Question

A square is inscribed in a circle. If the area of the square is 9 in2, what is the ratio of the circumference of the circle to the perimeter of the square?
answer should be put like this:

c°/p^

Answer 1

Hello!

Given the formula for the area of a square is:

A=s2 where A is the Area and s is the length of the side of the square, we can find the length of one side of the square by substituting and solving:

9 in2=s2

√9 in2=√s2

3 in=s

s=3 in

Using the Pythagorean Theorem we can find the length of the squares diagonal which is also the diameter of the circle:

The Pythagorean Theorem states:

a2+b=c2 where a and b are legs of the triangle and c is the hypotenuse of the right triangle. In this case, both legs of the triangle are sides of the square so the are both the same length. Substituting and solving gives:

(3i n)2+(3i n)2=c2

9 in2+9 in2=c2

9 in2×2=c2

√9 in2×2=√c2

√9 in2√2 in=c

3 in√2=c

c=3√2 in

We can now find the perimeter of the square and the circumference of the circle.

Formula for Perimeter of a square is:

p=4s where s is the length of a side of the square.

Substituting and calculating p gives:

p=4×3 in

p=12 in

Formula for the circumference of a circle is:

c=2πr where r is the radius of the circle.

Or,

c=dπ where d is the diameter of the circle. Remember: d=2r

Substituting and calculating c gives:

c=3√2π in

We can then write the ratio of the circumference to perimeter as:

3√2π in12 in⇒

3√2π in124 in⇒

√2π4