5 9 12 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 9   c = 12

Area: T = 20.39660780544
Perimeter: p = 26
Semiperimeter: s = 13

Angle ∠ A = α = 22.19216065663° = 22°11'30″ = 0.38773166009 rad
Angle ∠ B = β = 42.83334280661° = 42°50' = 0.74875843497 rad
Angle ∠ C = γ = 114.9754965368° = 114°58'30″ = 2.0076691703 rad

Height: ha = 8.15884312217
Height: hb = 4.53224617899
Height: hc = 3.39993463424

Median: ma = 10.3087764064
Median: mb = 8.01656097709
Median: mc = 4.12331056256

Inradius: r = 1.56989290811
Circumradius: R = 6.61989195609

Vertex coordinates: A[12; 0] B[0; 0] C[3.66766666667; 3.39993463424]
Centroid: CG[5.22222222222; 1.13331154475]
Coordinates of the circumscribed circle: U[6; -2.79546549257]
Coordinates of the inscribed circle: I[4; 1.56989290811]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.8088393434° = 157°48'30″ = 0.38773166009 rad
∠ B' = β' = 137.1676571934° = 137°10' = 0.74875843497 rad
∠ C' = γ' = 65.02550346323° = 65°1'30″ = 2.0076691703 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 = 5 ; ; b = 9 ; ; c = 12 ; ;

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

p = a+b+c = 5+9+12 = 26 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26 }{ 2 } = 13 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13 * (13-5)(13-9)(13-12) } ; ; T = sqrt{ 416 } = 20.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.4 }{ 5 } = 8.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.4 }{ 9 } = 4.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.4 }{ 12 } = 3.4 ; ;

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{ 5**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 22° 11'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-5**2-12**2 }{ 2 * 5 * 12 } ) = 42° 50' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-5**2-9**2 }{ 2 * 9 * 5 } ) = 114° 58'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.4 }{ 13 } = 1.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 22° 11'30" } = 6.62 ; ;




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