Jeremy Galvagni got me to bold the entries that are known to be optimal.

n=1area = 1
| n=2area = 1
| n=3 area = 1.195+ (MM) | n=4 area = 20√2–27 = 1.284+ (MM) | |||

n=5area = 5/4 = 1.25
| n=6area = 3√3/4 = 1.299+
| n=7 area = 1.332+ (MM) | n=8area = 4/3 = 1.333+ (BZ)
| |||

n=9area = 3√3/4 = 1.299+ (MM)
| n=10 area = 1.359+ (MM) | n=11area = 11/8 = 1.375 (MM)
| n=12 area = (24–7√3)/9 = 1.319+ (MM) | |||

n=13 area = 1.379+ (MM) | n=14area = 7/5 = 1.4 (MM)
| n=15 area = 1.340+ (MM) | n=16 area = (63–34√3)/3 = 1.370+ (MM) |

n=1area = π/4 = .785+
| n=2 area = .844+ (JD) | n=3 area = (3√3 + 4π)/16 = 1.110+ | n=4 area = 1.029+ (MM) | |||

n=5 area = 5π/16 = .981+ | n=6 area = (3√3 + 4π)/16 = 1.110+ | n=7 area = 7√3/12 = 1.013+ | n=8 area = 1.100+ (MM) | |||

n=9area = 3π/8 = 1.178+
| n=10area = 5π/14 = 1.121+ (MM)
| n=11 area = 11π/32 = 1.079+ (MM) | n=12 area = 1.150+ (MM) | |||

n=13 area = 1.131+ (AB) | n=14 area = 7π/20 = 1.099+ (AB) | n=15 area = 1.159+ (BZ) | n=16 area = 4π/11 = 1.142+ (MM) |

n=1area = 1
| n=2area = π/2 = 1.570+
| n=3 area = (3√3 + 2π)/8 = 1.434+ | n=4area = π/2 = 1.570+
| |||

n=5 area = 1.596+ (MM) | n=6area = π/2 = 1.570+
| n=7 area = 1.618+ (MM) | n=8area = π/2 = 1.570+
| |||

n=9 area = 1.664+ (JD) | n=10 area = 1.637+ (JD) | n=11 area = 55/36 = 1.527+ (MM) | n=12 area = π/2 = 1.570+ | |||

n=13 area = 1.537+ (JD) | n=14 area = 1.598+ (JD) | n=15 area = 1.650+ (BZ) | n=16 area = 8/5 = 1.6 (BZ) |

n=1area = 9
| n=2area = 9
| n=3 area = 21/2 = 10.5 | n=4 area = 11.293+ (MM) | |||

n=5area = 45/4 = 11.25
| n=6area = 12
| n=7 area = 49/4 = 12.25 (BZ) | n=8area = 12
| |||

n=9 area = 11.823 (JD) | n=10 area = 12.536+ (JD) | n=11area = 99/8 = 12.375 (MM)
| n=12area = 12
| |||

n=13area = 13
| n=14area = 63/5 = 12.6 (MM)
| n=15 area = 12.093+ (BZ) | n=16 area = 112/9 = 12.444+ (MM) |

n=1area = 9√3/4 = 3.897+
| n=2area = 9√3/4 = 3.897+
| n=3 area = 21√3/8 = 4.546+ | n=4area = 3√3 = 5.196+
| |||

n=5area = 45√3/16 = 4.871+
| n=6area = 3√3 = 5.196+
| n=7 area = 21(63–34√3)/16 = 5.394+ (MM) | n=8area = 3√3 = 5.196+ (MM)
| |||

n=9area = 3√3 = 5.196+ (MM)
| n=10 area = 20(2–√3) = 5.358+ (MM) | n=11 area = 11(45–17√3)/32 = 5.347+ (MM) | n=12area = 3√3 = 5.196+
| |||

n=13area = 13√3/4 = 5.629+
| n=14 area = 5.417+ (MM) | n=15 area = 5.155+ (MM) | n=16 area = 16(9–5√3) = 5.435+ (MM) |

n=1area = 9π/4 = 7.068+
| n=2area = 9π/4 = 7.068+
| n=3 area = 7.809+ (JG) | n=4 area = 8.666+ (JG) | |||

n=5 area = 8.796+ (JG) | n=6area = 3π = 9.424+ (JG)
| n=7 area = 9.568+ (MM) | n=8 area = 9.373+ (MM) | |||

n=9area = 3π = 9.424+ (AB)
| n=10 area = 9.658+ (JD) | n=11 area = 9.600+ (JD) | n=12area = 3π = 9.424+ (JD)
| |||

n=13 area = 9.759+ (JG) | n=14 area = 9.768+ (JG) | n=15 area = 9.606+ (JD) | n=16 area = 9.736+ (JD) |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/21/12.