# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...