(a)
The " vertex angle of measure " is \(\frac{360^{\circ}}n\) , and angle 1 is half of this angle. So...
m∠1 = \(\frac12*\frac{360^{\circ}}n=\frac{360^{\circ}}{2n}=\frac{180^{\circ}}{n}\)
(b)
sin (an angle) = (side opposite) / (hypotenuse)
sin (angle 1) = x / r
sin ( \(\frac{180^{\circ}}{n}\) ) = x/r
r * sin ( \(\frac{180^{\circ}}{n}\) ) = x
x = r * sin ( \(\frac{180^{\circ}}{n}\) )
(c)
Two x's fit on each side...and if there are n sides, then
perimeter = 2x * n
perimeter = 2 * r * sin ( \(\frac{180^{\circ}}{n}\) ) * n
(d)
The more sides you add, the closer the shape gets to becoming a circle. Think about a pentagon (5 sides) compared to a decagon (10 sides). The decagon is a lot more like a circle than the pentagon. So...the bigger "n" gets, the more circle-like the shape is, and the perimeter gets bigger and bigger the more sides you add.
Look: \(5\sin(\frac{180}{5})\approx 2.94 \\~\\ 10\sin(\frac{180}{10})\approx 3.09\)
(e)
Sorry Kakarot...I don't really think I know this one!
