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: ha = 17.2699187939
Height: hb = 12.47221912892
Height: hc = 11.81657601688

Median: ma = 17.32877234512
Median: mb = 13.56546599663
Median: mc = 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

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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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18**2+19**2-13**2 }{ 2 * 18 * 19 } ) = 41° 1'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+19**2-18**2 }{ 2 * 13 * 19 } ) = 65° 21'16" ; ; gamma = 180° - alpha - beta = 180° - 41° 1'42" - 65° 21'16" = 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 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 19**2 - 13**2 } }{ 2 } = 17.328 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 13**2 - 18**2 } }{ 2 } = 13.565 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 13**2 - 19**2 } }{ 2 } = 12.5 ; ;
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