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Solutions: 92
There are exactly 92 equable right prisms with an integer height and a base that is a cyclic Heronian quadrilateral (CHQ).
Proof. The analysis of equable quadrilaterals produced the following list of 107 equable CHQ prisms with integer heights, ordered by volume:
There is one equable cube, with all edges equal to 6. There are nine other equable square or rectangular prisms (cubiods), and the 15 duplicates identified in the table above are reorientations of them that produce different heights. For example, the flat cuboid with a 12 x 12 square base and a height of 3 is a reorientation of a narrow cuboid with a 12 x 3 rectangular base and a height of 12. None of the cuboids are Euler bricks. Ignoring duplicates, there are 92 unique equable CHQ prisms with integer heights, which is surprisingly close in quantity to the 94 equable triangular prisms with integer heights.
a b c d h V=S P A t p Description 6 6 6 6 6 216 24 36 1.5000 8.4853 Cube 5 5 5 5 10 250 20 25 1.2500 7.0711 Cuboid 10 10 5 5 5 250 30 50 1.6667 8.9443 Duplicate 8 8 4 4 8 256 24 32 1.3333 7.1554 Cuboid 8 8 8 8 4 256 32 64 2.0000 11.3137 Duplicate 6 6 4 4 12 288 20 24 1.2000 6.6564 Cuboid 12 12 4 4 6 288 32 48 1.5000 7.5895 Duplicate 12 12 6 6 4 288 36 72 2.0000 10.7331 Duplicate 12 9 8 1 7 294 30 42 1.4000 8.7198 9 7 6 2 10 300 24 30 1.2500 7.3756 14 13 6 3 5 300 36 60 1.6667 8.5907 15 10 8 3 5 300 36 60 1.6667 10.3807 Bicentric 13 7 5 5 8 320 30 40 1.3333 9.2848 16 10 10 4 4 320 40 80 2.0000 12.8062 Bicentric 11 10 5 2 9 324 28 36 1.2857 6.7082 12 6 5 5 9 324 28 36 1.2857 9.1381 18 12 5 5 6 360 40 60 1.5000 9.6623 18 10 10 2 6 360 40 60 1.5000 11.6619 Bicentric 16 12 6 2 8 384 36 48 1.3333 7.8215 Bicentric 10 5 5 4 14 392 24 28 1.1667 8.0623 5 5 4 4 20 400 18 20 1.1111 6.2470 Cuboid 20 20 4 4 5 400 48 80 1.6667 7.8446 Duplicate 22 14 8 4 5 400 48 80 1.6667 11.6821 20 20 5 5 4 400 50 100 2.0000 9.7014 Duplicate 12 12 3 3 12 432 30 36 1.2000 5.8209 Cuboid 12 12 12 12 3 432 48 144 3.0000 16.9706 Duplicate 24 18 9 3 4 432 54 108 2.0000 11.7323 Bicentric 10 10 3 3 15 450 26 30 1.1538 5.7470 Cuboid 11 8 4 3 15 450 26 30 1.1538 6.6483 15 15 3 3 10 450 36 45 1.2500 5.8835 Duplicate 15 15 10 10 3 450 50 150 3.0000 16.6410 Duplicate 25 18 9 2 5 450 54 90 1.6667 10.8734 Bicentric 18 13 13 8 3 468 52 156 3.0000 17.6918 Bicentric 27 22 6 5 4 480 60 120 2.0000 10.7882 28 12 10 10 4 480 60 120 2.0000 19.1565 9 9 3 3 18 486 24 27 1.1250 5.6921 Cuboid 18 18 3 3 9 486 42 54 1.2857 5.9184 Duplicate 18 18 9 9 3 486 54 162 3.0000 16.0997 Duplicate 28 22 5 5 5 500 60 100 1.6667 9.8744 21 16 12 7 3 504 56 168 3.0000 17.1406 Bicentric 28 24 5 3 6 540 60 90 1.5000 7.9342 22 21 12 5 3 540 60 180 3.0000 15.9968 22 20 15 3 3 540 60 180 3.0000 17.1554 24 15 15 6 3 540 60 180 3.0000 19.2094 Bicentric 32 17 17 2 4 544 68 136 2.0000 18.7883 Bicentric 8 8 3 3 24 576 22 24 1.0909 5.6180 Cuboid 9 5 5 3 24 576 22 24 1.0909 7.2111 18 16 4 2 12 576 40 48 1.2000 5.9275 Bicentric 24 24 3 3 8 576 54 72 1.3333 5.9537 Duplicate 24 24 8 8 3 576 64 192 3.0000 15.1789 Duplicate 29 16 9 6 7 588 60 84 1.4000 14.8041 27 22 11 6 3 594 66 198 3.0000 16.2049 Bicentric 23 12 9 4 10 600 48 60 1.2500 12.7852 30 21 14 5 3 630 70 210 3.0000 18.2483 Bicentric 27 16 12 1 9 648 56 72 1.2857 12.9500 Bicentric 25 13 13 1 10 650 52 65 1.2500 13.9284 Bicentric 38 14 13 13 5 650 78 130 1.6667 25.5322 40 35 7 2 5 700 84 140 1.6667 8.9691 Bicentric 41 22 14 7 5 700 84 140 1.6667 20.8125 32 22 11 1 8 704 66 88 1.3333 11.9718 Bicentric 24 16 5 5 12 720 50 60 1.2000 9.8894 24 15 10 1 12 720 50 60 1.2000 10.9507 Bicentric 36 24 10 10 3 720 80 240 3.0000 19.3247 36 20 20 4 3 720 80 240 3.0000 23.3238 Bicentric 26 18 5 5 11 726 54 66 1.2222 9.9083 29 21 5 5 10 750 60 75 1.2500 9.9288 20 12 5 5 16 768 42 48 1.1429 9.8287 19 11 5 5 18 810 40 45 1.1250 9.8058 34 26 5 5 9 810 70 90 1.2857 9.9504 42 18 15 15 3 810 90 270 3.0000 28.7348 50 34 17 1 5 850 102 170 1.6667 17.9813 Bicentric 29 22 6 3 12 864 60 72 1.2000 8.9511 7 7 3 3 42 882 20 21 1.0500 5.5149 Cuboid 18 10 5 5 21 882 38 42 1.1053 9.7780 42 42 3 3 7 882 90 126 1.4000 5.9848 Duplicate 42 42 7 7 3 882 98 294 3.0000 13.8095 Duplicate 25 18 8 1 15 900 52 60 1.1538 8.9689 Bicentric 45 40 10 5 3 900 100 300 3.0000 14.8187 Bicentric 48 39 13 4 3 936 104 312 3.0000 16.8338 Bicentric 44 36 5 5 8 960 90 120 1.3333 9.9720 49 40 10 1 7 980 100 140 1.4000 10.9908 Bicentric 62 36 21 7 4 1008 126 252 2.0000 27.8806 17 9 5 5 26 1014 36 39 1.0833 9.7439 54 38 19 3 3 1026 114 342 3.0000 21.8595 Bicentric 64 40 13 13 4 1040 130 260 2.0000 25.8806 56 48 10 6 3 1080 120 360 3.0000 15.8685 27 21 7 1 18 1134 56 63 1.1250 7.9786 Bicentric 63 26 26 15 3 1170 130 390 3.0000 40.2616 25 24 3 2 20 1200 54 60 1.1111 4.9762 Bicentric 66 36 17 17 3 1224 136 408 3.0000 33.5893 68 35 30 7 3 1260 140 420 3.0000 36.6919 16 8 5 5 36 1296 34 36 1.0588 9.7014 72 37 37 2 3 1332 148 444 3.0000 38.8973 Bicentric 74 66 5 5 7 1470 150 210 1.4000 9.9908 88 65 18 9 3 1620 180 540 3.0000 26.8978 32 27 6 1 24 1728 66 72 1.0909 6.9882 Bicentric 19 13 6 2 42 1764 40 42 1.0500 7.9185 27 22 4 3 30 1800 56 60 1.0714 6.9660 15 7 5 5 66 2178 32 33 1.0312 9.6476 122 42 41 41 3 2214 246 738 3.0000 81.5105 125 54 45 28 3 2268 252 756 3.0000 72.6890 129 65 40 26 3 2340 260 780 3.0000 65.7823 149 96 30 25 3 2700 300 900 3.0000 54.9021 169 120 34 17 3 3060 340 1020 3.0000 50.9456 174 126 25 25 3 3150 350 1050 3.0000 49.9456 49 45 5 1 42 4410 100 105 1.0500 5.9962 Bicentric 399 360 25 16 3 7200 800 2400 3.0000 40.9946
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