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The area of a circle is 16 pi square inches. What is the area, in square inches, of the largest square that can be drawn within this circle.

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4 Answers

The formula for the area of a circle is π r2  .  This means that radius of the given circle is sqrt(16) = 4 inches
 
Consequently, the diameter of the circle is 8 inches.      This means that the diagonal of the largest possible inscribed square is also 8 inches.     From the Pythagorean theorem, we figure the side of the largest square to be  (8 inches)/sqrt(2).      Since the area of a square is  (length of side)2  , the are of the square is 32 inches2  .
 
 

Comments

Typo in last line:
 
"the are of the square"
...........^
should be
"the area of the square"
...........^
Wow! So many similar solutions using the diagonal of the square.
 
The equation of the circle is 16π = πr2,   so r2 = 16 and r=4.  Great!  so let's assume the circle is centered at the origin.  Then x2 + y2 = r2 = 16
 
Assume we rotate the square so that it's sides are parallel with the x and y axes.   Then each side of the square is a chord which intersects a 90º arc, and the top side of the square intersects the arc at 45º and 135º, or at (-2√2, 2√2)  and (2√2, 2√2).  So the length of a side of the square is 4√2. 
 
So the area of the square with side 4√2 is 16*2 = 32.  Pythagorean theorem is not required.

Comments

You used the Pythagorean Theorem when you used the formula for a circle, which is derived using the Pythagorean Theorem.
And when you used the side lengths of the 45°-45°-90° Special Triangle; also derived using the Pythagorean Theorem.
 
[Re: "Pythagorean theorem is not required."]
Hi Julian;
Area of circle=(pi)(radius)2
Area=(pi)(16)
radius=√16=4
diameter is twice the radius.
diameter=(2)(4)=8
This is the diagonal of the square.
Pythagorean Theorem is...
a2+b2=c2
However, because this is a square, a=b...
2a2=c2
c is the diameter of the circle.
a2 is also the area of the square.
The formula for the area of the largest square within a circle is...
Area of square=a2=(1/2)(diameter of circle)2 
(1/2)(8)2
(1/2)(64)
32
Julian, to visualize this problem, draw a square within the circle where each of the four points are on the edge of the circle (like this http://www.yogaflavoredlife.com/wp-content/uploads/2010/09/square-circle.gif).
 
Area of a circle: A = πr2 
 
Thus,
16π = πr2
16 = r2
r = √16
r = 4
 
Double the radius, and you have the diameter (d=8), which also happens to be the distance from one corner of the square to the other (like this: http://mathcentral.uregina.ca/QQ/database/QQ.09.04/bob1.1.gif).
 
Note you've just made a triangle out of your square, and you know the length of the hypotenuse, 8/√n!
 
Using the Pythagorean Theorem (A2 + B2 = C2), you can solve for the length of the sides of the square (and since it's a square, and thus has equal sides, A2 + B2 = A2 + A2 = 2A2
 
Thus,
2A2 = C2
2A2 = 82
2A2 = 64
A2 = 32 in2
 
EDIT: Whoops, didn't notice the pi in the original problem.  Good call, Richard!