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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34 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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Distance of chord for given area?

For a circle w/ a unit radius, how far from the center of that circle must the center of a chord be for the area under that chord (the one that would not include the circle's center) to be A? If $...
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1answer
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Existence of a closed curve with zero area and infinite perimeter, but not conversely

I want to prove that we can find planar simple closed curves with arbitrarily small area and '$infinite$' perimeter, but not the other way around, i.e. we cannot find a simple closed curve with ...
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First variation of length

Let $M^3$ be a Riemannian manifold with nonempty boundary and let $\Sigma$ be a smooth surface with boundary. Consider $\Phi : \Sigma \times (-\varepsilon, \varepsilon) \to M$ a proper variation of a ...
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Prove that the area of the trangles are equal.

Prove that the area of all the traingles in the figure below are equal. I tried using geogebra to determine an arbitrary values of $a$ , $b$, and $c$. I found out that the triangles have equal ...
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How do I find the area under a wave/quadratic?

The roots A and B are (-1,0) and (2,0) respectively.
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32 views

Finding the area of the region bounded by $x=0$, $y=0$, $2x^2=\sqrt{x^2+y^2}$, $2y^3=\sqrt{x^2+y^2}$

I have 4 equations and need to find the area bounded by the corresponding curves. I don't know how to approach it.
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Ways to prove the area of an ellipse formula

One can prove the ellipse area formula $A=\pi a b$ ($a$, $b$ the major and minor semi-axis) either by integration or by the stretched-circle argument. See for instance here: https://proofwiki.org/wiki/...
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Ratio of surface area of two prisms given only volume of those prisms?

I'm trying to help my son with a math problem that totally has me stumped, and searching online is not helping. Given two prisms with volume 1536 and 375, what is the ratio of their surface areas? ...
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Area of the shaded region- Rotating Wheel

I wanted to compare answer for area of the region in the middle of the square, the leaf sort of shape. So the circle is like a wheel and as it turns to the other side, it basically draws the upper ...
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1answer
23 views

integral of chord length divided by surface area of sphere

I'm trying to find the integral of a chord length of a circle, divided by the surface area of a sphere. Imagine a circle, that is stuck between two layers of a spherical shell, and then take discrete ...
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What is the area of a region bounded by the curve $y=e^x$ and the lines $y=1$ and $x=1$?

What is the area of a region bounded by the curve $y=e^x$ and the lines $y=1$ and $x=1$? When $x=1$, $y=e$. When $y=1$, $x=0$. I tried to find the area by saying that $A=\int^1_0 e^x dx= e - 1$. ...
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Equation of a line that form a triangle of area 8

Find the equation of the line that passes through the intersection of two lines, $\ 3x-4y=0$ and $\ 2x-5y+7=0$, and form a triangle of area 8 with the coordinate axes. I know that the intersection ...
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1answer
44 views

Calculus 2 problems help [closed]

Ive been struggling with this problem. Thanks!
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2answers
50 views

Is there any difference in definition of Area in maths and physics?

I am a little bit confused after reading the definition of area : The area of a shape can be measured by comparing the shape to squares of a fixed size.[2] In the International System of Units (...
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Fraction represented by shaded area

What fraction of the area of square with side of length $a$ does the shaded area represent? I solved the problem of finding the fraction area of the triangle with sides of length $a$, $d$ and $e$; ...
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Finding the equation of the ellipse given the area and making a guess about the shape of the ellipse with maximum area

The standard form of the ellipse with the foci on the x-axis $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a+b=20$$ Given the formula for the area of ellipse is $$A=\pi ab$$ a) How do I find the area of ...
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How to prove $I_n = \int_0^{\pi/2} \sin^n(x)dx = \int_0^{\pi/2}\cos^n(x)dx$ without using induction.

$$I_n = \int_0^{\pi/2} \sin^n(x)dx = \int_0^{\pi/2}\cos^n(x)dx$$ I must show the above equation without using induction. I would simply refer to the visuals: area under the curve between 0 and $ \...
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1answer
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painting a wall randomly

Lets say I am trying to paint a wall by randomly (uniform) shooting at it with a paintball gun. On average, how many paintballs will it take so that 99% of it has been covered? The first paintball is ...
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Calculating area of a shape with circular boundaries with elementary methods

Question. $\square ABCD$ is a square with $AB = 10$. Circle $O$ inscribes the $\square ABCD$. The center of the arc is $A$. What is the area of the colored area? Explanation: This problem can be ...
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Minimum surface of revolution

Let us consider we need to find a curve between $(-x,y)$ to $(x,y)$ such that the surface of revolution of this curve has minimum surface area. I proceeded to find area by considering the infitesimal ...
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44 views

Radius of circle included in a certain area.

Area $A$ is defined as the area of intersection between $x^2+x-2$ and $3x+1$. If a circle $x^2+y^2=r^2$ fits inside the area $A$, then what is the range $0 < r < \dots$? At first, since it ...
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Australian Maths Competition Area Question [closed]

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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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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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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1answer
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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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19 views

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
37 views

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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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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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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4answers
479 views

How to find the shaded region [closed]

Find the area of the blue shaded region of the square in the following image: [Added by Jack:] The area of the triangle in the middle of the square is given by $$ 4.8\times 6=28.8\ (cm^2) $$ ...
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3answers
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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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782 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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1answer
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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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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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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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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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3answers
66 views

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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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
348 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
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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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4answers
80 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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1answer
56 views

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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1answer
144 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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4answers
131 views

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
31 views

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 ...