $h$ is inscribed in this sphere. How to express the volume and the surface area of the cylinder as functions of $r$? ...
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How can the area of a rectangle having base $x$ inscribed in a fixed circle of radius $a$ be expressed as a function of $x$?

We are given a circle of radius $a$, and a rectangle of base $x$ is inscribed in that circle. How to express the area of that rectangle as a function of $x$? My Attempt: The centroid of the rectangle ...
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Finding the area inside the circle $r=-3a\cos(\theta)$ and outside the cardioid $r=a(1-\cos(\theta))$ [closed]

I've calculated the area inside the circle $r=-3a\cos(\theta)$ and outside the cardioid $r=a(1-\cos(\theta))$ and I got the answer: $a^2\pi$. But this answer not seem to be correct. Can anyone help? ...
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Area under curve equals product of arc length and its projection

Find equation of a curve whose area under it equals product of arc length $L$ and its projection $(b-a)$ on the x-axis. $$ A = L (b-a)$$ I was trying to establish a corollary to the Amazing Catenary ...
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Showing that the graph of a function lies entirely above the x-axis

I have the following question here. a) Let $f(x)=x^4-x^3+1$. Show that the graph of the function $f$ lies above the $x$-axis Is there a way to approach this in a "nice" way? I could just ...
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Where does the error lie here?

Initially assume that the area $a$ and circumference $c$ of a radius $R>0$ circle are given axiomatically as: $a=πR^2$ and $c=2πR$ Now in order to calculate the surface area of the top hemisphere (...
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Distance between two circles with known size and intersection area

Caution - biologist at work I am trying to plot some circles and want to work out how far apart my circles should be with a target in mind. The ditance between the centre points of the two circles is ...
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Given a fixed volume. Can we find the minimum surface area of any shape? [duplicate]

For a given fixed volume, can we find the minimum surface area of any shape? We kind of know the minimum surface area is a sphere(restrict our problem in 3 dim space). (correct me if the sphere is NOT ...
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Calculate the area of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ inside of the cone $x^2+y^2=z^2$ where $z \geq 0$

Calculate the area of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ inside of the cone $x^2+y^2=z^2$ where $z \geq 0$ Attempt First we should use the elemental area formula given by $$\int_{S}f \dot dS=...
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Given two positive real numbers $A$ and $l$, does there exist a plane, closed, simple curve with area $A$ and length $l$?

We are given two real positive numbers $A$ and $l$, such that $4\pi A \leqslant l^2$ (so the Isoperimetric inequality holds). Is it true we can always find a plane, simple, closed curve so that curve ...
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Does surface area of a torus depends on its center

I need to solve the following problem: Let S be a torus parallel to the xy plane, centered at (2,3,3), with major radius 6 and minor radius 5. Calculate the surface area of S, using appropriate ...
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Astroid area with Green Theorem

I know that the area of an astroid can be calculated by Green's theorem through the integral: $$ A = \frac{3a^2}{2}\int_{0}^{2\pi}[\cos^4t \sin^2t+\sin^4t\cos^2t]dt$$. However, I found the following ...
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Calculate the area of the part of the cone $x^2+y^2=z^2$ with $z \geq 0$ that is inside of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$

Calculate the area of the part of the cone $x^2+y^2=z^2$ with $z \geq 0$ that is inside of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ Attempt Notice that we should apply the formula $$\int_{S}f \dot ...
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Calculate the surface area bounded by the curves $3x^2−1$ and its tangents that pass through point (0,1/4).

Ok here i have a calculus problem to solve to find the area of the curve: Calculate the surface area bounded by the curves $ 3x^2-1 $ and its tangents that pass through point $(0,1/4)$. My question is ...
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Laboratory Flask geometrical shape.

https://en.wikipedia.org/wiki/Laboratory_flask How one can classify Geometrical shape of Laboratory flasks especially applied in Chemistry? Is it a Cylinder, Spherical, Oval, Cone shape? How one can ...
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In the following right $\Delta ABC$, $AC = BC = 1$ and $DEF$ is an arc of a circle with center $A$.

In the following right $\Delta ABC$, $AC = BC = 1$ and $DEF$ is an arc of a circle with center $A$. Suppose the shaded areas $BDE$ and $CEF$ are equal. Then $AD = \frac{x}{\sqrt{x}}.$ Find $x$ . What ...
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Algorithm for Determining Maximum Size for Squares in a Rectangle

I'm trying to create a program to draw a specific number of equally-sized squares within a rectangle. I want to set my squares' sizes to the maximum size possible while still fitting within a ...
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Find the length of rope when horse grazes double area

A field of grass $12$ ft by $20$ ft has a horse tied to a corner of the field. The initial length of the rope was $10$ ft but when the horse needed twice as much area over which to graze, the rope ...
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A geometry question…

In the given figure, $ABCD$ is a square of side $3$cm. If $BEMN$ is another square of side $5$cm & $BCE$ is a triangle right angled at $C$. Then the length of $CN$ will be:- I plotted this on ...
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1answer
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Calculus 1 - Find the area of shaded region

I have been working at this for like 3 hours and can find nothing online like it and there are no examples in the text book. I must be missing a simple step somewhere. The correct answer is given as $$...
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The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $\frac{[\Delta ADE]}{[\Delta DEC]}$ .

The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$ . What is the value of $\frac{[\Delta ADE]}{[\Delta DEC]}$ . What I Tried: Here is the diagram :- You can see I marked the ...
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area of the region bounded by the latus recta of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} =1$ and the tangents to the ellipse drawn at their ends.

I tried making a diagram of this (please don't look at the numerical values as the tool I used did not let me use the general a, b values. ) So the 2 latus recta here are: BC and FE and I think what ...
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1answer
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If the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ intersects the axes at points $A,B,C$ then Area of Triangle $= \sqrt{b^2c^2+c^2a^2+a^2b^2}$

I'm working through this problem, Compute a surface area by integration to show that if the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ intersects the axes at points $A,B,C$ then Area of Triangle $= ...
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Area under a curve with two functions given

Find the area of the region bounded by the curve and line in the diagram I've found the upper function to be ($-x + 6$) and the lower function is ($x^2+4$) I did $-x + 6 = x^2 + 4 $ to get the upper ...
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Does a cube give the maximum volume for a fixed surface area?

If I am making a fish tank with an open lid and $1600 \,\rm{cm}^2$ of material, why is the maximum volume that the tank can occupy not given by a cube? My work Suppose the base of the fish tank is a ...
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Finding circle tiling area using limit

So I want to find the green area which denotes the "waste" when using circle tiling. Circle radius may be assumed to be 1. Sure, there are multiple ways of doing it, but I want to try ...
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Find the area of the next polygon

The problem didn't give me any angles. I was approaching the angle of 640 and 650 as a right angle to calculate the hypotenuse and thus have an edge, but I don't think it is the most correct
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The perimeter of a segment of a circle is 22 inches. The arc is 2 radians. What is the area of the segment?

The perimeter of a segment of a circle is 22 inches. The arc is 2 radians. What is the area of the segment? I am having a hard time answering this question since the given are the perimeter of the ...
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42 views

Area of a triangle using determinants of side lengths (not coordinates of vertices)

Given the side lengths of a triangle (and not the coordinates of vertices) is there a way to find the area of a triangle using determinants? For example, if the three side lengths are $a$, $b$, and $...
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1answer
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Finding the area between $f(x)=e^{x/2}-2$ and $g(x)=x^2+2x-1$, from $x=0$ to $x=2$

I need to find the area between the graphs of $$f(x)=e^{x/2}-2 \quad\text{and}\quad g(x)=x^2+2x-1$$ and the lines $x = 0$ and $x = 2$. I think I need to do $g(x)-f(x)$ with $0$ and $2$ as the lower ...
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1answer
34 views

Area of triangle bounded by a tangent to a curve

Find a curve such that the surface of the triangle bounded by the line going through the tangent point and perpendicular to the x-axis and the tangent line to the graph is equal to $a^2$. I didn't ...
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1answer
47 views

Finding errors in the triangle area

The lengths of sides b and c of a triangle ABC are measured accurately, while angle A is measured with an error of 1/2 degree. If b = 12cm, c = 15cm and angle A is 30 degrees, what is the approximate ...
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2answers
38 views

Volume of polygon-based pyramid

I read in a book a few months ago that the volume of a (solid) pyramid with a base that is ANY polygon (I'm not sure if it mentioned it being regular or not) is equal to $$\frac{1}{3}\times A\times h$$...
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I need help finding algebraic area and perimeter, please

Could you help me solve some or all of the problems please? I can't understand the procedure for solving them... solve the following problems determines algebraically the perimeter and area of the ...

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