8.49 21.1 27.29 triangle

Obtuse scalene triangle.

Sides: a = 8.49   b = 21.1   c = 27.29

Area: T = 69.20443184924
Perimeter: p = 56.88
Semiperimeter: s = 28.44

Angle ∠ A = α = 13.90882790303° = 13°54'30″ = 0.24327452624 rad
Angle ∠ B = β = 36.68326100365° = 36°40'57″ = 0.64402323234 rad
Angle ∠ C = γ = 129.4099110933° = 129°24'33″ = 2.25986150679 rad

Height: ha = 16.30325485259
Height: hb = 6.56596510419
Height: hc = 5.07217712343

Median: ma = 24.02199297459
Median: mb = 17.23768674648
Median: mc = 8.5122286708

Inradius: r = 2.43333445321
Circumradius: R = 17.66603982833

Vertex coordinates: A[27.29; 0] B[0; 0] C[6.80986148772; 5.07217712343]
Centroid: CG[11.36662049591; 1.69105904114]
Coordinates of the circumscribed circle: U[13.645; -11.21217635778]
Coordinates of the inscribed circle: I[7.34; 2.43333445321]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.092172097° = 166°5'30″ = 0.24327452624 rad
∠ B' = β' = 143.3177389963° = 143°19'3″ = 0.64402323234 rad
∠ C' = γ' = 50.59108890668° = 50°35'27″ = 2.25986150679 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 = 8.49+21.1+27.29 = 56.88 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56.88 }{ 2 } = 28.44 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.44 * (28.44-8.49)(28.44-21.1)(28.44-27.29) } ; ; T = sqrt{ 4789.24 } = 69.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.2 }{ 8.49 } = 16.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.2 }{ 21.1 } = 6.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.2 }{ 27.29 } = 5.07 ; ;

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{ 21.1**2+27.29**2-8.49**2 }{ 2 * 21.1 * 27.29 } ) = 13° 54'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.49**2+27.29**2-21.1**2 }{ 2 * 8.49 * 27.29 } ) = 36° 40'57" ; ;
 gamma = 180° - alpha - beta = 180° - 13° 54'30" - 36° 40'57" = 129° 24'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.2 }{ 28.44 } = 2.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.49 }{ 2 * sin 13° 54'30" } = 17.66 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.1**2+2 * 27.29**2 - 8.49**2 } }{ 2 } = 24.02 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.29**2+2 * 8.49**2 - 21.1**2 } }{ 2 } = 17.237 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.1**2+2 * 8.49**2 - 27.29**2 } }{ 2 } = 8.512 ; ;
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