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Trirectangular tetrahedron
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<h2 id="metric-formulas">Metric formulas</h2> <p>If the legs have lengths <i>a, b, c</i>, then the trirectangular tetrahedron has the <a href="/facts/Volume/aeAXCBUd">volume</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> V = a b c 6 . {\displaystyle V={\frac {abc}{6}}.} <p>The altitude <i>h</i> satisfies<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> 1 h 2 = 1 a 2 + 1 b 2 + 1 c 2 . {\displaystyle {\frac {1}{h^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}.} <p>The area T 0 {\displaystyle T_{0}} of the base is given by<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> T 0 = a b c 2 h . {\displaystyle T_{0}={\frac {abc}{2h}}.} <p>The <a href="/facts/Solid_angle/se8IXufg">solid angle</a> at the right-angled vertex, from which the opposite face (the base) subtends an <a href="/facts/Octant_of_a_sphere/PnNATHCv">octant</a>, has measure π/2 <a href="/facts/Steradian/PRCqT4n1">steradians</a>, one eighth of the surface area of a <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a>. </p> <h2 id="de-guas-theorem">De Gua's theorem</h2> <p class="note">Main article: <a href="/facts/De_Gua%27s_theorem/8yD3XQap">De Gua's theorem</a></p> <p>If the <a href="/facts/Area/KsNG1G9t">area</a> of the base is T 0 {\displaystyle T_{0}} and the areas of the three other (right-angled) faces are T 1 {\displaystyle T_{1}} , T 2 {\displaystyle T_{2}} and T 3 {\displaystyle T_{3}} , then </p> T 0 2 = T 1 2 + T 2 2 + T 3 2 . {\displaystyle T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.} <p>This is a generalization of the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a> to a tetrahedron. </p> <h2 id="integer-solution">Integer solution</h2> <h3>Integer edges</h3> <p>Trirectangular tetrahedrons with integer legs a , b , c {\displaystyle a,b,c} and sides d = b 2 + c 2 , e = a 2 + c 2 , f = a 2 + b 2 {\displaystyle d={\sqrt {b^{2}+c^{2}}},e={\sqrt {a^{2}+c^{2}}},f={\sqrt {a^{2}+b^{2}}}} of the base triangle exist, e.g. a = 240 , b = 117 , c = 44 , d = 125 , e = 244 , f = 267 {\displaystyle a=240,b=117,c=44,d=125,e=244,f=267} (discovered 1719 by Halcke). Here are a few more examples with integer legs and sides. </p> a b c d e f 240 117 44 125 244 267 275 252 240 348 365 373 480 234 88 250 488 534 550 504 480 696 730 746 693 480 140 500 707 843 720 351 132 375 732 801 720 132 85 157 725 732 792 231 160 281 808 825 825 756 720 1044 1095 1119 960 468 176 500 976 1068 1100 1008 960 1392 1460 1492 1155 1100 1008 1492 1533 1595 1200 585 220 625 1220 1335 1375 1260 1200 1740 1825 1865 1386 960 280 1000 1414 1686 1440 702 264 750 1464 1602 1440 264 170 314 1450 1464 <p>Notice that some of these are multiples of smaller ones. Note also A031173. </p> <h3>Integer faces</h3> <p>Trirectangular tetrahedrons with integer faces T c , T a , T b , T 0 {\displaystyle T_{c},T_{a},T_{b},T_{0}} and altitude <i>h</i> exist, e.g. a = 42 , b = 28 , c = 14 , T c = 588 , T a = 196 , T b = 294 , T 0 = 686 , h = 12 {\displaystyle a=42,b=28,c=14,T_{c}=588,T_{a}=196,T_{b}=294,T_{0}=686,h=12} without or a = 156 , b = 80 , c = 65 , T c = 6240 , T a = 2600 , T b = 5070 , T 0 = 8450 , h = 48 {\displaystyle a=156,b=80,c=65,T_{c}=6240,T_{a}=2600,T_{b}=5070,T_{0}=8450,h=48} with coprime a , b , c {\displaystyle a,b,c} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Tetrahedron/7mkSrl6j">Irregular tetrahedra</a></li> <li><a href="/facts/Standard_simplex/0FCfNRN8">Standard simplex</a></li> <li><a href="/facts/Euler_Brick/9h5MIJrK">Euler Brick</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/TrirectangularTetrahedron.html">"Trirectangular tetrahedron"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kepler, Johannes (1619). Harmonices Mundi (in Latin). p. 181. <a href="/wiki/Johannes_Kepler" target="_blank">/wiki/Johannes_Kepler</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Antonio Caminha Muniz Neto (2018). An Excursion through Elementary Mathematics, Volume II: Euclidean Geometry. Springer. p. 437. ISBN 978-3-319-77974-4. Problem 3 on page 437 <a href="978-3-319-77974-4" target="_blank">978-3-319-77974-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Gutierrez, Antonio, "Right Triangle Formulas" <a href="http://gogeometry.com/pythagoras/right_triangle_formulas_facts.htm" target="_blank">http://gogeometry.com/pythagoras/right_triangle_formulas_facts.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>