正8角形?应该是正八边形吧?
已知什么?所求的直径是外接圆直径还是内切圆直径?
1、如果已知边长a,所求为内切圆直径d
d=a/tg(360°/16)
=a/tg(45°/2)
tg(45°/2)
=sin45°/(1+cos45°)
=(√2/2)/(1+√2/2)
=(√2/2)/[(2+√2)/2]
=(√2/2)×2/(2+√2)
=√2/(2+√2)
=√2(2-√2)/[(2+√2)(2-√2)]
=(2√2-2)/2
=√2-1
即：d=a/(√2-1)=a(√2+1)
所求内切圆直径为：d=a(√2+1)
2、如果已知边长a,所求为外接圆直径D：
D=a/sin(360°/16)
=a/sin(45°/2)
sin(45°/2)
=√[(1-cos45°)/2]
=√[(1-√2/2)/2]
=√{[(2-√2)/2]/2}
=[√(2-√2)]/2
即：D=a/{[√(2-√2)]/2}=2a/√(2-√2)]
所求外接圆直径为：D=2a/√(2-√2)]