What is the maximum area of a rectangle inscribed in a circle?

In summary: Since the tangent line at a point is perpendicular to the radius at that point, we can use this to find c, such that (sqrtc)^2 + (sqrtc)^2 = R^2 , i.e. c = (1/2) R^2, giving the maximum area of 2R^2.In summary, the conversation discusses how to show that the maximum possible area for a rectangle inscribed in a circle of radius R is 2R^2. The conversation suggests setting up a relationship and using the concept of derivatives to maximize the area. It also mentions that a trigonometric approach can be used and that a geometrical solution is proposed. Finally, the conversation concludes with a suggestion to use a
  • #1
The_Prime_Number
3
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The problem asks me to show that the maximum possible area for a rectangle
inscribed in a circle of radius R is 2R^2. It also gives a hint saying that I should first maximize the square of the area.

I set the problem up as xy = 2R^2. I decided to work on the left side and wrote it as x(2R). But, I don't know where to go from here. Thanks.
 
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  • #2
You need to start up with a relationship such like

[tex] x^2 + y^2 = (2R)^2 [/tex]

and like you noted

[tex] xy = Area[/tex]

The use the concept of derivate to maximize for the area.

The standard procedure is to find a or more relations F(x,y) and then turn it into a function F(x), so you can use the concept of derivative.
 

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  • #3
If You know some trigonometry i.e. 2 sinA cosA = sin2A.

Solution : Since the rectangle is inscribed in a circle hence, its diagonal,D, is equal to the diamter,2R, of the given circle. i.e. D = 2R.
Now consider the triangle formed by the diagonal and two adjacent sides of our rectangle. It is a right triangle. Hence its sides can be expressed as D(diagonal the hypotenuse) , D*sinA and D*cosA(the perpendicular and base ).
Note : they also satisfy the pythagoras theorem D^2 = (D*sinA)^2 + (D*cosA)^2.

hence area of rectangle = product of adjacent sides
= DsinA * DcosA
= D*D* (sinA*cosA)
= 2R*2R*(sinA cosA)
= 2(R^2)*(sin 2A )
now R is fixed for a given circle. and A may vary.
for maximum area, sin2A must be maximum. which is 1
hence maximum area is 2R^2.
Note : sin 2A is 1 only if A =45 degree
which means that sinA = cos A
hence adjacent sides of the rectangle D*sinA and D*cosA become equal.
This justifies that in order to maximise its area the rectangle must be made into a square.
 
  • #4
I like the geometrical solution proposed in post #3,but,to carry on with the idea in post #2,i'd say the simplest & most elegant way is to use a Lagrange multiplier.

Daniel.
 
  • #5
this is a trivial calculus problem, to maximize A= xy when x^2 + y^2 = 4R^2, as x runs from 0 to 2R.

If we take the hint and maximize A^2 = x^2 y^2 = x^2 [ 4R^2 - x^2]

= 4R^2 x^2 - x^4, the derivative is 8R^2 x - 4x^3, which is zero when x = 0 and x = sqrt(2)R.

the max is obviously not at x= 0 hence occurs at x = sqrt(2)R (since on a closed interval there must be a max and it must occur at a critical point.
 
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  • #6
the solution is also obvious geometrically as follows: picture the level curves of the function xy, i.e. the hyperbolas xy = c. if you know what these look like, it is obvious that they are symmetrical about the line x=y, and the point on each hyperbola which is nearest the origin lies on this line.

Hence if we assume also that (x,y) lies on a circle of given radius, then the hyperbola xy = c which meets the circle and has largest value of c, is the one tangent to the circle at the point (sqrtc,sqrtc).
 
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What is a rectangle in a circle?

A rectangle in a circle is a geometric shape in which a rectangle is inscribed within a circle, so that all four of its corners touch the circle's circumference.

What is the area of a rectangle in a circle?

The area of a rectangle in a circle can be calculated by multiplying the length and width of the rectangle and then multiplying that by pi (π).

What is the perimeter of a rectangle in a circle?

The perimeter of a rectangle in a circle can be calculated by adding the length of all four sides of the rectangle, which includes the length of the curved part of the rectangle that touches the circle.

Can a rectangle in a circle have a larger area than the circle itself?

No, the area of a rectangle in a circle will always be smaller than the area of the circle itself. This is because a circle has a larger perimeter than a rectangle, so the rectangle cannot fully fill the circle's area.

What real-life objects can be represented by a rectangle in a circle?

Many objects in our daily lives can be represented by a rectangle in a circle, such as wheels, plates, and clocks. These objects often have a circular shape with a rectangular component inside.

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