5 8 12 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 8   c = 12

Area: T = 14.52436875483
Perimeter: p = 25
Semiperimeter: s = 12.5

Angle ∠ A = α = 17.61224390704° = 17°36'45″ = 0.30773950511 rad
Angle ∠ B = β = 28.95550243719° = 28°57'18″ = 0.50553605103 rad
Angle ∠ C = γ = 133.4332536558° = 133°25'57″ = 2.32988370922 rad

Height: ha = 5.80994750193
Height: hb = 3.63109218871
Height: hc = 2.42106145914

Median: ma = 9.88768599666
Median: mb = 8.27664726786
Median: mc = 2.91554759474

Inradius: r = 1.16218950039
Circumradius: R = 8.26223644719

Vertex coordinates: A[12; 0] B[0; 0] C[4.375; 2.42106145914]
Centroid: CG[5.45883333333; 0.80768715305]
Coordinates of the circumscribed circle: U[6; -5.68803755744]
Coordinates of the inscribed circle: I[4.5; 1.16218950039]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.388756093° = 162°23'15″ = 0.30773950511 rad
∠ B' = β' = 151.0454975628° = 151°2'42″ = 0.50553605103 rad
∠ C' = γ' = 46.56774634422° = 46°34'3″ = 2.32988370922 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 = 8 ; ; c = 12 ; ;

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

p = a+b+c = 5+8+12 = 25 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25 }{ 2 } = 12.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.5 * (12.5-5)(12.5-8)(12.5-12) } ; ; T = sqrt{ 210.94 } = 14.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.52 }{ 5 } = 5.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.52 }{ 8 } = 3.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.52 }{ 12 } = 2.42 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8**2+12**2-5**2 }{ 2 * 8 * 12 } ) = 17° 36'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+12**2-8**2 }{ 2 * 5 * 12 } ) = 28° 57'18" ; ; gamma = 180° - alpha - beta = 180° - 17° 36'45" - 28° 57'18" = 133° 25'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.52 }{ 12.5 } = 1.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 17° 36'45" } = 8.26 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 12**2 - 5**2 } }{ 2 } = 9.887 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 5**2 - 8**2 } }{ 2 } = 8.276 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 5**2 - 12**2 } }{ 2 } = 2.915 ; ;
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