47 45 55 triangle

Acute scalene triangle.

Sides: a = 47   b = 45   c = 55

Area: T = 1013.386600124
Perimeter: p = 147
Semiperimeter: s = 73.5

Angle ∠ A = α = 54.97546089071° = 54°58'29″ = 0.95994879304 rad
Angle ∠ B = β = 51.6332998341° = 51°37'59″ = 0.90111658237 rad
Angle ∠ C = γ = 73.39223927519° = 73°23'33″ = 1.28109388994 rad

Height: ha = 43.12328085632
Height: hb = 45.03993778327
Height: hc = 36.85504000449

Median: ma = 44.41656503949
Median: mb = 45.94328993426
Median: mc = 36.88883450428

Inradius: r = 13.78875646427
Circumradius: R = 28.69771104441

Vertex coordinates: A[55; 0] B[0; 0] C[29.17327272727; 36.85504000449]
Centroid: CG[28.05875757576; 12.28334666816]
Coordinates of the circumscribed circle: U[27.5; 8.20220819213]
Coordinates of the inscribed circle: I[28.5; 13.78875646427]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.0255391093° = 125°1'31″ = 0.95994879304 rad
∠ B' = β' = 128.3677001659° = 128°22'1″ = 0.90111658237 rad
∠ C' = γ' = 106.6087607248° = 106°36'27″ = 1.28109388994 rad

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How did we calculate this triangle?

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

p = a+b+c = 47+45+55 = 147 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 147 }{ 2 } = 73.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 73.5 * (73.5-47)(73.5-45)(73.5-55) } ; ; T = sqrt{ 1026951.19 } = 1013.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1013.39 }{ 47 } = 43.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1013.39 }{ 45 } = 45.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1013.39 }{ 55 } = 36.85 ; ;

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{ 45**2+55**2-47**2 }{ 2 * 45 * 55 } ) = 54° 58'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+55**2-45**2 }{ 2 * 47 * 55 } ) = 51° 37'59" ; ;
 gamma = 180° - alpha - beta = 180° - 54° 58'29" - 51° 37'59" = 73° 23'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1013.39 }{ 73.5 } = 13.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 47 }{ 2 * sin 54° 58'29" } = 28.7 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 55**2 - 47**2 } }{ 2 } = 44.416 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 47**2 - 45**2 } }{ 2 } = 45.943 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 47**2 - 55**2 } }{ 2 } = 36.888 ; ;
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