Equable Right Quadrilateral Prism

EQUABLE RIGHT QUADRILATERAL PRISM

Balmoral Software

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:

abcdhV=SPAtpDescription
6666621624361.50008.4853Cube
55551025020251.25007.0711Cuboid
101055525030501.66678.9443Duplicate
8844825624321.33337.1554Cuboid
8888425632642.000011.3137Duplicate
66441228820241.20006.6564Cuboid
121244628832481.50007.5895Duplicate
121266428836722.000010.7331Duplicate
12981729430421.40008.7198
97621030024301.25007.3756
141363530036601.66678.5907
151083530036601.666710.3807Bicentric
13755832030401.33339.2848
1610104432040802.000012.8062Bicentric
111052932428361.28576.7082
12655932428361.28579.1381
181255636040601.50009.6623
1810102636040601.500011.6619Bicentric
161262838436481.33337.8215Bicentric
105541439224281.16678.0623
55442040018201.11116.2470Cuboid
202044540048801.66677.8446Duplicate
221484540048801.666711.6821
2020554400501002.00009.7014Duplicate
1212331243230361.20005.8209Cuboid
121212123432481443.000016.9706Duplicate
2418934432541082.000011.7323Bicentric
1010331545026301.15385.7470Cuboid
118431545026301.15386.6483
1515331045036451.25005.8835Duplicate
151510103450501503.000016.6410Duplicate
251892545054901.666710.8734Bicentric
18131383468521563.000017.6918Bicentric
2722654480601202.000010.7882
281210104480601202.000019.1565
99331848624271.12505.6921Cuboid
181833948642541.28575.9184Duplicate
1818993486541623.000016.0997Duplicate
2822555500601001.66679.8744
21161273504561683.000017.1406Bicentric
282453654060901.50007.9342
22211253540601803.000015.9968
22201533540601803.000017.1554
24151563540601803.000019.2094Bicentric
32171724544681362.000018.7883Bicentric
88332457622241.09095.6180Cuboid
95532457622241.09097.2111
1816421257640481.20005.9275Bicentric
242433857654721.33335.9537Duplicate
2424883576641923.000015.1789Duplicate
291696758860841.400014.8041
27221163594661983.000016.2049Bicentric
2312941060048601.250012.7852
30211453630702103.000018.2483Bicentric
2716121964856721.285712.9500Bicentric
25131311065052651.250013.9284Bicentric
381413135650781301.666725.5322
4035725700841401.66678.9691Bicentric
41221475700841401.666720.8125
3222111870466881.333311.9718Bicentric
2416551272050601.20009.8894
24151011272050601.200010.9507Bicentric
362410103720802403.000019.3247
36202043720802403.000023.3238Bicentric
2618551172654661.22229.9083
2921551075060751.25009.9288
2012551676842481.14299.8287
1911551881040451.12509.8058
342655981070901.28579.9504
421815153810902703.000028.7348
503417158501021701.666717.9813Bicentric
2922631286460721.20008.9511
77334288220211.05005.5149Cuboid
1810552188238421.10539.7780
4242337882901261.40005.9848Duplicate
4242773882982943.000013.8095Duplicate
2518811590052601.15388.9689Bicentric
454010539001003003.000014.8187Bicentric
483913439361043123.000016.8338Bicentric
4436558960901201.33339.9720
494010179801001401.400010.9908Bicentric
6236217410081262522.000027.8806
1795526101436391.08339.7439
5438193310261143423.000021.8595Bicentric
64401313410401302602.000025.8806
5648106310801203603.000015.8685
27217118113456631.12507.9786Bicentric
63262615311701303903.000040.2616
25243220120054601.11114.9762Bicentric
66361717312241364083.000033.5893
6835307312601404203.000036.6919
1685536129634361.05889.7014
7237372313321484443.000038.8973Bicentric
746655714701502101.40009.9908
8865189316201805403.000026.8978
32276124172866721.09096.9882Bicentric
19136242176440421.05007.9185
27224330180056601.07146.9660
1575566217832331.03129.6476
122424141322142467383.000081.5105
125544528322682527563.000072.6890
129654026323402607803.000065.7823
149963025327003009003.000054.9021
16912034173306034010203.000050.9456
17412625253315035010503.000049.9456
4945514244101001051.05005.9962Bicentric
39936025163720080024003.000040.9946
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.


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