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Perimeter of Equilateral triangle calculation of formula and proof by Maarif Online Classes
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Denoting the common length of the sides of the equilateral triangle as {\displaystyle a}, we can determine using the Pythagorean theorem that:
The area is {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}},
The perimeter is {\displaystyle p=3a\,\!}
The radius of the circumscribed circle is {\displaystyle R={\frac {a}{\sqrt {3}}}}
The radius of the inscribed circle is {\displaystyle r={\frac {\sqrt {3}}{6}}a} or {\displaystyle r={\frac {R}{2}}}
The geometric center of the triangle is the center of the circumscribed and inscribed circles
The altitude (height) from any side is {\displaystyle h={\frac {\sqrt {3}}{2}}a}
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is {\displaystyle \mathrm {A} ={\frac {3{\sqrt {3}}}{4}}R^{2}}
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is {\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}
The height of the center from each side, or apothem, is {\displaystyle {\frac {h}{3}}}
The radius of the circle circumscribing the three vertices is {\displaystyle R={\frac {2h}{3}}}
The radius of the inscribed circle is {\displaystyle r={\frac {h}{3}}}
In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.
CharacterizationsEdit
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
SidesEdit
{\displaystyle \displaystyle a=b=c}
{\displaystyle \displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}}[1]
SemiperimeterEdit
{\displaystyle \displaystyle s=2R+(3{\sqrt {3}}-4)r\quad {\text{(Blundon)}}}[2]
{\displaystyle \displaystyle s^{2}=3r^{2}+12Rr}[3]
{\displaystyle \displaystyle s^{2}=3{\sqrt {3}}T}[4]
{\displaystyle \displaystyle s=3{\sqrt {3}}r}
{\displaystyle \displaystyle s={\frac {3{\sqrt {3}}}{2}}R}
AnglesEdit
{\displaystyle \displaystyle A=B=C=60^{\circ }}
{\displaystyle \displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}
{\displaystyle \displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}}[5]
AreaEdit
{\displaystyle \displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad } (Weitzenböck)[6]
{\displaystyle \displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{^{\frac {2}{3}}}}[4]
Circumradius, inradius, and exradiiEdit
{\displaystyle \displaystyle R=2r\quad {\text{(Chapple-Euler)}}}[7]
{\displaystyle \displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}}[7]
{\displaystyle \displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}}[5]
{\displaystyle \displaystyle r_{a}=r_{b}=r_{c}}
Equal ceviansEdit
Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:[8]
The three altitudes have equal lengths.
The three medians have equal lengths.
The three angle bisectors have equal lengths.
Coincident triangle centers
Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:
A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide.
The area is {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}},
The perimeter is {\displaystyle p=3a\,\!}
The radius of the circumscribed circle is {\displaystyle R={\frac {a}{\sqrt {3}}}}
The radius of the inscribed circle is {\displaystyle r={\frac {\sqrt {3}}{6}}a} or {\displaystyle r={\frac {R}{2}}}
The geometric center of the triangle is the center of the circumscribed and inscribed circles
The altitude (height) from any side is {\displaystyle h={\frac {\sqrt {3}}{2}}a}
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is {\displaystyle \mathrm {A} ={\frac {3{\sqrt {3}}}{4}}R^{2}}
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is {\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}
The height of the center from each side, or apothem, is {\displaystyle {\frac {h}{3}}}
The radius of the circle circumscribing the three vertices is {\displaystyle R={\frac {2h}{3}}}
The radius of the inscribed circle is {\displaystyle r={\frac {h}{3}}}
In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.
CharacterizationsEdit
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
SidesEdit
{\displaystyle \displaystyle a=b=c}
{\displaystyle \displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}}[1]
SemiperimeterEdit
{\displaystyle \displaystyle s=2R+(3{\sqrt {3}}-4)r\quad {\text{(Blundon)}}}[2]
{\displaystyle \displaystyle s^{2}=3r^{2}+12Rr}[3]
{\displaystyle \displaystyle s^{2}=3{\sqrt {3}}T}[4]
{\displaystyle \displaystyle s=3{\sqrt {3}}r}
{\displaystyle \displaystyle s={\frac {3{\sqrt {3}}}{2}}R}
AnglesEdit
{\displaystyle \displaystyle A=B=C=60^{\circ }}
{\displaystyle \displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}
{\displaystyle \displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}}[5]
AreaEdit
{\displaystyle \displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad } (Weitzenböck)[6]
{\displaystyle \displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{^{\frac {2}{3}}}}[4]
Circumradius, inradius, and exradiiEdit
{\displaystyle \displaystyle R=2r\quad {\text{(Chapple-Euler)}}}[7]
{\displaystyle \displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}}[7]
{\displaystyle \displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}}[5]
{\displaystyle \displaystyle r_{a}=r_{b}=r_{c}}
Equal ceviansEdit
Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:[8]
The three altitudes have equal lengths.
The three medians have equal lengths.
The three angle bisectors have equal lengths.
Coincident triangle centers
Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:
A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide.
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