Trójkąt równoboczny

Trójkąt równoboczny – trójkąt, którego wszystkie boki mają taką samą długość (oznaczmy ją $a\,$ $a\,$ ). Taki trójkąt ma następujące własności:

każdy jego kąt wewnętrzny ma miarę:

\alpha ={\frac {\pi }{3}}=60^{\circ }

;

obwód wynosi:

L=3a\,

;

wysokość ma długość:

h={\frac {a~{\sqrt {3}}}{2}}\approx 0,866~a

;

pole powierzchni jest równe:

S={\frac {a^{2}~{\sqrt {3}}}{4}}\approx 0,433~a^{2}

;

długość promienia okręgu wpisanego wynosi:

r={\frac {1}{3}}h={\frac {a{\sqrt {3}}}{6}}\approx 0,289~a

;

długość promienia okręgu opisanego wynosi:

R={\frac {2}{3}}h={\frac {a{\sqrt {3}}}{3}}\approx 0,577~a

;

jego wysokości pokrywają się z dwusiecznymi, symetralnymi i środkowymi oraz dzielą się w stosunku 1 : 2;
jest to szczególny przypadek trójkąta równoramiennego;
jest to wielokąt foremny.

Podstawowe zależności w trójkącie równobocznym[edytuj]

$=\,$ $=\,$	$a\,$ $a\,$	$h\,$ $h\,$	$S\,$ $S\,$	$r\,$ $r\,$	$R\,$ $R\,$	$L_{r}\,$ $L_{r}\,$	$L_{R}\,$ $L_{R}\,$	$S_{r}\,$ $S_{r}\,$	$S_{R}\,$ $S_{R}\,$
$a\,$ $a\,$	$a\,$ $a\,$	${\frac {2h{\sqrt {3}}}{3}}$ ${\frac {2h{\sqrt {3}}}{3}}$	$2{\sqrt {\frac {S{\sqrt {3}}}{3}}}$ $2{\sqrt {{\frac {S{\sqrt {3}}}{3}}}}$	$2r{\sqrt {3}}$ $2r{\sqrt {3}}$	$R{\sqrt {3}}$ $R{\sqrt {3}}$	${\frac {L_{r}{\sqrt {3}}}{\pi }}$ ${\frac {L_{r}{\sqrt {3}}}{\pi }}$	${\frac {L_{R}{\sqrt {3}}}{2\pi }}$ ${\frac {L_{R}{\sqrt {3}}}{2\pi }}$	$2{\sqrt {\frac {3S_{r}}{\pi }}}$ $2{\sqrt {{\frac {3S_{r}}{\pi }}}}$	${\sqrt {\frac {3S_{R}}{\pi }}}$ ${\sqrt {{\frac {3S_{R}}{\pi }}}}$
$h\,$ $h\,$	${\frac {a{\sqrt {3}}}{2}}$ ${\frac {a{\sqrt {3}}}{2}}$	$h\,$ $h\,$	${\sqrt {S{\sqrt {3}}}}$ ${\sqrt {S{\sqrt {3}}}}$	$3r\,$ $3r\,$	${\frac {3}{2}}R$ ${\frac {3}{2}}R$	${\frac {3L_{r}}{2\pi }}$ ${\frac {3L_{r}}{2\pi }}$	${\frac {3L_{R}}{4\pi }}$ ${\frac {3L_{R}}{4\pi }}$	$3{\sqrt {\frac {S_{r}}{\pi }}}$ $3{\sqrt {{\frac {S_{r}}{\pi }}}}$	${\frac {3}{2}}{\sqrt {\frac {S_{R}}{\pi }}}$ ${\frac {3}{2}}{\sqrt {{\frac {S_{R}}{\pi }}}}$
$S\,$ $S\,$	${\frac {a^{2}{\sqrt {3}}}{4}}$ ${\frac {a^{2}{\sqrt {3}}}{4}}$	${\frac {h^{2}{\sqrt {3}}}{3}}$ ${\frac {h^{2}{\sqrt {3}}}{3}}$	$S\,$ $S\,$	$3r^{2}{\sqrt {3}}$ $3r^{2}{\sqrt {3}}$	${\frac {3R^{2}{\sqrt {3}}}{4}}$ ${\frac {3R^{2}{\sqrt {3}}}{4}}$	${\frac {3{L_{r}}^{2}{\sqrt {3}}}{4\pi ^{2}}}$ ${\frac {3{L_{r}}^{2}{\sqrt {3}}}{4\pi ^{2}}}$	${\frac {3{L_{R}}^{2}{\sqrt {3}}}{16\pi ^{2}}}$ ${\frac {3{L_{R}}^{2}{\sqrt {3}}}{16\pi ^{2}}}$	${\frac {3S_{r}{\sqrt {3}}}{\pi }}$ ${\frac {3S_{r}{\sqrt {3}}}{\pi }}$	${\frac {3S_{R}{\sqrt {3}}}{4\pi }}$ ${\frac {3S_{R}{\sqrt {3}}}{4\pi }}$
$r\,$ $r\,$	${\frac {a{\sqrt {3}}}{6}}$ ${\frac {a{\sqrt {3}}}{6}}$	${\frac {1}{3}}h$ ${\frac {1}{3}}h$	${\frac {\sqrt {S{\sqrt {3}}}}{3}}$ ${\frac {{\sqrt {S{\sqrt {3}}}}}{3}}$	$r\,$ $r\,$	${\frac {1}{2}}R$ ${\frac {1}{2}}R$	${\frac {L_{r}}{2\pi }}$ ${\frac {L_{r}}{2\pi }}$	${\frac {L_{R}}{4\pi }}$ ${\frac {L_{R}}{4\pi }}$	${\sqrt {\frac {S_{r}}{\pi }}}$ ${\sqrt {{\frac {S_{r}}{\pi }}}}$	${\sqrt {\frac {S_{R}}{4\pi }}}$ ${\sqrt {{\frac {S_{R}}{4\pi }}}}$
$R\,$ $R\,$	${\frac {a{\sqrt {3}}}{3}}$ ${\frac {a{\sqrt {3}}}{3}}$	${\frac {2}{3}}h$ ${\frac {2}{3}}h$	${\frac {2{\sqrt {S{\sqrt {3}}}}}{3}}$ ${\frac {2{\sqrt {S{\sqrt {3}}}}}{3}}$	$2r\,$ $2r\,$	$R\,$ $R\,$	${\frac {L_{r}}{\pi }}$ ${\frac {L_{r}}{\pi }}$	${\frac {L_{R}}{2\pi }}$ ${\frac {L_{R}}{2\pi }}$	$2{\sqrt {\frac {S_{r}}{\pi }}}$ $2{\sqrt {{\frac {S_{r}}{\pi }}}}$	${\sqrt {\frac {S_{R}}{\pi }}}$ ${\sqrt {{\frac {S_{R}}{\pi }}}}$
$L_{r}\,$ $L_{r}\,$	${\frac {\pi a{\sqrt {3}}}{3}}$ ${\frac {\pi a{\sqrt {3}}}{3}}$	${\frac {2\pi h}{3}}$ ${\frac {2\pi h}{3}}$	${\frac {2\pi {\sqrt {S{\sqrt {3}}}}}{3}}$ ${\frac {2\pi {\sqrt {S{\sqrt {3}}}}}{3}}$	$2\pi r\,$ $2\pi r\,$	$\pi R\,$ $\pi R\,$	$L_{r}\,$ $L_{r}\,$	${\frac {L_{R}}{2}}$ ${\frac {L_{R}}{2}}$	$2{\sqrt {\pi S_{r}}}$ $2{\sqrt {\pi S_{r}}}$	${\sqrt {\pi S_{R}}}$ ${\sqrt {\pi S_{R}}}$
$L_{R}\,$ $L_{R}\,$	${\frac {2\pi a{\sqrt {3}}}{3}}$ ${\frac {2\pi a{\sqrt {3}}}{3}}$	${\frac {4\pi h}{3}}$ ${\frac {4\pi h}{3}}$	${\frac {4\pi {\sqrt {S{\sqrt {3}}}}}{3}}$ ${\frac {4\pi {\sqrt {S{\sqrt {3}}}}}{3}}$	$4\pi r\,$ $4\pi r\,$	$2\pi R\,$ $2\pi R\,$	$2L_{r}\,$ $2L_{r}\,$	$L_{R}\,$ $L_{R}\,$	$4{\sqrt {\pi S_{r}}}$ $4{\sqrt {\pi S_{r}}}$	$2{\sqrt {\pi S_{R}}}$ $2{\sqrt {\pi S_{R}}}$
$S_{r}\,$ $S_{r}\,$	${\frac {\pi a^{2}}{12}}$ ${\frac {\pi a^{2}}{12}}$	${\frac {\pi h^{2}}{9}}$ ${\frac {\pi h^{2}}{9}}$	${\frac {\pi S{\sqrt {3}}}{9}}$ ${\frac {\pi S{\sqrt {3}}}{9}}$	$\pi r^{2}\,$ $\pi r^{2}\,$	${\frac {\pi R^{2}}{4}}$ ${\frac {\pi R^{2}}{4}}$	${\frac {{L_{r}}^{2}}{4\pi }}$ ${\frac {{L_{r}}^{2}}{4\pi }}$	${\frac {{L_{R}}^{2}}{16\pi }}$ ${\frac {{L_{R}}^{2}}{16\pi }}$	$S_{r}\,$ $S_{r}\,$	${\frac {S_{R}}{4}}$ ${\frac {S_{R}}{4}}$
$S_{R}\,$ $S_{R}\,$	${\frac {\pi a^{2}}{3}}$ ${\frac {\pi a^{2}}{3}}$	${\frac {4\pi h^{2}}{9}}$ ${\frac {4\pi h^{2}}{9}}$	${\frac {4\pi S{\sqrt {3}}}{9}}$ ${\frac {4\pi S{\sqrt {3}}}{9}}$	$4\pi r^{2}\,$ $4\pi r^{2}\,$	$\pi R^{2}\,$ $\pi R^{2}\,$	${\frac {{L_{r}}^{2}}{\pi }}$ ${\frac {{L_{r}}^{2}}{\pi }}$	${\frac {{L_{R}}^{2}}{4\pi }}$ ${\frac {{L_{R}}^{2}}{4\pi }}$	$4S_{r}\,$ $4S_{r}\,$	$S_{R}\,$ $S_{R}\,$