13 18 19 triangle

Acute scalene triangle.

Sides: a = 13 b = 18 c = 19

Area: T = 112.2549721603
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 41.02882543699° = 41°1'42″ = 0.71660781251 rad
Angle ∠ B = β = 65.35444154842° = 65°21'16″ = 1.14106497309 rad
Angle ∠ C = γ = 73.61773301459° = 73°37'2″ = 1.28548647976 rad

Height: h_a = 17.2699187939
Height: h_b = 12.47221912892
Height: h_c = 11.81657601688

Median: m_a = 17.32877234512
Median: m_b = 13.56546599663
Median: m_c = 12.5

Inradius: r = 4.49899888641
Circumradius: R = 9.90220290129

Vertex coordinates: A[19; 0] B[0; 0] C[5.42110526316; 11.81657601688]
Centroid: CG[8.14403508772; 3.93985867229]
Coordinates of the circumscribed circle: U[9.5; 2.7932879978]
Coordinates of the inscribed circle: I[7; 4.49899888641]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.972174563° = 138°58'18″ = 0.71660781251 rad
∠ B' = β' = 114.6465584516° = 114°38'44″ = 1.14106497309 rad
∠ C' = γ' = 106.3832669854° = 106°22'58″ = 1.28548647976 rad

Calculate another triangle

How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

$a = 13 ; ; b = 18 ; ; c = 19 ; ;$

1. The triangle circumference is the sum of the lengths of its three sides

$p = a+b+c = 13+18+19 = 50 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-13)(25-18)(25-19) } ; ; T = sqrt{ 12600 } = 112.25 ; ;$

4. Calculate the heights of the triangle from its area.

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 112.25 }{ 13 } = 17.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 112.25 }{ 18 } = 12.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 112.25 }{ 19 } = 11.82 ; ;$

5. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13**2-18**2-19**2 }{ 2 * 18 * 19 } ) = 41° 1'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 65° 21'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-13**2-18**2 }{ 2 * 18 * 13 } ) = 73° 37'2" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 112.25 }{ 25 } = 4.49 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 41° 1'42" } = 9.9 ; ;$

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