1041:
1049:
878:
94:
38:
806:
2356:
2452:
870:
1033:
818:
859:
1477:
the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the
Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".
1476:
the side of the other." The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the
Pythagorean theorem." Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find
2550:
2701:
2176:
1422:, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. It was first conjectured by the historian
2466:
108:
special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90
1480:
The following are all the
Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
73:
that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of
2622:
2758:
2631:
2574:
2906:
2886:
2866:
2846:
2826:
2806:
2786:
2436:. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.
81:. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in
1426:
in 1882. It is known that right angles were laid out accurately in
Ancient Egypt; that their surveyors did use ropes for measurement; that
2082:
1277:
of the sides produces the same relationship. Using Euclid's formula for generating
Pythagorean triples, the sides must be in the ratio
3124:
2237:
1959:
Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is
17:
2579:
3006:
192:
1444:(before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is
3272:
1064:
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (
187:
182:
177:
1980:
right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the
1967:
3215:
3277:
2545:{\displaystyle a=2\sin {\frac {\pi }{10}}={\frac {-1+{\sqrt {5}}}{2}}={\frac {1}{\varphi }}\approx 0.618}
2710:
1337:
There are several
Pythagorean triples which are well-known, including those with sides in the ratios:
1436:(around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle; and that the
1048:
1040:
877:
2951:
2344:
843:
97:
Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering
3231:
2363:
is a right triangle formed by three squares with areas in geometric progression according to the
1441:
145:
98:
3031:
2760:, so these three lengths form the sides of a right triangle. The same triangle forms half of a
2440:
1404:
1274:
1197:
2559:
2456:
2380:
955:
Of all right triangles, such 45° - 45° - 90° degree triangles have the smallest ratio of the
3202:
3173:
3157:
3104:
831:
151:
Special triangles are used to aid in calculating common trigonometric functions, as below:
50:
8:
2928:
2765:
2696:{\displaystyle c=2\sin {\frac {\pi }{5}}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}\approx 1.176}
1984:
differ by one. Such almost-isosceles right-angled triangles can be obtained recursively,
1437:
1432:
1189:
1021:
981:
949:
3190:
3024:
2891:
2871:
2851:
2831:
2811:
2791:
2771:
1403:
The 3 : 4 : 5 triangles are the only right triangles with edges in
1332:
1270:
1258:
1017:
1013:
1009:
46:
923:
radians) and two other congruent angles each measuring half of a right angle (45°, or
835:
3085:
3002:
2966:
1408:
30:"90-45-45 triangle" and "30-60-90 triangle" redirect here. For the drawing tool, see
3182:
3046:
2923:
2917:
2761:
2439:
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in
1262:
1180:
1119:
93:
69:
easier, or for which simple formulas exist. For example, a right triangle may have
941:
radians). The sides in this triangle are in the ratio 1 : 1 :
37:
3256:
3251:
3198:
3153:
3100:
2996:
2374:
172:
77:, such as 3 : 4 : 5, or of other special numbers such as the
3138:
1008:
Triangles with these angles are the only possible right triangles that are also
2828:, and the endpoints of this line segment together with any of the neighbors of
1266:
1053:
894:
882:
847:
157:
109:
74:
62:
3246:
3266:
2190:
1423:
1419:
42:
3171:
Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles",
1245:
must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
2417:
2365:
2208:
1196:
The 30°–60°–90° triangle is the only right triangle whose angles are in an
1131:
1024:, there are infinitely many different shapes of right isosceles triangles.
78:
167:
137:
2294:
2244:
1714:
1600:
1484:
1200:. The proof of this fact is simple and follows on from the fact that if
27:
Right triangle with a feature making calculations on the triangle easier
3194:
1057:
956:
886:
862:
839:
144:
methods. This approach may be used to rapidly reproduce the values of
31:
1273:.) They are most useful in that they may be easily remembered and any
2171:{\displaystyle ({\tfrac {x-1}{2}})^{2}+({\tfrac {x+1}{2}})^{2}=y^{2}}
3186:
805:
2704:
1981:
1427:
1135:
141:
82:
66:
2355:
1418:
The possible use of the 3 : 4 : 5 triangle in
2625:
2553:
2451:
1254:
869:
1166:
is a 30°–60°–90° triangle with hypotenuse of length 2, and base
1032:
898:
162:
130:
3139:"A note on the set of almost-isosceles right-angled triangles"
1224:
are the angles in the progression then the sum of the angles
858:
136:
The side lengths are generally deduced from the basis of the
70:
2455:
The sides of a pentagon, hexagon, and decagon, inscribed in
830:
The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the
817:
3119:
2379:
The Kepler triangle is a right triangle whose sides are in
2232:
1828:
1412:
1341:
2343:
Alternatively, the same triangles can be derived from the
3001:(2nd ed.). John Wiley & Sons. pp. 237–238.
2383:. If the sides are formed from the geometric progression
65:
with some regular feature that makes calculations on the
2350:
1954:
3170:
2126:
2090:
2894:
2874:
2854:
2834:
2814:
2794:
2774:
2713:
2634:
2582:
2562:
2469:
2240:).. The smallest Pythagorean triples resulting are:
2085:
85:
problems without resorting to more advanced methods.
3021:
984:from the hypotenuse to the sum of the legs, namely
3023:
2900:
2880:
2860:
2840:
2820:
2800:
2780:
2752:
2695:
2616:
2568:
2544:
2170:
834:/equiangular (60°–60°–60°) triangle are the three
2848:form the vertices of a right triangle with sides
45:of types of triangles, using the definition that
3264:
1036:Set square, shaped as 30° - 60° - 90°° triangle
133:, is equal to the sum of the other two angles.
3259: – with interactive animations
2808:to the plane of its five neighbors has length
1974:cannot be expressed as a ratio of two integers
1257:lengths, with the sides collectively known as
873:The side lengths of a 45° - 45° - 90° triangle
1407:. Triangles based on Pythagorean triples are
1326:
3044:
2949:Posamentier, Alfred S., and Lehman, Ingmar.
2788:: the shortest line segment from any vertex
2446:
1118:). The sides are in the ratio 1 :
2990:
2988:
2986:
2984:
2982:
2980:
2920:, combining several special right triangles
2617:{\displaystyle b=2\sin {\frac {\pi }{6}}=1}
1027:
853:
2998:The History of Mathematics: A Brief Course
1044:The side lengths of a 30°–60°–90° triangle
3093:The Australasian Journal of Combinatorics
3086:"Almost-isosceles right-angled triangles"
41:Position of some special triangles in an
3083:
2977:
2450:
2354:
1047:
1039:
1031:
876:
868:
857:
92:
36:
3232:nLab: pentagon decagon hexagon identity
3026:Mathematics in the Time of the Pharaohs
2420:. Its sides are therefore in the ratio
1235:= 180°. After dividing by 3, the angle
14:
3265:
3247:3 : 4 : 5 triangle
3136:
1130:The proof of this fact is clear using
3045:Forget, T. W.; Larkin, T. A. (1968),
2994:
2964:
2351:Arithmetic and geometric progressions
948:, which follows immediately from the
2945:
2943:
2556:inscribed in the unit circle, where
1955:Almost-isosceles Pythagorean triples
1313:are any positive integers such that
1261:, possess angles that cannot all be
49:have at least two equal sides, i.e.
1253:Right triangles whose sides are of
24:
3059:described by recurrence sequences"
1146:with side length 2 and with point
905:, each with one right angle (90°,
901:along its diagonal results in two
865:shaped as 45° - 45° - 90° triangle
148:for the angles 30°, 45°, and 60°.
101:of multiples of 30 and 45 degrees.
25:
3289:
3240:
3084:Chen, C. C.; Peng, T. A. (1995),
2940:
2753:{\displaystyle a^{2}+b^{2}=c^{2}}
3047:"Pythagorean triads of the form
2764:. It may also be found within a
2703:be the side length of a regular
2624:be the side length of a regular
2552:be the side length of a regular
1175:The fact that the remaining leg
980:. and the greatest ratio of the
838:in the plane, meaning that they
816:
804:
3225:
3209:
959:to the sum of the legs, namely
3164:
3130:
3111:
3077:
3038:
3015:
2958:
2459:circles, form a right triangle
2146:
2122:
2110:
2086:
2076:= 1, 2, 3, .... Equivalently,
1154:. Draw an altitude line from
88:
13:
1:
3022:Gillings, Richard J. (1982).
2934:
1248:
1188:follows immediately from the
1142:Draw an equilateral triangle
2628:in the unit circle, and let
1411:, meaning they have integer
7:
3220:, Book XIII, Proposition 10
2911:
2336:
2333:
2330:
2325:
2322:
2319:
2314:
2311:
2308:
2303:
2300:
2297:
2286:
2283:
2280:
2275:
2272:
2269:
2264:
2261:
2258:
2253:
2250:
2247:
2207:being the odd terms of the
1976:. However, infinitely many
1947:
1944:
1941:
1936:
1933:
1930:
1925:
1922:
1919:
1914:
1911:
1908:
1903:
1900:
1897:
1892:
1889:
1886:
1881:
1878:
1875:
1870:
1867:
1864:
1859:
1856:
1853:
1848:
1845:
1842:
1837:
1834:
1831:
1822:
1819:
1816:
1811:
1808:
1805:
1800:
1797:
1794:
1789:
1786:
1783:
1778:
1775:
1772:
1767:
1764:
1761:
1756:
1753:
1750:
1745:
1742:
1739:
1734:
1731:
1728:
1723:
1720:
1717:
1708:
1705:
1702:
1697:
1694:
1691:
1686:
1683:
1680:
1675:
1672:
1669:
1664:
1661:
1658:
1653:
1650:
1647:
1642:
1639:
1636:
1631:
1628:
1625:
1620:
1617:
1614:
1609:
1606:
1603:
1592:
1589:
1586:
1581:
1578:
1575:
1570:
1567:
1564:
1559:
1556:
1553:
1548:
1545:
1542:
1537:
1534:
1531:
1526:
1523:
1520:
1515:
1512:
1509:
1504:
1501:
1498:
1493:
1490:
1487:
1394:
1391:
1388:
1383:
1380:
1377:
1372:
1369:
1366:
1361:
1358:
1355:
1350:
1347:
1344:
1150:as the midpoint of segment
10:
3294:
2372:
1415:as well as integer sides.
1330:
1327:Common Pythagorean triples
29:
2955:. Prometheus Books, 2012.
2707:in the unit circle. Then
2576:is the golden ratio. Let
2447:Sides of regular polygons
2345:square triangular numbers
2072:is length of hypotenuse,
903:isosceles right triangles
3273:Euclidean plane geometry
2995:Cooke, Roger L. (2011).
2952:The Secrets of Triangles
2569:{\displaystyle \varphi }
1028:30° - 60° - 90° triangle
854:45° - 45° - 90° triangle
18:Isosceles right triangle
3146:The Fibonacci Quarterly
2189:} are solutions to the
1442:Middle Kingdom of Egypt
146:trigonometric functions
99:trigonometric functions
3137:Nyblom, M. A. (1998),
2902:
2882:
2862:
2842:
2822:
2802:
2782:
2754:
2697:
2618:
2570:
2546:
2460:
2441:arithmetic progression
2395:then its common ratio
2370:
2203:, with the hypotenuse
2172:
1405:arithmetic progression
1269:. (This follows from
1198:arithmetic progression
1061:
1045:
1037:
890:
874:
866:
102:
59:special right triangle
54:
2903:
2883:
2863:
2843:
2823:
2803:
2783:
2755:
2698:
2619:
2571:
2547:
2454:
2381:geometric progression
2358:
2173:
1051:
1043:
1035:
880:
872:
861:
96:
51:equilateral triangles
40:
3174:Mathematics Magazine
2892:
2872:
2852:
2832:
2812:
2792:
2772:
2711:
2632:
2580:
2560:
2467:
2230:, 2378... (sequence
2083:
1982:non-hypotenuse edges
846:in their sides; see
3066:Fibonacci Quarterly
2967:"Rational Triangle"
2965:Weisstein, Eric W.
2929:Spiral of Theodorus
2766:regular icosahedron
1438:Berlin Papyrus 6619
1259:Pythagorean triples
1190:Pythagorean theorem
1022:hyperbolic geometry
1010:isosceles triangles
950:Pythagorean theorem
47:isosceles triangles
3278:Types of triangles
2898:
2878:
2858:
2838:
2818:
2798:
2778:
2750:
2693:
2614:
2566:
2542:
2461:
2371:
2168:
2143:
2107:
1333:Pythagorean triple
1062:
1046:
1038:
1018:spherical geometry
1014:Euclidean geometry
891:
875:
867:
103:
55:
3257:45–45–90 triangle
3252:30–60–90 triangle
3030:. Dover. p.
3008:978-1-118-03024-0
2901:{\displaystyle c}
2881:{\displaystyle b}
2861:{\displaystyle a}
2841:{\displaystyle V}
2821:{\displaystyle a}
2801:{\displaystyle V}
2781:{\displaystyle c}
2685:
2684:
2678:
2658:
2606:
2534:
2521:
2515:
2493:
2341:
2340:
2291:
2290:
2142:
2106:
1952:
1951:
1827:
1826:
1713:
1712:
1597:
1596:
1399:
1398:
798:
797:
16:(Redirected from
3285:
3235:
3229:
3223:
3213:
3207:
3205:
3168:
3162:
3160:
3143:
3134:
3128:
3122:
3115:
3109:
3107:
3090:
3081:
3075:
3073:
3063:
3055: + 1,
3042:
3036:
3035:
3029:
3019:
3013:
3012:
2992:
2975:
2974:
2962:
2956:
2947:
2924:Integer triangle
2918:Ailles rectangle
2907:
2905:
2904:
2899:
2887:
2885:
2884:
2879:
2867:
2865:
2864:
2859:
2847:
2845:
2844:
2839:
2827:
2825:
2824:
2819:
2807:
2805:
2804:
2799:
2787:
2785:
2784:
2779:
2762:golden rectangle
2759:
2757:
2756:
2751:
2749:
2748:
2736:
2735:
2723:
2722:
2702:
2700:
2699:
2694:
2686:
2680:
2679:
2674:
2665:
2664:
2659:
2651:
2623:
2621:
2620:
2615:
2607:
2599:
2575:
2573:
2572:
2567:
2551:
2549:
2548:
2543:
2535:
2527:
2522:
2517:
2516:
2511:
2499:
2494:
2486:
2435:
2430:
2429:
2411:
2410:
2295:
2245:
2235:
2202:
2177:
2175:
2174:
2169:
2167:
2166:
2154:
2153:
2144:
2138:
2127:
2118:
2117:
2108:
2102:
2091:
1978:almost-isosceles
1973:
1972:
1965:
1964:
1829:
1715:
1601:
1485:
1475:
1473:
1472:
1469:
1466:
1459:
1457:
1456:
1453:
1450:
1342:
1322:
1301:
1263:rational numbers
1244:
1234:
1223:
1213:
1186:
1185:
1127: : 2.
1125:
1124:
1117:
1115:
1114:
1111:
1108:
1107:
1099:
1097:
1096:
1093:
1090:
1089:
1081:
1079:
1078:
1075:
1072:
1071:
1052:30° - 60° - 90°
1004:
1002:
1001:
998:
995:
994:
993:
979:
977:
976:
973:
970:
969:
968:
947:
946:
940:
938:
937:
934:
931:
930:
922:
920:
919:
916:
913:
912:
881:45° - 45° - 90°
836:Möbius triangles
820:
808:
787:
785:
784:
781:
778:
777:
776:
764:
762:
761:
758:
755:
754:
753:
742:
740:
739:
736:
733:
722:
720:
719:
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698:
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687:
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630:
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624:
621:
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619:
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605:
602:
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591:
589:
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585:
582:
578:
571:
569:
568:
565:
562:
561:
541:
539:
538:
537:
536:
530:
527:
520:
518:
517:
514:
511:
510:
509:
498:
496:
495:
494:
493:
487:
484:
477:
475:
474:
471:
468:
467:
466:
455:
453:
452:
449:
446:
435:
433:
432:
429:
426:
425:
411:
410:
403:
401:
400:
399:
398:
392:
389:
381:
379:
378:
375:
372:
371:
370:
359:
357:
356:
353:
350:
343:
341:
340:
337:
334:
333:
332:
321:
319:
318:
315:
312:
304:
302:
301:
298:
295:
291:
284:
282:
281:
278:
275:
274:
253:
251:
250:
247:
244:
243:
242:
230:
228:
227:
224:
221:
220:
219:
154:
153:
129:
127:
126:
123:
120:
119:
21:
3293:
3292:
3288:
3287:
3286:
3284:
3283:
3282:
3263:
3262:
3243:
3238:
3230:
3226:
3214:
3210:
3187:10.2307/2691431
3169:
3165:
3141:
3135:
3131:
3118:
3116:
3112:
3088:
3082:
3078:
3061:
3043:
3039:
3020:
3016:
3009:
2993:
2978:
2963:
2959:
2948:
2941:
2937:
2914:
2893:
2890:
2889:
2873:
2870:
2869:
2853:
2850:
2849:
2833:
2830:
2829:
2813:
2810:
2809:
2793:
2790:
2789:
2773:
2770:
2769:
2768:of side length
2744:
2740:
2731:
2727:
2718:
2714:
2712:
2709:
2708:
2673:
2666:
2663:
2650:
2633:
2630:
2629:
2598:
2581:
2578:
2577:
2561:
2558:
2557:
2526:
2510:
2500:
2498:
2485:
2468:
2465:
2464:
2449:
2425:
2423:
2421:
2406:
2404:
2377:
2375:Kepler triangle
2361:Kepler triangle
2353:
2231:
2193:
2162:
2158:
2149:
2145:
2128:
2125:
2113:
2109:
2092:
2089:
2084:
2081:
2080:
2071:
2060:
2050:
2041:
2031:
2021:
2011:
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1993:
1970:
1968:
1962:
1960:
1957:
1470:
1467:
1464:
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1461:
1454:
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1433:Isis and Osiris
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1271:Niven's theorem
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3241:External links
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3181:(2): 105–115,
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2373:Main article:
2352:
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1328:
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1250:
1247:
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1193:
1172:
1171:
1054:right triangle
1029:
1026:
1016:. However, in
895:plane geometry
883:right triangle
855:
852:
848:Triangle group
842:the plane via
822:
815:
814:
810:
803:
802:
801:
800:
799:
796:
795:
792:
789:
766:
743:
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723:
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700:
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677:
669:
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609:
592:
572:
553:
549:
548:
545:
542:
499:
456:
439:
436:
417:
413:
412:
404:
382:
360:
322:
305:
285:
266:
262:
261:
258:
255:
232:
209:
206:
203:
200:
196:
195:
190:
185:
180:
175:
170:
165:
160:
90:
87:
63:right triangle
53:are isosceles.
26:
9:
6:
4:
3:
2:
3290:
3279:
3276:
3274:
3271:
3270:
3268:
3258:
3255:
3253:
3250:
3248:
3245:
3244:
3233:
3228:
3221:
3219:
3212:
3204:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3175:
3167:
3159:
3155:
3151:
3147:
3140:
3133:
3126:
3121:
3114:
3106:
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3098:
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2875:
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2835:
2815:
2795:
2775:
2767:
2763:
2745:
2741:
2737:
2732:
2728:
2724:
2719:
2715:
2706:
2690:
2687:
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2675:
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2660:
2655:
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2647:
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2641:
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2329:
2318:
2312:23,661 :
2309:23,660 :
2307:
2296:
2279:
2268:
2257:
2246:
2243:
2242:
2241:
2239:
2234:
2229:
2225:
2221:
2217:
2213:
2210:
2206:
2200:
2196:
2192:
2191:Pell equation
2188:
2184:
2163:
2159:
2155:
2150:
2139:
2135:
2132:
2129:
2119:
2114:
2103:
2099:
2096:
2093:
2079:
2078:
2077:
2075:
2070:
2066:
2058:
2054:
2049:
2045:
2040:
2036:
2033:
2029:
2025:
2019:
2015:
2010:
2006:
2003:
1997:
1990:
1987:
1986:
1985:
1983:
1979:
1975:
1940:
1929:
1918:
1907:
1896:
1885:
1874:
1863:
1852:
1841:
1830:
1815:
1804:
1793:
1782:
1771:
1760:
1749:
1738:
1727:
1716:
1701:
1690:
1679:
1668:
1657:
1646:
1635:
1624:
1613:
1602:
1585:
1574:
1563:
1552:
1541:
1530:
1519:
1508:
1497:
1486:
1483:
1482:
1481:
1478:
1443:
1439:
1435:
1434:
1429:
1425:
1424:Moritz Cantor
1421:
1420:Ancient Egypt
1416:
1414:
1410:
1406:
1387:
1376:
1365:
1354:
1343:
1340:
1339:
1338:
1334:
1324:
1321:
1317:
1312:
1308:
1300:
1296:
1292:
1288:
1284:
1280:
1279:
1278:
1276:
1272:
1268:
1264:
1260:
1256:
1246:
1243:
1239:
1233:
1229:
1222:
1218:
1212:
1208:
1203:
1199:
1191:
1187:
1178:
1174:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1140:
1139:
1137:
1133:
1128:
1126:
1059:
1055:
1050:
1042:
1034:
1025:
1023:
1019:
1015:
1011:
1006:
983:
958:
953:
951:
904:
900:
897:, dividing a
896:
888:
884:
879:
871:
864:
860:
851:
849:
845:
841:
837:
833:
819:
807:
793:
790:
767:
744:
727:
724:
705:
702:
701:
678:
670:
632:
610:
593:
573:
554:
551:
550:
546:
543:
500:
457:
440:
437:
418:
415:
414:
405:
383:
361:
323:
306:
286:
267:
264:
263:
259:
256:
233:
210:
207:
204:
201:
198:
197:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
155:
152:
149:
147:
143:
139:
134:
132:
111:
107:
100:
95:
86:
84:
80:
76:
75:whole numbers
72:
68:
64:
60:
52:
48:
44:
43:Euler diagram
39:
33:
19:
3227:
3217:
3211:
3178:
3172:
3166:
3149:
3145:
3132:
3113:
3096:
3092:
3079:
3069:
3065:
3056:
3052:
3048:
3040:
3025:
3017:
2997:
2970:
2960:
2950:
2462:
2438:
2432:
2426:
2418:golden ratio
2413:
2407:
2400:
2399:is given by
2396:
2392:
2388:
2384:
2378:
2366:golden ratio
2364:
2360:
2342:
2301:4,060 :
2298:4,059 :
2227:
2223:
2219:
2215:
2211:
2209:Pell numbers
2204:
2198:
2194:
2186:
2182:
2180:
2073:
2068:
2064:
2063:
2056:
2052:
2047:
2043:
2038:
2034:
2027:
2023:
2017:
2013:
2008:
2004:
1995:
1988:
1977:
1958:
1479:
1431:
1430:recorded in
1417:
1402:
1336:
1319:
1315:
1310:
1306:
1304:
1298:
1294:
1290:
1286:
1282:
1252:
1241:
1237:
1231:
1227:
1220:
1216:
1210:
1206:
1201:
1195:
1176:
1170:of length 1.
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1132:trigonometry
1129:
1100:), and 90° (
1063:
1007:
954:
902:
892:
829:
150:
135:
105:
104:
79:golden ratio
58:
56:
3099:: 263–267,
3072:(3): 94–104
1179:has length
844:reflections
832:equilateral
823:30°–60°–90°
811:45°–45°–90°
138:unit circle
106:Angle-based
89:Angle-based
3267:Categories
3117:(sequence
2935:References
2337:1,136,689
2284:697 :
2281:696 :
2273:120 :
2270:119 :
1945:224 :
1942:207 :
1934:253 :
1931:204 :
1923:240 :
1920:161 :
1912:231 :
1909:160 :
1901:171 :
1898:140 :
1890:156 :
1887:133 :
1879:209 :
1876:120 :
1868:120 :
1865:119 :
1857:252 :
1854:115 :
1846:208 :
1843:105 :
1835:153 :
1832:104 :
1820:247 :
1809:168 :
1798:105 :
1787:132 :
1776:187 :
1754:221 :
1732:176 :
1721:165 :
1706:140 :
1684:117 :
1640:255 :
1629:195 :
1607:143 :
1590:220 :
1557:180 :
1546:144 :
1524:112 :
1249:Side-based
1138:proof is:
1058:hypotenuse
957:hypotenuse
887:hypotenuse
863:Set square
840:tessellate
260:undefined
32:Set square
3216:Euclid's
2971:MathWorld
2688:≈
2671:−
2653:π
2648:
2601:π
2596:
2564:φ
2537:≈
2532:φ
2502:−
2488:π
2483:
2457:congruent
2422:1 :
2262:21 :
2259:20 :
2097:−
1817:96 :
1806:95 :
1795:88 :
1784:85 :
1773:84 :
1765:72 :
1762:65 :
1751:60 :
1743:91 :
1740:60 :
1729:57 :
1718:52 :
1703:51 :
1695:55 :
1692:48 :
1681:44 :
1673:80 :
1670:39 :
1662:77 :
1659:36 :
1651:56 :
1648:33 :
1637:32 :
1626:28 :
1618:45 :
1615:28 :
1604:24 :
1587:21 :
1579:99 :
1576:20 :
1568:21 :
1565:20 :
1554:19 :
1543:17 :
1535:63 :
1532:16 :
1521:15 :
1513:84 :
1510:13 :
1502:35 :
1499:12 :
1491:60 :
1488:11 :
1440:from the
1392:40 :
1381:24 :
1370:15 :
1359:12 :
1289: : 2
1136:geometric
1060:length 1.
889:length 1.
791:undefined
575:66
288:33
142:geometric
140:or other
83:geometric
3218:Elements
2912:See also
2705:pentagon
2431: :
2326:195,025
2251:4 :
2248:3 :
1428:Plutarch
1409:Heronian
1389:9 :
1378:7 :
1367:8 :
1356:5 :
1348:4 :
1345:3 :
1293: :
1275:multiple
1162:. Then
1082:), 60° (
982:altitude
67:triangle
3203:1448883
3195:2691431
3158:1640364
3123:in the
3120:A001652
3105:1327342
2626:hexagon
2554:decagon
2424:√
2416:is the
2405:√
2315:33,461
2236:in the
2233:A000129
2226:, 408,
2181:where {
1969:√
1961:√
1474:
1462:
1458:
1446:
1267:degrees
1255:integer
1182:√
1134:. The
1121:√
1116:
1102:
1098:
1084:
1080:
1066:
1003:
990:√
986:
978:
965:√
961:
943:√
939:
925:
921:
907:
786:
773:√
769:
763:
750:√
746:
741:
729:
721:
707:
697:
690:√
680:
672:√
667:
655:
651:
638:√
634:
629:
616:√
612:
607:
595:
590:
570:
556:
540:
533:√
523:
519:
506:√
502:
497:
490:√
480:
476:
463:√
459:
454:
442:
434:
420:
407:√
402:
395:√
385:
380:
367:√
363:
358:
346:
342:
329:√
325:
320:
308:
303:
283:
269:
252:
239:√
235:
229:
216:√
212:
163:radians
158:degrees
131:radians
128:
114:
110:degrees
3201:
3193:
3156:
3103:
3005:
2888:, and
2412:where
2304:5,741
2222:, 70,
2218:, 12,
1305:where
899:square
71:angles
3191:JSTOR
3142:(PDF)
3089:(PDF)
3062:(PDF)
2691:1.176
2540:0.618
2214:, 2,
1994:= 1,
1593::221
1318:>
193:cotan
173:turns
61:is a
3125:OEIS
3003:ISBN
2463:Let
2287:985
2276:169
2238:OEIS
2201:= −1
1966:and
1948:305
1937:325
1926:289
1915:281
1904:221
1893:205
1882:241
1871:169
1860:277
1849:233
1838:185
1823:265
1812:193
1801:137
1790:157
1779:205
1757:229
1746:109
1735:185
1724:173
1709:149
1687:125
1643:257
1632:197
1610:145
1582:101
1560:181
1549:145
1527:113
1413:area
1309:and
1020:and
168:gons
3183:doi
3032:161
2645:sin
2593:sin
2480:sin
2265:29
2228:985
2224:169
2197:− 2
2042:= 2
2012:= 2
2001:= 2
1768:97
1698:73
1676:89
1665:85
1654:65
1621:53
1571:29
1538:65
1516:85
1505:37
1494:61
1395:41
1384:25
1373:17
1362:13
1265:of
1230:+ 3
1219:+ 2
1164:ABD
1158:to
1144:ABC
1056:of
1012:in
893:In
885:of
788:= 0
765:= 1
725:100
703:90°
552:60°
416:45°
265:30°
254:= 1
231:= 0
188:tan
183:cos
178:sin
112:or
3269::
3199:MR
3197:,
3189:,
3179:70
3177:,
3154:MR
3150:36
3148:,
3144:,
3101:MR
3097:11
3095:,
3091:,
3068:,
3064:,
3051:,
2979:^
2969:.
2942:^
2908:.
2868:,
2491:10
2443:.
2403:=
2393:ar
2391:,
2389:ar
2387:,
2359:A
2347:.
2254:5
2220:29
2185:,
2059:−1
2051:+
2030:−1
2022:+
2020:−1
1460:+
1351:5
1323:.
1297:+
1291:mn
1285:−
1240:+
1214:,
1209:+
1204:,
1177:AD
1168:BD
1152:BC
1005:.
952:.
850:.
794:0
653:=
547:1
521:=
478:=
438:50
344:=
317:12
199:0°
57:A
3234:.
3222:.
3206:.
3185::
3161:.
3127:)
3108:.
3074:.
3070:6
3057:z
3053:x
3049:x
3034:.
3011:.
2973:.
2896:c
2876:b
2856:a
2836:V
2816:a
2796:V
2776:c
2746:2
2742:c
2738:=
2733:2
2729:b
2725:+
2720:2
2716:a
2682:2
2676:5
2668:5
2661:=
2656:5
2642:2
2639:=
2636:c
2612:1
2609:=
2604:6
2590:2
2587:=
2584:b
2529:1
2524:=
2519:2
2513:5
2508:+
2505:1
2496:=
2477:2
2474:=
2471:a
2433:φ
2427:φ
2414:φ
2408:φ
2401:r
2397:r
2385:a
2369:.
2216:5
2212:1
2205:y
2199:y
2195:x
2187:y
2183:x
2164:2
2160:y
2156:=
2151:2
2147:)
2140:2
2136:1
2133:+
2130:x
2123:(
2120:+
2115:2
2111:)
2104:2
2100:1
2094:x
2087:(
2074:n
2069:n
2065:a
2057:n
2053:b
2048:n
2044:a
2039:n
2035:b
2028:n
2024:a
2018:n
2014:b
2009:n
2005:a
1999:0
1996:b
1992:0
1989:a
1971:2
1963:2
1471:4
1468:/
1465:1
1455:2
1452:/
1449:1
1320:n
1316:m
1311:n
1307:m
1299:n
1295:m
1287:n
1283:m
1242:δ
1238:α
1232:δ
1228:α
1226:3
1221:δ
1217:α
1211:δ
1207:α
1202:α
1192:.
1184:3
1160:D
1156:A
1148:D
1123:3
1113:2
1110:/
1106:π
1095:3
1092:/
1088:π
1077:6
1074:/
1070:π
1000:4
997:/
992:2
975:2
972:/
967:2
945:2
936:4
933:/
929:π
918:2
915:/
911:π
783:2
780:/
775:0
760:2
757:/
752:4
738:4
735:/
732:1
718:2
715:/
711:π
692:3
686:/
683:1
674:3
664:2
661:/
658:1
648:2
645:/
640:1
626:2
623:/
618:3
604:6
601:/
598:1
587:3
584:/
581:2
577:+
567:3
564:/
560:π
544:1
535:2
529:/
526:1
516:2
513:/
508:2
492:2
486:/
483:1
473:2
470:/
465:2
451:8
448:/
445:1
431:4
428:/
424:π
409:3
397:3
391:/
388:1
377:2
374:/
369:3
355:2
352:/
349:1
339:2
336:/
331:1
314:/
311:1
300:3
297:/
294:1
290:+
280:6
277:/
273:π
257:0
249:2
246:/
241:4
226:2
223:/
218:0
208:0
205:0
202:0
125:2
122:/
118:π
34:.
20:)
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