circumscribing a circle

Question

How much larger is the side of a square circumscribing a circle 173 cm in diameter than a square inscribed in the same circle? Round the answer to one decimal place

Answer 1

Draw a circle, then draw the squares. For the inscribing square draw a diagonal line connecting two opposite corners.

 

The circumscribing square is the easiest, it has side length of 173cm, since the diameter of the circle is 173cm and each side of the square touches the circle. 

 

The inscribing square has a diagonal length of 173cm, since the diameter of the circle is 173cm and each corner of the square touches the circle, so you'll need to use the pythagorean theorem to find the side length. 

 

a^2 + b^2 = c^2

 

We know c^2 = 173^2 and a^2 = b^2, so we can rewrite our equation:

 

2a^2 = 173^2

 

Solving for a: a = sqrrt((173^2)/2) = 122.33

 

So the side length of the inscribing square is 122.33.

 

Now we just subtract the smaller side length from the bigger to get the difference.

 

173 cm - 122.33 cm = 50.67 cm

 

So, the circumscribing square is 50.7 cm larger than the inscribing square.

Answer 2

Circumscribe means to draw inside, with the object on the inside drawn as large as possible while none of its area is outside of the object by which it is being circumscribed. In the first case of your problem, a circle with a diameter of 173 cm is inside a square. Since the sides of the square would be touching the circle, a diameter can be drawn from one point where it touches to an opposite point, so the measure of the side of the square is the same as the diameter of the circle, or 173 cm.

 

Now, if another square were to be circumscribed by (inside of) the same circle, that means only the corners of the square would be touching the circle. This means a diagonal can be drawn from one corner to the opposite corner, and this diagonal has the same measure as the diameter of the circle, or 173 cm.

 

Now, since a side of a square and the diagonal of the same square have a relationship of:

 

s = s√(2)

 

where s is the measure of one side of the square, and s√(2) is the measure of the diagonal,

 

We can say that the diagonal of the square inside the circle is:

 

s√(2) = 173

 

solving for s gives us:

 

s = 122.3

 

And so the measure of one side of the square is 122.3 cm. (you could have also used the Pythagorean Theorem, since drawing a diagonal from one corner of a square to the opposite corner creates two identical right triangles).

 

So to find how much larger the side of the first square is, we do:

 

173 - 122.3 = 50.7