12 13 20 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 13   c = 20

Area: T = 74.90661913329
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 35.18438154883° = 35°11'2″ = 0.61440734237 rad
Angle ∠ B = β = 38.62548328731° = 38°37'29″ = 0.67441305067 rad
Angle ∠ C = γ = 106.1911351639° = 106°11'29″ = 1.85333887232 rad

Height: ha = 12.48443652221
Height: hb = 11.52440294358
Height: hc = 7.49106191333

Median: ma = 15.76438827704
Median: mb = 15.15875063912
Median: mc = 7.51766481892

Inradius: r = 3.32991640592
Circumradius: R = 10.41330244259

Vertex coordinates: A[20; 0] B[0; 0] C[9.375; 7.49106191333]
Centroid: CG[9.79216666667; 2.49768730444]
Coordinates of the circumscribed circle: U[10; -2.90436318111]
Coordinates of the inscribed circle: I[9.5; 3.32991640592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.8166184512° = 144°48'58″ = 0.61440734237 rad
∠ B' = β' = 141.3755167127° = 141°22'31″ = 0.67441305067 rad
∠ C' = γ' = 73.80986483614° = 73°48'31″ = 1.85333887232 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 = 13 ; ; c = 20 ; ;

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

p = a+b+c = 12+13+20 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-12)(22.5-13)(22.5-20) } ; ; T = sqrt{ 5610.94 } = 74.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74.91 }{ 12 } = 12.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74.91 }{ 13 } = 11.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74.91 }{ 20 } = 7.49 ; ;

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**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 35° 11'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 38° 37'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 106° 11'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74.91 }{ 22.5 } = 3.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 35° 11'2" } = 10.41 ; ;




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