400 599 320 triangle

Obtuse scalene triangle.

Sides: a = 400   b = 599   c = 320

Area: T = 59288.93661512
Perimeter: p = 1319
Semiperimeter: s = 659.5

Angle ∠ A = α = 38.21657302689° = 38°12'57″ = 0.66769903192 rad
Angle ∠ B = β = 112.121115944° = 112°7'16″ = 1.95768833934 rad
Angle ∠ C = γ = 29.66331102915° = 29°39'47″ = 0.5187718941 rad

Height: ha = 296.4454680756
Height: hb = 197.9659720037
Height: hc = 370.5565850945

Median: ma = 436.5788171694
Median: mb = 203.7154874273
Median: mc = 483.5299213182

Inradius: r = 89.9899827371
Circumradius: R = 323.2988093107

Vertex coordinates: A[320; 0] B[0; 0] C[-150.62765625; 370.5565850945]
Centroid: CG[56.45878125; 123.5198616982]
Coordinates of the circumscribed circle: U[160; 280.9329985951]
Coordinates of the inscribed circle: I[60.5; 89.9899827371]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.7844269731° = 141°47'3″ = 0.66769903192 rad
∠ B' = β' = 67.87988405604° = 67°52'44″ = 1.95768833934 rad
∠ C' = γ' = 150.3376889708° = 150°20'13″ = 0.5187718941 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 = 400 ; ; b = 599 ; ; c = 320 ; ;

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

p = a+b+c = 400+599+320 = 1319 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1319 }{ 2 } = 659.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 659.5 * (659.5-400)(659.5-599)(659.5-320) } ; ; T = sqrt{ 3515177949.94 } = 59288.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59288.94 }{ 400 } = 296.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59288.94 }{ 599 } = 197.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59288.94 }{ 320 } = 370.56 ; ;

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{ 599**2+320**2-400**2 }{ 2 * 599 * 320 } ) = 38° 12'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 400**2+320**2-599**2 }{ 2 * 400 * 320 } ) = 112° 7'16" ; ; gamma = 180° - alpha - beta = 180° - 38° 12'57" - 112° 7'16" = 29° 39'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59288.94 }{ 659.5 } = 89.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 400 }{ 2 * sin 38° 12'57" } = 323.3 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 599**2+2 * 320**2 - 400**2 } }{ 2 } = 436.578 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 320**2+2 * 400**2 - 599**2 } }{ 2 } = 203.715 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 599**2+2 * 400**2 - 320**2 } }{ 2 } = 483.529 ; ;
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