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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.

I don't know how to start.

### 4 Answers by Expert Tutors

Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
4.9 4.9 (629 lesson ratings) (629)
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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  .

Typo in last line:

"the are of the square"
...........^
should be
"the area of the square"
...........^
Kenneth G. | Experienced Tutor of Mathematics and StatisticsExperienced Tutor of Mathematics and Sta...
1
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.

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."]
Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
3.0 3.0 (1 lesson ratings) (1)
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Hi Julian;
Area=(pi)(16)
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
Curt J. | Math/Science/General Ed Tutor in WaikeleMath/Science/General Ed Tutor in Waikele
5.0 5.0 (5 lesson ratings) (5)
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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!