12 19 20 triangle

Acute scalene triangle.

Sides: a = 12   b = 19   c = 20

Area: T = 110.9376637321
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 35.72436974884° = 35°43'25″ = 0.62334961422 rad
Angle ∠ B = β = 67.58988679538° = 67°35'20″ = 1.18796482835 rad
Angle ∠ C = γ = 76.68774345579° = 76°41'15″ = 1.33884482279 rad

Height: ha = 18.48994395534
Height: hb = 11.67875407706
Height: hc = 11.09436637321

Median: ma = 18.56107111933
Median: mb = 13.48114687627
Median: mc = 12.34990890352

Inradius: r = 4.35504563655
Circumradius: R = 10.2766136248

Vertex coordinates: A[20; 0] B[0; 0] C[4.575; 11.09436637321]
Centroid: CG[8.19216666667; 3.69878879107]
Coordinates of the circumscribed circle: U[10; 2.36662155834]
Coordinates of the inscribed circle: I[6.5; 4.35504563655]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.2766302512° = 144°16'35″ = 0.62334961422 rad
∠ B' = β' = 112.4111132046° = 112°24'40″ = 1.18796482835 rad
∠ C' = γ' = 103.3132565442° = 103°18'45″ = 1.33884482279 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 = 20 ; ;

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

p = a+b+c = 12+19+20 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-12)(25.5-19)(25.5-20) } ; ; T = sqrt{ 12306.94 } = 110.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110.94 }{ 12 } = 18.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110.94 }{ 19 } = 11.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110.94 }{ 20 } = 11.09 ; ;

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**2-20**2 }{ 2 * 19 * 20 } ) = 35° 43'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 67° 35'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 76° 41'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110.94 }{ 25.5 } = 4.35 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 35° 43'25" } = 10.28 ; ;




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