Suppose you have a circle of radius 25 inches If you inscrib

Suppose you have a circle of radius 25 inches. If you inscribe a square in the circle, draw and label a sketch. Then find the exact length of each side. Then add this information to your sketch. What is the total area of the four \"lemon wedges\" in your figure? Be sure to answer this question in a complete English sentence. Use (ii) to find the area of your square. Now repeat (i) and (ii) for an inscribed pentagon, ending with the necessary English sentence. (There will now be five lemon wedges.) Use (iv) to write an exact formula for the area of your pentagon. Now find a formula if you draw an n-gon. (where n is any positive integer above 2 Your answer must have an n in it, so do not just plug in a particular number here, but think in a general way.) See if you can find the trend, as n becomes very large Then explain in a complete English sentence.

Solution

i) to iii)

Since two opposing corners of the square touch the circle, that diagonal in the square is equivalent to the diameter (2r) of the circle.

Now to find one side of the square we will set up pythag theorem:

a^2 + b^2 = c^2
Since it\'s a square a = b, which means the sides on all 4 sides are same , ya?
2a^2 = c^2
We know c, which is 2r, the diagonal.
2a^2 = (2r)^2
2a^2 = 4r^2
a^2 = 2r^2

We can leave it as a^2 since a is one side of the square and a^2 wouldbe the area of the square.

So the area of the square is 2r^2

For the perimeter, we found that
a^2 = 2r^2, so solve for 4a.
a = (2r^2)
a = r2
4a = 4r2

So the area of the square is 2r^2
And the perimiter of the square is 42*r