Equable Right Triangular Prism

EQUABLE RIGHT TRIANGULAR PRISM

Balmoral Software

Solutions: 94

There are exactly 94 equable right prisms with an integer height and a base that is a Heronian triangle (HT).

Proof. The analysis of equable triangles produced the following list of 94 equable HT prisms with integer heights, ordered by volume:

 a b c h V=S P A t Description 12 10 10 6 288 32 48 1.5000 Acute Isosceles 13 13 10 5 300 36 60 1.6667 Acute Isosceles 15 12 9 6 324 36 54 1.5000 Right 15 14 13 4 336 42 84 2.0000 Acute 17 15 8 6 360 40 60 1.5000 Right 16 10 10 8 384 36 48 1.3333 Isosceles 20 16 12 4 384 48 96 2.0000 Right 20 13 11 6 396 44 66 1.5000 24 15 15 4 432 54 108 2.0000 Isosceles 25 17 12 5 450 54 90 1.6667 26 24 10 4 480 60 120 2.0000 Right 25 24 7 6 504 56 84 1.5000 Right 30 25 11 4 528 66 132 2.0000 30 26 8 6 576 64 96 1.5000 24 20 20 3 576 64 192 3.0000 Acute Isosceles 34 20 18 4 576 72 144 2.0000 26 25 17 3 612 68 204 3.0000 Acute 37 26 15 4 624 78 156 2.0000 28 25 17 3 630 70 210 3.0000 Acute 29 21 20 3 630 70 210 3.0000 Right 30 24 18 3 648 72 216 3.0000 Right 40 30 14 4 672 84 168 2.0000 39 35 10 4 672 84 168 2.0000 37 20 19 6 684 76 114 1.5000 24 13 13 12 720 50 60 1.2000 Isosceles 39 25 16 6 720 80 120 1.5000 34 30 16 3 720 80 240 3.0000 Right 41 40 9 4 720 90 180 2.0000 Right 41 28 15 6 756 84 126 1.5000 35 34 15 3 756 84 252 3.0000 Acute 40 26 22 3 792 88 264 3.0000 39 36 15 3 810 90 270 3.0000 Right 51 40 13 6 936 104 156 1.5000 58 50 12 4 960 120 240 2.0000 51 30 27 3 972 108 324 3.0000 52 33 25 3 990 110 330 3.0000 53 35 24 3 1008 112 336 3.0000 50 48 14 3 1008 112 336 3.0000 Right 65 34 33 4 1056 132 264 2.0000 60 45 21 3 1134 126 378 3.0000 52 50 6 8 1152 108 144 1.3333 60 52 16 3 1152 128 384 3.0000 35 29 8 14 1176 72 84 1.1667 65 55 12 6 1188 132 198 1.5000 74 51 25 4 1200 150 300 2.0000 65 51 20 3 1224 136 408 3.0000 68 65 7 6 1260 140 210 1.5000 65 61 14 3 1260 140 420 3.0000 30 29 5 18 1296 64 72 1.1250 78 75 9 4 1296 162 324 2.0000 74 40 38 3 1368 152 456 3.0000 73 60 19 3 1368 152 456 3.0000 75 44 35 3 1386 154 462 3.0000 78 50 32 3 1440 160 480 3.0000 82 56 30 3 1512 168 504 3.0000 85 60 29 3 1566 174 522 3.0000 97 90 11 4 1584 198 396 2.0000 87 75 18 3 1620 180 540 3.0000 89 65 28 3 1638 182 546 3.0000 85 84 13 3 1638 182 546 3.0000 Right 104 85 21 4 1680 210 420 2.0000 80 73 9 8 1728 162 216 1.3333 52 51 5 14 1764 108 126 1.1667 106 102 8 5 1800 216 360 1.6667 102 80 26 3 1872 208 624 3.0000 109 100 11 6 1980 220 330 1.5000 113 92 25 3 2070 230 690 3.0000 116 105 17 3 2142 238 714 3.0000 125 122 13 3 2340 260 780 3.0000 130 110 24 3 2376 264 792 3.0000 136 130 14 3 2520 280 840 3.0000 145 74 73 3 2628 292 876 3.0000 170 153 19 4 2736 342 684 2.0000 159 140 23 3 2898 322 966 3.0000 164 111 55 3 2970 330 990 3.0000 195 148 49 3 3528 392 1176 3.0000 123 122 5 12 3600 250 300 1.2000 205 195 16 3 3744 416 1248 3.0000 218 200 22 3 3960 440 1320 3.0000 229 185 46 3 4140 460 1380 3.0000 305 289 18 4 4896 612 1224 2.0000 174 169 7 12 5040 350 420 1.2000 300 259 43 3 5418 602 1806 3.0000 298 296 6 7 5880 600 840 1.4000 397 380 21 3 7182 798 2394 3.0000 409 370 41 3 7380 820 2460 3.0000 449 442 9 7 8820 900 1260 1.4000 519 481 40 3 9360 1040 3120 3.0000 740 703 39 3 13338 1482 4446 3.0000 485 481 6 20 21600 972 1080 1.1111 1405 1369 38 3 25308 2812 8436 3.0000 1213 1212 5 11 32670 2430 2970 1.2222 1700 1695 7 11 45738 3402 4158 1.2222 8669 8665 6 19 368220 17340 19380 1.1176
Six of these prisms have isosceles bases. Twelve have bases that are right triangles, seven have bases that are acute, and the remainder have bases that are obtuse. These 94 prisms are surprisingly close in quantity to the 92 equable quadrilateral prisms with integer heights.