13 19 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 19   c = 27

Area: T = 113.0366222071
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 26.14877242894° = 26°8'52″ = 0.45663638808 rad
Angle ∠ B = β = 40.09768097336° = 40°5'49″ = 0.76998213494 rad
Angle ∠ C = γ = 113.7555465977° = 113°45'20″ = 1.98554074234 rad

Height: ha = 17.39901880109
Height: hb = 11.89985496916
Height: hc = 8.37330534867

Median: ma = 22.42220873248
Median: mb = 18.94106969249
Median: mc = 9.09767026993

Inradius: r = 3.83217363414
Circumradius: R = 14.75496967738

Vertex coordinates: A[27; 0] B[0; 0] C[9.94444444444; 8.37330534867]
Centroid: CG[12.31548148148; 2.79110178289]
Coordinates of the circumscribed circle: U[13.5; -5.94216794696]
Coordinates of the inscribed circle: I[10.5; 3.83217363414]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.8522275711° = 153°51'8″ = 0.45663638808 rad
∠ B' = β' = 139.9033190266° = 139°54'11″ = 0.76998213494 rad
∠ C' = γ' = 66.24545340229° = 66°14'40″ = 1.98554074234 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 = 19 ; ; c = 27 ; ;

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

p = a+b+c = 13+19+27 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-13)(29.5-19)(29.5-27) } ; ; T = sqrt{ 12777.19 } = 113.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.04 }{ 13 } = 17.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.04 }{ 19 } = 11.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.04 }{ 27 } = 8.37 ; ;

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**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 26° 8'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 40° 5'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 113° 45'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.04 }{ 29.5 } = 3.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 26° 8'52" } = 14.75 ; ;




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