13 22 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 22   c = 27

Area: T = 141.7322141732
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 28.50435101171° = 28°30'13″ = 0.49774800999 rad
Angle ∠ B = β = 53.86111853369° = 53°51'40″ = 0.94400550232 rad
Angle ∠ C = γ = 97.63553045461° = 97°38'7″ = 1.70440575305 rad

Height: ha = 21.80549448819
Height: hb = 12.88547401575
Height: hc = 10.49986771653

Median: ma = 23.75439470404
Median: mb = 18.11107702763
Median: mc = 12.01104121495

Inradius: r = 4.5722004572
Circumradius: R = 13.62107636208

Vertex coordinates: A[27; 0] B[0; 0] C[7.66766666667; 10.49986771653]
Centroid: CG[11.55655555556; 3.54995590551]
Coordinates of the circumscribed circle: U[13.5; -1.81097518098]
Coordinates of the inscribed circle: I[9; 4.5722004572]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.4966489883° = 151°29'47″ = 0.49774800999 rad
∠ B' = β' = 126.1398814663° = 126°8'20″ = 0.94400550232 rad
∠ C' = γ' = 82.36546954539° = 82°21'53″ = 1.70440575305 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 = 22 ; ; c = 27 ; ;

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

p = a+b+c = 13+22+27 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-13)(31-22)(31-27) } ; ; T = sqrt{ 20088 } = 141.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 141.73 }{ 13 } = 21.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 141.73 }{ 22 } = 12.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 141.73 }{ 27 } = 10.5 ; ;

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**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 28° 30'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 53° 51'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-22**2 }{ 2 * 22 * 13 } ) = 97° 38'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 141.73 }{ 31 } = 4.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 28° 30'13" } = 13.62 ; ;




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