40.7 52.5 84.49 triangle

Obtuse scalene triangle.

Sides: a = 40.7   b = 52.5   c = 84.49

Area: T = 822.8276949928
Perimeter: p = 177.69
Semiperimeter: s = 88.845

Angle ∠ A = α = 21.77773020487° = 21°46'38″ = 0.3880085623 rad
Angle ∠ B = β = 28.5921567239° = 28°35'30″ = 0.49990169866 rad
Angle ∠ C = γ = 129.6311130712° = 129°37'52″ = 2.2622490044 rad

Height: ha = 40.43437567532
Height: hb = 31.34657885687
Height: hc = 19.47774991106

Median: ma = 67.33296557989
Median: mb = 60.89771473059
Median: mc = 20.53660652268

Inradius: r = 9.26113759911
Circumradius: R = 54.85217545262

Vertex coordinates: A[84.49; 0] B[0; 0] C[35.73767741745; 19.47774991106]
Centroid: CG[40.07655913915; 6.49224997035]
Coordinates of the circumscribed circle: U[42.245; -34.98767824986]
Coordinates of the inscribed circle: I[36.345; 9.26113759911]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2232697951° = 158°13'22″ = 0.3880085623 rad
∠ B' = β' = 151.4088432761° = 151°24'30″ = 0.49990169866 rad
∠ C' = γ' = 50.36988692877° = 50°22'8″ = 2.2622490044 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 = 40.7 ; ; b = 52.5 ; ; c = 84.49 ; ;

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

p = a+b+c = 40.7+52.5+84.49 = 177.69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 177.69 }{ 2 } = 88.85 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 88.85 * (88.85-40.7)(88.85-52.5)(88.85-84.49) } ; ; T = sqrt{ 677044.19 } = 822.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 822.83 }{ 40.7 } = 40.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 822.83 }{ 52.5 } = 31.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 822.83 }{ 84.49 } = 19.48 ; ;

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{ 52.5**2+84.49**2-40.7**2 }{ 2 * 52.5 * 84.49 } ) = 21° 46'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40.7**2+84.49**2-52.5**2 }{ 2 * 40.7 * 84.49 } ) = 28° 35'30" ; ;
 gamma = 180° - alpha - beta = 180° - 21° 46'38" - 28° 35'30" = 129° 37'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 822.83 }{ 88.85 } = 9.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40.7 }{ 2 * sin 21° 46'38" } = 54.85 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 52.5**2+2 * 84.49**2 - 40.7**2 } }{ 2 } = 67.33 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 84.49**2+2 * 40.7**2 - 52.5**2 } }{ 2 } = 60.897 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 52.5**2+2 * 40.7**2 - 84.49**2 } }{ 2 } = 20.536 ; ;
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