A3=2Rsin( 180 degrees /3)

=2Rsin60 degrees

= (root number 3) R,

A4=2Rsin( 180 degrees /4)

= (radical number 2)R,

So the perimeter of a regular triangle inscribed in a circle =(3 root numbers 3)R,

Area = [(root number 3)/4][ (root number 3) r] 2

=(3 radicals 3) r 2/4.

Circumference of a square inscribed in a circle = (radical number 2 of 4) r

Area = [(root number 2) r] 2

=2R^2.

(Note: area formula of regular triangle: area of regular triangle = [(root number 3)/4] times square of side length. )

7。 Because r/R=cos( 180 degrees /3)=cos60 degrees = 1/2.

So R=2r

And because r-r = 2.

So R=4.

So the side length of this regular triangle a3=4 root number 3.

Area = [(root number 3)/4] (root number 4 3) 2

= 12 root number 3.

8。 It is proved that: (1) Because all sides of a regular Pentagon are equal, all internal angles are equal to 108 degrees.

So angle AEB= angle ABE=36 degrees, angle BAC= angle BCA=36 degrees,

So angle EAP= angle EAB-angle BAC=72 degrees,

Angle EPA= angle ABE+ angle BAC=72 degrees,

So angle EAP= angle EPA,

So PE=AE=AB.

(2) Because angle AEB= angle BAC=36 degrees and angle ABE= angle PBA,

Triangle AEB is similar to triangle PAB,

So AB/BP=BE/AB

So ab 2 = be * BP

Because PE=AE=AB.

So PE 2 = be * BP.