Q:

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^

Accepted Solution

A:
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=√s23 in=ss=3 inUsing 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=c29 in2+9 in2=c29 in2×2=c2√9 in2×2=√c2√9 in2√2 in=c3 in√2=cc=3√2 inWe 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 inp=12 inFormula 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=2rSubstituting and calculating c gives:c=3√2π inWe can then write the ratio of the circumference to perimeter as:3√2π in12 in⇒3√2π in124 in⇒√2π4