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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Australian Maths Competition Area Question

PQRS is a square. T and U are midpoints of the sides PS and PQ respectively. TQ, SU and PR intersect at V. This is a question from the 2009 Intermediate Division AMC paper. I was given it in a ...
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Bisection of a a triangular area

Here's the sketch: From the inner point P of a triangle ABC the three connecting lines to the corner points are drawn. In addition, the lines PE, PD and PF are each drawn parallel to a median of ABC. ...
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Geometry : what is the $\phi$ angle, if area of yellow rectangle is equal with area of red triangle?

I have a right triangle and in it area of yellow rectangle is equal with area of red triangle. How could prove that $\phi=45^{\circ}$? $$\text{Area of Yellow Rectangle}=\text{Area of Red Triangle}...
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How is area defined?

Thinking about area in the context of the Lebesgue measure, I have an intuitive understanding of how area is constructed in $\mathbb{R}^2$: define all rectangles to have the area $length \times width$...
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1answer
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Is it possible to solve the two unknowns of this function given the area, the base length and a ratio between the start and end points?

I am trying to calculate the values of $a$ and $b$ in the following function: $$ f(x) = -e^{ax} + b + 1. $$ There are a few "rules" in play: $\int\limits_{0}^{n} f(x)\, \mathrm{d}x= v$ $f(n) = rb$ ...
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2k views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
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1answer
41 views

Flat geometry, What is the value of BC Side?

In a triangle ABC have AB =10cm and AC=12cm. The incentro(I) and the baricenter(B) are in the same parallel to BC. The BC side measurement is equal to: I have developed so far: I did not calculate ...
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Divide circle and subcircle evenly by area

How do you divide a circle into a certain number of shapes with the same area while having at least one sub circle dividing the whole figure? I know that if we divide along the center of the circle in ...
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2answers
726 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
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Finding the area of a curve where one of the curves is a line

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 4x, y = 4x - 2 $$ Answer: \...
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Determine the area of a regular octagon with vertices on the unit circle [closed]

How would I determine the area? Help please
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741 views

Square covered with circles

I have a square 800x800 and i need to fully cover it with the least number of circles possible, each circle has a radius of 150. QUESTIONS: - What pattern would be the best to use? Clover, diamon or ...
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Calculating the area between two curves

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 9x, y = \frac{3x^2}{8} $$ ...
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742 views

How to maximize area of a square inscribed in a equilateral triangle?

We have an equilateral triangle and want to inscribe a square, but want to do so in the way that maximizes the area of the square. I sketched two possible ways, not to scale and not perfect. Note I ...
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Surface area of Knitted S-yarn [closed]

I am working on modeling and simulation of Knitted S-yarn, my question is how I can find the surface area of this shape.
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2answers
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points $R$ and $T$ lie on the side $CD$ of the parallelogram $ABCD$ such that $DR= RT= TC$ what is the area, in $cm^2$ , of the shaded region?

points $R$ and $T$ lie on the side $CD$ of the parallelogram $ABCD$ such that $DR= RT= TC$ . Lines $AR$ and $AT$ intersect the extension of $BC$ at points $M$ and $L$ respectively, and the lines $BT$ ...
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1answer
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Finding the area under a curve when the area is bounded by 3 curves.

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves and lines. $$ y = x, y = \frac{1}{ \sqrt{...
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3answers
58 views

Find integral area with respect to x

The question is finding the area enclosed by the curves x=$y^2$ and x+2y=8 using both x and y integrals Graph for reference Purple is x+2y=8, red is x=$y^2$ First I found the limits by letting x=8-...
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Area Between Three Curves With Two Similar Lines

I have a little Calculus problem which is confusing me quite a bit, so I thought to ask you guys for help. The problem consists in calculating the area between three curves, they are: $$ - y = x² - ...
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How to find the area of a region bounded by a simple closed curve?

I have the following equation: $$ \frac{p}{(a-x)^2+y^2}+\frac{1-p}{(b-x)^2+y^2}=1 \text{ where } 0\leq p\leq 1 $$ Which represent a simple close curve. Obviously, when $p=0,p=1$ or $a=b$ we recover a ...
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1answer
386 views

How to calculate an area enclosed by two parametric curves?

I know the area under the curve given by parametric equations$x=f(t),y=g(t),\alpha\leq t\leq \beta$ is given by$$A=\int_{\alpha}^{\beta}g(t)f'(t)dt$$ That is in $\int_{a}^{b}ydx$ we have substituted $...
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1answer
39 views

Inequality of areas in elementary geometry

During the IMC 2019 contest i ended up with the following question in elementary 2D Euclidean geometry: Let $\mathrm{CDE}$ be any nondegenerate triangle inside a circle, consider the regions with ...
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37 views

What is the total area enclosed between the curve $y=x^2-1$, the x-axis and the lines $x=-2$ and $x=2$?

What is the total area enclosed between the curve $y=x^2-1$, the x-axis and the lines $x=-2$ and $x=2$? I tried to find the area by using the integrals $\int_1^2$ and $\int_{-1}^{-2}$ . $x^2-1$ ...
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Area between two polar curves using iterated integrals?

The question is from a practice exam I am currently trying to do: I am really not sure how to go about this one. In essence, I'd imagine that the idea is to find the area of the greater curve, and ...
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1answer
2k views

Area of a Triangle Formed by a Line Tangent to f(x) and the Axis

Original question Find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis and a tangent line to the graph of $f = (x + 2)^{-2}$ So far I've looked here and ...
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3answers
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Find the area bounded by $x= at^{2}$ and $y = 2at$ from $t=1$ to $t=2$

Find the area bounded by $x= at^{2}$ and $y = 2at$ from $t=1$ to $t=2$ I tried to solve this by integrating $\int_{1}^2 y \frac{dx}{dt} dt$ $\int_{1}^2 (4a^{2}t^{2}) dt$ $= (28/3)a^{2}$ What is ...
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4answers
72 views

The maximum possible area bounded by the parabola $y = x^2 + x + 10$ and a chord of the parabola of length $1$ is?

The maximum possible area bounded by the parabola $y = x^2 + x + 10$ and a chord of the parabola of length $1$ is? $(y-39/4)=(x+1/2)^2$, Vertex: $(-1/2, 3/4)$ How do I find the equation of the chord ...
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Calculate the equivalent of Principle Component Analysis for Center of Pressure Sway Area but for a 3d object?

I am trying to find the total COP "sway area" of an object, but in 3D Space. I'm aware that Area describes a 2 dimensional coordinate system (such as X,Y) and have used a PCA analysis to do so, but I ...
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Problem involving the square root of a trigonometric term

I was trying to find the shaded area in this figure: And no, it isn't homework. I just chanced upon it on Facebook and had a go at it. I managed to find it using a very simple method. I now want to ...
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1answer
340 views

Does there exist an area-preserving map from the hyperbolic plane to the Euclidean plane?

Fairly simple question: does there exist an area-preserving map from the hyperbolic plane to the Euclidean plane? If not, does there exist an area-preserving map from an arbitrarily large subset of ...
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1answer
64 views

To find the area of a given curve.

The curve is given $$x =(1-t^{3})/(1+t^{2})$$ and $$y= 2t/(1+t^{2})$$ I know the method for finding the area, but I'm having problem with the tracing of curves. In exams, I won't really have time to ...
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1answer
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Calculating area bounded by polar curves

Find the area bounded by the following polar curves $r<3|sin(2t)|$ Well, I know that the formula used to calculate an area bounded by $g(t) \le r \le f(t)$ with $\alpha \le t \le \beta$ is $A=\...
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1answer
129 views

The largest equilateral triangle circumscribing a given triangle

Seven years ago, one of my many contributions to the March 2010 edition of Erich Friedman's Math Magic was a packing of eight circles of unit diameter and one equilateral triangle of unit side length ...
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1answer
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Intersection area of concentric ellipses

I need to find the area of intersection of two concentric ellipses. I imagine the concentricity should make this simpler than the general case, but the ellipses could be rotated. Alternatively, if ...
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4answers
375 views

How to show that $\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$

I was reading Advanced Integration Techniques, and found that$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$ The book provides one method using residue theorem and Laurent expansion. ...
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1answer
685 views

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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Finding the area under the curve from a graph

What is the area of the shared region in the figure above that is bounded by the $x$-axis and the curve with the equation $y=x\sqrt{1-x^2}$? This is the problem I was given. I assumed the answer ...
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1answer
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Finding the length of a semi-ellipse (Calculus)

A fireplace is to be constructed in the shape of a semi-ellipse (half of the ellipse). The opening is to have a height of 2 feet at the center and a width of 5 feet along the base. The contractor who ...
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Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
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4answers
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The Curve $y = \frac{x^2}2$ divides the curve $x^2 + y^2 = 8$ in two partss. Find the area of smaller part.

Question: The Curve $y = \frac{x^2}2$ divides the curve $x^2 + y^2 = 8$ in two parts. Find the area of smaller part. This question is from Basic Mathematics. Please explain how I can solve it ...
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Calculating area of a rectangle strip

I would like to calculate the red marked area . How to do it? with steps?
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optimised combined area of a rectangle and a square [closed]

Consider a rectangle of dimensions $2x$ by $x$ and a square of dimensions $y$ by $y$. If the sum of the perimeters of the rectangle and square is $l$, find the value of $x$ and $y$ (in terms of $l$) ...
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Standard Ellipse Area

I am writting my PhD about Spatial Analysis in football. I am using Standard Ellipse Areas and Prediction Ellipse Areas in a database (x,y). I am defining its shape and size, and the means of the x ...
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What is the ratio the areas of △$PST$ and quadrilateral $STRQ$?

I tried to answer this question: What is the ratio of the areas of △$PST$ and quadrilateral $STRQ$ if $∠1\cong ∠2$ Since $∠1\cong ∠2$, △$PST\sim$ △$PQR$. And since the two triangles are similar, the ...
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1answer
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Calculate area of a crescent

My math teacher gave me this question as a challenge, but I can't seem to solve it. If anyone could kindly assist me in how to go about solving it, and providing a set of detailed steps required to ...
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8answers
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V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
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4answers
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Finding the area of a trapezoid given vertices

I encountered this problem and did not know how to solve it: I graphed it out - making a point for each vertice. I then decided there must be a 3 because when $x=0$ then $y=3$. But, according to that ...
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2answers
791 views

Bernoulli Lemniscate - surface area and volume

How do I calculate the surface area and volume of the solid obtained by rotating the Bernoulli lemniscate $$(x^2+y^2)^2=2a^2(x^2-y^2)$$ around the $x$-axis? It is not like I'm lazy and asking for ...
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Find the area between the following curves and their asymptotes.

The curve is given by the polar equation $ r = a( \sec \theta + \cos \theta) $ At $\theta = \pi /2, r$ goes to infinity, so asymptotes should be at $\theta = \pi /2$ To find the area I integrated $2$ ...
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Moment of inertia of a square (single integral)

My goal is to determine the moment of inertia of a square with the side length $a$. I know I could do this like this: $$J = \frac{m}{a^2}\int_{\frac{-a}{2}}^{\frac{a}{2}} \int_{\frac{-a}{2}}^{\frac{...