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Find The Area (in Terms Of Ac) Of A Square Inscribed Within A Circle Of Area `ac`

Here is an exemplar solution for the above question.

Date : 25/04/2019

Priyash

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Uploaded by : Priyash
Uploaded on : 25/04/2019
Subject : Maths

VISUALISE: We must begin by visualizing a Square inscribed inside a Circle. Note that owing to the symmetry of the problem the Diagonals of the Square also act as the Diameter of the Circle. This is a key aspect that will help relate the two areas and get to the ultimate answer.

SideNote: Additionally by another line of logic we can arrive at the equivalence between the Diagonal and the Diameter - Because the Diagonals subtend right angles onto the circle the must be Diameters.


SOLVE: Let the side of the square be `s` and the diameter of the circle be `d`.


Thus, using the information in the problem statement the area of the circle, we have `Ac` = (pi)*((d/2)2). This is just by invoking the expression for the area of a circle which is pi*(radius^2) where we have substituted the radius as half of the diametrical length i.e `radius` = d/2.


Or, Ac = (pi*d*d)/4 ............Equation.1


Now, the diagonal `d` of the square is related to the side `s` by the Pythagorean Theorem. This gives:


d2 = s2 + s2, because the sum of the squares of the base `s` and the perpendicular `s` is equal to the square fo the hypotenuse `d`. This eventually yields,


2*s*s = d*d ............Equation.2


As the area of the square is nothing but the length into its height which in this case are the same as `s` we have, Area of Square (say, As) = s*s. Substituting this value of As in Equation.2 we get,


2*As = d*d ............Equation.3


Now expressing `d*d` from Equation.1 and substituting in Equation.3 we get,


Ac = (pi*(2*As))/4. Or,


Ac = pi*As/2. Now, making As the subject of the equation we get,


As = (2/pi)*Ac. & ............Equation.4


ANSWER: Equation.4 is the required expression relating the area of a circle to that of an inscribed square.

:)


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