SOLUTION: A circle is inscribed in a square and a regular hexagon is inscribed in that circle. The area of the square is 36 square cm. Find the following: A) The perimeter of the square

Question 139968: A circle is inscribed in a square and a regular hexagon is inscribed in that circle. The area of the square is 36 square cm. Find the following:
A) The perimeter of the square
B) The area of the circle
C) The perimeter of the circle
D) The perimeter of the hexagon
E) The area of the hexagon.
I am just confused on where exactly to begin. Please help me, thankyou for your time. Bill
Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
A) The perimeter of the square whose sides (s) are 6 cm. each () is:
s = 4(6) = 24 cm.
B) The area of the inscribed circle is where r = half of s, the side of the square, so:

cm.
approx.

sq.cm.
C) The perimeter (AKA the circumference) of the circle is:

cm.
D) The perimeter of the regular hexagon is 6 times the length of one side.
To find the length of one side, draw the 6 diagonals of the hexagon. This will divide the hexagon into 6 equilateral triangles. The diagonals are equal in length to the side of the square (6 cm.) and the sides of the equilateral triangles are just half of this (3 cm.), so the perimeter is:

cm.
E) The area of the regular hexagon can be found in a couple of ways:
1) Find the area of one of the equilateral triangles and multiply this by 6.
2) Use the formula: where: n = number of sides (6), r = radius of circumscribed circle (3).
1) Using Heron's formula to find the area of a triangle when only the lengths of the sides are known:
where: s = the semi perimeter (4.5) of the triangle, a, b, and c, are the lengths of the sides of the triangle, a = 3, b = 3, and c = 3.

Now multiply this by 6 for the 6 equilateral triangles inside the hexagon:

sq.cm.
2) Using the formula:
) Substitute n = 6, r = 3

sq.cm.
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