一个正方体和球的表面积相等 求体积比

问题描述:

一个正方体和球的表面积相等 求体积比

6a^2=4πr^2
a=√(2πr^2/3)
则体积比:
a^3:(4πr^3/3)
=[√(2πr^2/3)]^3*3/(4πr^3)
=r^3/3*√(2π/3)*3/(4πr^3)
=√(π/6) .

设正方体的棱长是a ,球体的最大圆半径是r ,根据正方体和球的表面积相等,得:
6a^2=4πr^2
a=√(2πr^2/3)
则体积比:
a^3:(4πr^3/3)
=[√(2πr^2/3)]^3*3/(4πr^3)
=r^3/3*√(2π/3)*3/(4πr^3)
=√(π/6) .

6a² = 4πR²
a²/R² = 2π/3
a/R = √(2π/3)
V方 / V球
= a³ / (4πR³/3)
= (3 /4π) × a³ / R³
= (3 /4π) × (2π/3)^(3/2)
= (3 /4π) × (2π/3)×√(2π/3)
= √(2π/3) / 2
= √(6π) / 6