设正三棱锥的高为h，则底面三角形所在的圆的半径r = √{3²-(3-h)²} = √(6h-h²)底面积 V = 3×1/2×r²×sin120° = 3√3/4(6h-h²)V = 1/3Sh = 1/3×3√3/4(6h-h²)×h = √3/4(6h²-h³)V ′ = √3/4(12h-3h²) = (3√3/4)h(4-h)当h=4时，体积有极大值Vmax= √3/4(6×4²-4³) = 8√3
