Trójkąt równoboczny – trójkąt, którego wszystkie boki mają taką samą długość; szczególny przypadek trójkąta równoramiennego. Jest przykładem wielokąta foremnego.
Niech a {\displaystyle a}
oznacza bok trójkąta równobocznego. Wówczas trójkąt ma następujące własności:Niech dodatkowo L r , L R {\displaystyle L_{r},L_{R}}
oznaczają odpowiednio obwód okręgu wpisanego i opisanego, zaś S r , S R {\displaystyle S_{r},S_{R}} – odpowiednio pole koła wpisanego i opisanego.= {\displaystyle =} | a {\displaystyle a} | h {\displaystyle h} | S {\displaystyle S} | r {\displaystyle r} | R {\displaystyle R} | L r {\displaystyle L_{r}} | L R {\displaystyle L_{R}} | S r {\displaystyle S_{r}} | S R {\displaystyle S_{R}} |
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a {\displaystyle a} | a {\displaystyle a} | 2 h 3 3 {\displaystyle {\frac {2h{\sqrt {3}}}{3}}} | 2 S 3 3 {\displaystyle 2{\sqrt {\frac {S{\sqrt {3}}}{3}}}} | 2 r 3 {\displaystyle 2r{\sqrt {3}}} | R 3 {\displaystyle R{\sqrt {3}}} | L r 3 π {\displaystyle {\frac {L_{r}{\sqrt {3}}}{\pi }}} | L R 3 2 π {\displaystyle {\frac {L_{R}{\sqrt {3}}}{2\pi }}} | 2 3 S r π {\displaystyle 2{\sqrt {\frac {3S_{r}}{\pi }}}} | 3 S R π {\displaystyle {\sqrt {\frac {3S_{R}}{\pi }}}} |
h {\displaystyle h} | a 3 2 {\displaystyle {\frac {a{\sqrt {3}}}{2}}} | h {\displaystyle h} | S 3 {\displaystyle {\sqrt {S{\sqrt {3}}}}} | 3 r {\displaystyle 3r} | 3 2 R {\displaystyle {\frac {3}{2}}R} | 3 L r 2 π {\displaystyle {\frac {3L_{r}}{2\pi }}} | 3 L R 4 π {\displaystyle {\frac {3L_{R}}{4\pi }}} | 3 S r π {\displaystyle 3{\sqrt {\frac {S_{r}}{\pi }}}} | 3 2 S R π {\displaystyle {\frac {3}{2}}{\sqrt {\frac {S_{R}}{\pi }}}} |
S {\displaystyle S} | a 2 3 4 {\displaystyle {\frac {a^{2}{\sqrt {3}}}{4}}} | h 2 3 3 {\displaystyle {\frac {h^{2}{\sqrt {3}}}{3}}} | S {\displaystyle S} | 3 r 2 3 {\displaystyle 3r^{2}{\sqrt {3}}} | 3 R 2 3 4 {\displaystyle {\frac {3R^{2}{\sqrt {3}}}{4}}} | 3 L r 2 3 4 π 2 {\displaystyle {\frac {3{L_{r}}^{2}{\sqrt {3}}}{4\pi ^{2}}}} | 3 L R 2 3 16 π 2 {\displaystyle {\frac {3{L_{R}}^{2}{\sqrt {3}}}{16\pi ^{2}}}} | 3 S r 3 π {\displaystyle {\frac {3S_{r}{\sqrt {3}}}{\pi }}} | 3 S R 3 4 π {\displaystyle {\frac {3S_{R}{\sqrt {3}}}{4\pi }}} |
r {\displaystyle r} | a 3 6 {\displaystyle {\frac {a{\sqrt {3}}}{6}}} | 1 3 h {\displaystyle {\frac {1}{3}}h} | S 3 3 {\displaystyle {\frac {\sqrt {S{\sqrt {3}}}}{3}}} | r {\displaystyle r} | 1 2 R {\displaystyle {\frac {1}{2}}R} | L r 2 π {\displaystyle {\frac {L_{r}}{2\pi }}} | L R 4 π {\displaystyle {\frac {L_{R}}{4\pi }}} | S r π {\displaystyle {\sqrt {\frac {S_{r}}{\pi }}}} | S R 4 π {\displaystyle {\sqrt {\frac {S_{R}}{4\pi }}}} |
R {\displaystyle R} | a 3 3 {\displaystyle {\frac {a{\sqrt {3}}}{3}}} | 2 3 h {\displaystyle {\frac {2}{3}}h} | 2 S 3 3 {\displaystyle {\frac {2{\sqrt {S{\sqrt {3}}}}}{3}}} | 2 r {\displaystyle 2r} | R {\displaystyle R} | L r π {\displaystyle {\frac {L_{r}}{\pi }}} | L R 2 π {\displaystyle {\frac {L_{R}}{2\pi }}} | 2 S r π {\displaystyle 2{\sqrt {\frac {S_{r}}{\pi }}}} | S R π {\displaystyle {\sqrt {\frac {S_{R}}{\pi }}}} |
L r {\displaystyle L_{r}} | π a 3 3 {\displaystyle {\frac {\pi a{\sqrt {3}}}{3}}} | 2 π h 3 {\displaystyle {\frac {2\pi h}{3}}} | 2 π S 3 3 {\displaystyle {\frac {2\pi {\sqrt {S{\sqrt {3}}}}}{3}}} | 2 π r {\displaystyle 2\pi r} | π R {\displaystyle \pi R} | L r {\displaystyle L_{r}} | L R 2 {\displaystyle {\frac {L_{R}}{2}}} | 2 π S r {\displaystyle 2{\sqrt {\pi S_{r}}}} | π S R {\displaystyle {\sqrt {\pi S_{R}}}} |
L R {\displaystyle L_{R}} | 2 π a 3 3 {\displaystyle {\frac {2\pi a{\sqrt {3}}}{3}}} | 4 π h 3 {\displaystyle {\frac {4\pi h}{3}}} | 4 π S 3 3 {\displaystyle {\frac {4\pi {\sqrt {S{\sqrt {3}}}}}{3}}} | 4 π r {\displaystyle 4\pi r} | 2 π R {\displaystyle 2\pi R} | 2 L r {\displaystyle 2L_{r}} | L R {\displaystyle L_{R}} | 4 π S r {\displaystyle 4{\sqrt {\pi S_{r}}}} | 2 π S R {\displaystyle 2{\sqrt {\pi S_{R}}}} |
S r {\displaystyle S_{r}} | π a 2 12 {\displaystyle {\frac {\pi a^{2}}{12}}} | π h 2 9 {\displaystyle {\frac {\pi h^{2}}{9}}} | π S 3 9 {\displaystyle {\frac {\pi S{\sqrt {3}}}{9}}} | π r 2 {\displaystyle \pi r^{2}} | π R 2 4 {\displaystyle {\frac {\pi R^{2}}{4}}} | L r 2 4 π {\displaystyle {\frac {{L_{r}}^{2}}{4\pi }}} | L R 2 16 π {\displaystyle {\frac {{L_{R}}^{2}}{16\pi }}} | S r {\displaystyle S_{r}} | S R 4 {\displaystyle {\frac {S_{R}}{4}}} |
S R {\displaystyle S_{R}} | π a 2 3 {\displaystyle {\frac {\pi a^{2}}{3}}} | 4 π h 2 9 {\displaystyle {\frac {4\pi h^{2}}{9}}} | 4 π S 3 9 {\displaystyle {\frac {4\pi S{\sqrt {3}}}{9}}} | 4 π r 2 {\displaystyle 4\pi r^{2}} | π R 2 {\displaystyle \pi R^{2}} | L r 2 π {\displaystyle {\frac {{L_{r}}^{2}}{\pi }}} | L R 2 4 π {\displaystyle {\frac {{L_{R}}^{2}}{4\pi }}} | 4 S r {\displaystyle 4S_{r}} | S R {\displaystyle S_{R}} |
trójkąty |
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czworokąty |
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inne grupy z ustaloną liczbą boków |
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wielokąty foremne |
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wielokąty gwiaździste | |||||||
inne | |||||||
obiekty nazywane jak wielokąty |
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uogólnienia |