# Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

2,226 questions
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### Area of the common region of three circles.

Find the area of the region that is common to the circles $r = 1$, $r = 2 \cos θ$, and $r = 2 \sin θ$. I tried many ways to get the common region, but it seems impossible to eliminate to that point, ...
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### Calculating the area between two functions expressed in polar coördinates

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$. The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to ...
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### Area of cut-out of three circles

Let three circles at different centers with different radii be given. They might intersect as shown in the picture. How to derive the area of the set (blue) in terms of the centers and radii? ...
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### Area of one of four regions within a rectangle

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the ...
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### Area under curve given a set $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$

Evaluate area under the curve for: $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$ I know that to find the area under a curve of a function from a to b, ...
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### find area of kite given side length

Circle with two tangent lines Above is the picture in question. A circle is given, center (-2,4) and a point outside the circle (0,10) is shown. Asked to calculate the area of the quadrilateral ABCD, ...
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### calculating the area in polar coördinates

I have difficulties calculating the area and setting the right boundaries of the following polar coördinates: $$r=2(1+cos(\theta) )$$ Thanks in advance
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### Find the area of a sphere inside a function

I want to find the area of the portion of the sphere $x^2+y^2+z^2=4z$ inside the function $x^2+y^2=z$ using double integrals. The graph would be something like this: Because of the nature of this ...
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### I Need Help in a Challenge [closed]

My teacher challenged me with the question below: \sqrt{\frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2}\ast \left ( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \frac{\sqrt{41}}{1} \right )\ast \left ( \frac{\...
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### How can i Prove that the gray area is the same as white area? [duplicate]

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? I just want any hint for solving this question....
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### Find area of region that is common to both squares.

A 3-meter square and a 4-meter square overlap as shown in the diagram.D is the center of the 3-meter square. Find the area of the region DGFE. I 've tried to form right triangles in such region but ...
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### How to calculate area of triangle-like structure of blocks? (Pick's Theorem seems insufficient.)

Given the following structure, is there a formula to calculate the number of blocks? (EDIT: and I am really looking for a solution for any BASE and HEIGHT.) At first, it would seem that this is a ...
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### Calculating density of individuals within an area (tree stand density)

Any help greatly appreciated! I want to find the density of trees surrounding a sample point (measured in trees/m$^2$). The distance from the sample point to the three nearest trees has been ...
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### How do you find area of the loop in the graph of $x(x^2+y^2)=(x^2-y^2)$

The graph of the given equation is $x(x^2+y^2)=(x^2-y^2)$"> I believe I have to use (r,θ) coordinates but I do not know how to integrate this in (r,θ).
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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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### Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$

Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$ One very obvious way would be to find the points of intersection which would be messy and subject to many conditions. I was trying to solve ...