20/12/08 15:39:04.60 em4cCnZq.net
円弧Oの半径をr, 中心角を180°,両端に長さhの線分を追加
円弧Aの半径をR, 中心角を240°
とおくと
d = 2r = (√3)R,
L = ∂O + ∂A - d
= (2+π)r + 2h + (4π/3 + √3)R - d,
= {(√3)(1+π/2) + 4π/3}R + 2h,
面積条件
S_o = (π/2)rr + 2rh = 1,
S_a = (2π/3 + √3)RR = 2,
から
R = (√3)/√{π + 3(√3)/8} = 0.88956428208625
r = (√3)/2・R = 0.77038526658596
これより
∂O = (2+π)r + 2h = (2+π/2)r + 1/r
= 4.04894070352596 = 2√π + 0.50403300
∂A = (4π/3 + √3)R
= 5.26696868450265 = 2√(2π) + 0.253712135
L = (4π/3 + 2√3)R + (4π/3 - √3)r = 7.77513885485668
(大意)
共有境界が線分の場合
(180,180) 7.86233808183232 >>816
(180,240) 7.77513885485668
(240,240) 7.74842993556412 >>876
(無条件) 7.71391 >>872
いずれも >>801 より大きい