12 19 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 19   c = 27

Area: T = 99.29875326985
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 22.77657191519° = 22°46'33″ = 0.39875112887 rad
Angle ∠ B = β = 37.80329497815° = 37°48'11″ = 0.66597859407 rad
Angle ∠ C = γ = 119.4211331067° = 119°25'17″ = 2.08442954242 rad

Height: ha = 16.55495887831
Height: hb = 10.4522371863
Height: hc = 7.35553727925

Median: ma = 22.56110283454
Median: mb = 18.60877940659
Median: mc = 8.38215273071

Inradius: r = 3.42440528517
Circumradius: R = 15.49988745257

Vertex coordinates: A[27; 0] B[0; 0] C[9.48114814815; 7.35553727925]
Centroid: CG[12.16604938272; 2.45217909308]
Coordinates of the circumscribed circle: U[13.5; -7.61334822231]
Coordinates of the inscribed circle: I[10; 3.42440528517]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.2244280848° = 157°13'27″ = 0.39875112887 rad
∠ B' = β' = 142.1977050219° = 142°11'49″ = 0.66597859407 rad
∠ C' = γ' = 60.57986689334° = 60°34'43″ = 2.08442954242 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 = 12 ; ; b = 19 ; ; c = 27 ; ;

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

p = a+b+c = 12+19+27 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-12)(29-19)(29-27) } ; ; T = sqrt{ 9860 } = 99.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 99.3 }{ 12 } = 16.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 99.3 }{ 19 } = 10.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 99.3 }{ 27 } = 7.36 ; ;

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{ 12**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 22° 46'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 37° 48'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 119° 25'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 99.3 }{ 29 } = 3.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 22° 46'33" } = 15.5 ; ;




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