SOLUTION: Tarry is a farmer. She decides to make a circular pen for her sheep. She uses all of the fencing she has to make the fence (i.e. circumference =100) She then decides to make a squa
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Question 504167: Tarry is a farmer. She decides to make a circular pen for her sheep. She uses all of the fencing she has to make the fence (i.e. circumference =100) She then decides to make a square enclosure inside the circle, so that the corners of the square just touch the circle. What would be the squares perimeter?
She then decides to make a square enclosure around the outside of the circle. What would the outside squares perimeter be?
I found the radius of the circle=15.92.
I found the diameter of the circle=31.8.
The area of the circle 795.
But im lost on getting any information of the square inside the circles.
My thoughts on a diameter is diagnol line inside the circle where it can also go straight across and i think that is whats throwing me off.
My guess id the perimeter of the inner square is 10.
Please help. Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! you're on the right track.
here's a picture of what you're trying to do.
the inscribed square will have a diagonal that is a diameter of the circle.
the circumscribed square will have an apothem that is a radius of the circle.
the circumference of your circle is 100
the formula for the circumference is 2*pi*r
you get:
2*pi*r = 100
solve for r to get:
r = 100/(2*pi) = 15.91549431
the questions you are being asked are:
Tarry is a farmer. She decides to make a circular pen for her sheep. She uses all of the fencing she has to make the fence (i.e. circumference =100) She then decides to make a square enclosure inside the circle, so that the corners of the square just touch the circle. What would be the squares perimeter?
She then decides to make a square enclosure around the outside of the circle. What would the outside squares perimeter be?
without getting into the details, the perimeter of the inscribed square will be equal to 4 * 22.5079079 = 90.03163162
without getting into details again, the perimeter of the circumscribed square will be equal to 4 * 31.83098862 = 127.3239545
now the details.....
the diagonal of the inscribed square is the diameter of the circle.
this makes the diagonal equal to 2 times the radius you found earlier which makes it equal to 15.92 * 2 = 31.83.
it's not 31.84 because i used number from the calculator that haven't been rounded yet.
this diagonal is the hypotenuse of a right triangle whose legs are the sides of the square.
we use the pythagorean formula to find the length of the legs.
the formula becomes:
2*L^2 = 31.83^2.
we divide this formula by 2 to get:
L^2 = 15.92^2 / 2
that gets us:
L = 22.5079079
4 times that is the perimeter of the square which equals 90.03163162.
L stands for Leg of the right triangle.
L is also the side of the square.
i will follow up with another picture to show you.
the circumscribed square is tangent to the circle at it's apothem.
the apothem turns out to be the length of the radius of the circle.
extend that across and you have the length of a side of the circumscribed square.
times that by 4 and you get the perimeter of the circumscribed square.
it turns out to be 4 * 31.83 = 127.3239545
the picture below shows you the inscribed square measurements and the circumscribed square measurements.
