12 19 28 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 19   c = 28

Area: T = 90.17217111959
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 19.81554363505° = 19°48'56″ = 0.3465844607 rad
Angle ∠ B = β = 32.46217449672° = 32°27'42″ = 0.56765643306 rad
Angle ∠ C = γ = 127.7232818682° = 127°43'22″ = 2.22991837159 rad

Height: ha = 15.02986185327
Height: hb = 9.49217590733
Height: hc = 6.4410836514

Median: ma = 23.16224696438
Median: mb = 19.33326149292
Median: mc = 7.51766481892

Inradius: r = 3.05766681761
Circumradius: R = 17.76995642961

Vertex coordinates: A[28; 0] B[0; 0] C[10.125; 6.4410836514]
Centroid: CG[12.70883333333; 2.14769455047]
Coordinates of the circumscribed circle: U[14; -10.82993386812]
Coordinates of the inscribed circle: I[10.5; 3.05766681761]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.185456365° = 160°11'4″ = 0.3465844607 rad
∠ B' = β' = 147.5388255033° = 147°32'18″ = 0.56765643306 rad
∠ C' = γ' = 52.27771813177° = 52°16'38″ = 2.22991837159 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 = 28 ; ;

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

p = a+b+c = 12+19+28 = 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 * (29.5-12)(29.5-19)(29.5-28) } ; ; T = sqrt{ 8130.94 } = 90.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90.17 }{ 12 } = 15.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90.17 }{ 19 } = 9.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90.17 }{ 28 } = 6.44 ; ;

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**2-28**2 }{ 2 * 19 * 28 } ) = 19° 48'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-28**2 }{ 2 * 12 * 28 } ) = 32° 27'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 127° 43'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90.17 }{ 29.5 } = 3.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 19° 48'56" } = 17.7 ; ;




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