503 415 365 triangle

Acute scalene triangle.

Sides: a = 503   b = 415   c = 365

Area: T = 74594.1770356587
Perimeter: p = 1283
Semiperimeter: s = 641.5

Angle ∠ A = α = 80.03218230577° = 80°1'55″ = 1.39768188187 rad
Angle ∠ B = β = 54.35502298339° = 54°21'1″ = 0.94985904598 rad
Angle ∠ C = γ = 45.61879471084° = 45°37'5″ = 0.7966183375 rad

Height: ha = 296.59770988333
Height: hb = 359.49899776221
Height: hc = 408.73551800361

Median: ma = 299.12199592137
Median: mb = 387.37767545943
Median: mc = 423.45110007073

Inradius: r = 116.28108579214
Circumradius: R = 255.35548240961

Vertex coordinates: A[365; 0] B[0; 0] C[293.16330136986; 408.73551800361]
Centroid: CG[219.38876712329; 136.2455060012]
Coordinates of the circumscribed circle: U[182.5; 178.6055252412]
Coordinates of the inscribed circle: I[226.5; 116.28108579214]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 99.96881769423° = 99°58'5″ = 1.39768188187 rad
∠ B' = β' = 125.65497701661° = 125°38'59″ = 0.94985904598 rad
∠ C' = γ' = 134.38220528916° = 134°22'55″ = 0.7966183375 rad

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How did we calculate this triangle?

a = 503 ; ; b = 415 ; ; c = 365 ; ;

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

p = a+b+c = 503+415+365 = 1283 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1283 }{ 2 } = 641.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 641.5 * (641.5-503)(641.5-415)(641.5-365) } ; ; T = sqrt{ 5564290251.19 } = 74594.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74594.17 }{ 503 } = 296.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74594.17 }{ 415 } = 359.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74594.17 }{ 365 } = 408.74 ; ;

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{ 415**2+365**2-503**2 }{ 2 * 415 * 365 } ) = 80° 1'55" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 503**2+365**2-415**2 }{ 2 * 503 * 365 } ) = 54° 21'1" ; ; gamma = 180° - alpha - beta = 180° - 80° 1'55" - 54° 21'1" = 45° 37'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74594.17 }{ 641.5 } = 116.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 503 }{ 2 * sin 80° 1'55" } = 255.35 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 415**2+2 * 365**2 - 503**2 } }{ 2 } = 299.12 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 365**2+2 * 503**2 - 415**2 } }{ 2 } = 387.377 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 415**2+2 * 503**2 - 365**2 } }{ 2 } = 423.451 ; ;
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