12 19 24 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 19   c = 24

Area: T = 112.6109668768
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 29.59773766968° = 29°35'51″ = 0.51765716733 rad
Angle ∠ B = β = 51.44551253397° = 51°26'42″ = 0.89878868213 rad
Angle ∠ C = γ = 98.95774979635° = 98°57'27″ = 1.7277134159 rad

Height: ha = 18.7688278128
Height: hb = 11.8543649344
Height: hc = 9.3844139064

Median: ma = 20.79766343431
Median: mb = 16.42440677057
Median: mc = 10.4166333328

Inradius: r = 4.09548970461
Circumradius: R = 12.14881575691

Vertex coordinates: A[24; 0] B[0; 0] C[7.47991666667; 9.3844139064]
Centroid: CG[10.49330555556; 3.12880463547]
Coordinates of the circumscribed circle: U[12; -1.89114894461]
Coordinates of the inscribed circle: I[8.5; 4.09548970461]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.4032623303° = 150°24'9″ = 0.51765716733 rad
∠ B' = β' = 128.555487466° = 128°33'18″ = 0.89878868213 rad
∠ C' = γ' = 81.04325020365° = 81°2'33″ = 1.7277134159 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 = 24 ; ;

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

p = a+b+c = 12+19+24 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-12)(27.5-19)(27.5-24) } ; ; T = sqrt{ 12680.94 } = 112.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 112.61 }{ 12 } = 18.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 112.61 }{ 19 } = 11.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 112.61 }{ 24 } = 9.38 ; ;

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**2-24**2 }{ 2 * 19 * 24 } ) = 29° 35'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 51° 26'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 98° 57'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 112.61 }{ 27.5 } = 4.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 29° 35'51" } = 12.15 ; ;




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