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.

Question

I don't know how to start.

 

Answer 1

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  .

 

 

Answer 2

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.

Answer 3

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

Answer 4

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!