# Questions tagged [area]

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

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### Areas versus volumes of revolution: why does the area require approximation by a cone?

Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. ...
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### Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
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### Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation for ...
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### Why determinant of a 2 by 2 matrix is the area of a parallelogram?

Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$. How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$? Are the areas of the ...
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### What is the maximum area of a square inscribed in an equilateral triangle?

What is the maximum area of a square inscribed in an equilateral triangle? Please post the approach to solve the above question.
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### Circle areas on squared grid

There is a circle. On 9 equal squares. Every square has some value assigned to it. Every square gets weight, depending of what percentage of it is circle (area-wise). I need to find circle radius, ...
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### How calculate the shaded area in this picture?

Let the centers of four circles with the radius $R=a$ be on 4 vertexs a square with edge size $a$. How calculate the shaded area in this picture?
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### Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
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### Area of parallelogram = Area of square. Shear transform

Below the parallelogram is obtained from square by stretching the top side while fixing the bottom. Since area of parallelogram is base times height, both square and parallelogram have the same area. ...
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### Which area is larger, the blue area, or the white area?

In the square below, two semicircles are overlapping in a symmetrical pattern. Which is greater: the area shaded blue or the area shaded white? My Solution Let the length of each side of the square ...
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### Find the area where dog can roam [duplicate]

A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam? I tried to find the area of all semicircles and then to ...
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### Do two right triangles with the same length hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
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### Check if a point is inside a rectangular shaped area (3D)?

I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example ...
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### Why does the magnitude of the cross-product of a and b give the area of a parallelogram spanned by a and b?

I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working ...
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### Finding the area of the shaded square inside a square created by connecting point-opposite midpoint [closed]

If the lines intersect the vertices of the square. The area of the square is $1$ and the lines also intersect the midpoints of the square lines. How to find the area of shaded region?
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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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### Calculate the area of an irregular cyclic convex polygon

I want to write a program in C++ to calculate the area of irregular cyclic convex polygons. However, the inputs are in the form corner point angles. I am just not sure what the inputs mean and what ...
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### A way to find this shaded area without calculus?

This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600) $ABCD$ is a square with a side of $10$, $APD$ and $CPD$ are ...
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### How was the area formula for a circle ($A = \pi r^2$) derived before the introduction of calculus?

How did mathematicians prior to the coming of calculus derive the area of the circle from scratch, without the use of calculus? The area, $A$, of a circle is $\pi r^2$. Given radius $r$, diameter $d$ ...
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### Where is the mass of a hypercube?

Question: Consider a hypercube $C := \{\vec{x}\in\mathbb{R}^n : \forall(i)\: |x_i|\leq 1\}$ and hypersphere $S_r := \{\vec{x}\in\mathbb{R}^n : |\vec{x}| = r\}$. Let $R(n)$ be the radius of the ...
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### How to maximize the area of a triangle, given two sides?

I am having a question that is it possible to find what will be the maximum area of a triangle if suppose we know any two of the sides and none of the angles? We don't know whether one of these sides ...
### Why is the area of the circle $Ļr^2$? [duplicate]
I searched many times about the cause of the circle area formula but I did not know anything so ... Why is the area of the circle $\pi r^2$? Thanks for all here.
### Show that the Area of image = Area of object $\cdot |\det(T)|$? Where $T$ is a linear transformation from $R^2 \rightarrow R^2$
Prove that the area of an image in $2d$ cartesian coordinates is equal to the determinant of the linear transformation times the area of the initial shape. I've tried to formulate general expression ...