Triangle calculator SSA

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Obtuse scalene triangle.

Sides: a = 48   b = 89   c = 127.2722194826

Area: T = 1527.266633791
Perimeter: p = 264.2722194826
Semiperimeter: s = 132.1366097413

Angle ∠ A = α = 15.64442096986° = 15°38'39″ = 0.27330429681 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 134.3565790301° = 134°21'21″ = 2.34549509099 rad

Height: ha = 63.6366097413
Height: hb = 34.32105918632
Height: hc = 24

Median: ma = 107.1621587278
Median: mb = 85.26993132839
Median: mc = 32.60328695984

Inradius: r = 11.55882824664
Circumradius: R = 89

Vertex coordinates: A[127.2722194826; 0] B[0; 0] C[41.56992193817; 24]
Centroid: CG[56.28804714025; 8]
Coordinates of the circumscribed circle: U[63.6366097413; -62.22109539146]
Coordinates of the inscribed circle: I[43.1366097413; 11.55882824664]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.3565790301° = 164°21'21″ = 0.27330429681 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 45.64442096986° = 45°38'39″ = 2.34549509099 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 48 ; ; b = 89 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 89**2 = 48**2 + c**2 -2 * 48 * c * cos (30° ) ; ; ; ; c**2 -83.138c -5617 =0 ; ; p=1; q=-83.138; r=-5617 ; ; D = q**2 - 4pr = 83.138**2 - 4 * 1 * (-5617) = 29380 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 83.14 ± sqrt{ 29380 } }{ 2 } = fraction{ 83.14 ± 2 sqrt{ 7345 } }{ 2 } ; ; c_{1,2} = 41.56921938 ± 85.7029754443 ; ; c_{1} = 127.272194824 ; ; c_{2} = -44.1337560643 ; ; ; ; text{ Factored form: } ; ; (c -127.272194824) (c +44.1337560643) = 0 ; ; ; ; c>0 ; ;


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 = 48 ; ; b = 89 ; ; c = 127.27 ; ;

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

p = a+b+c = 48+89+127.27 = 264.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 264.27 }{ 2 } = 132.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.14 * (132.14-48)(132.14-89)(132.14-127.27) } ; ; T = sqrt{ 2332542.47 } = 1527.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1527.27 }{ 48 } = 63.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1527.27 }{ 89 } = 34.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1527.27 }{ 127.27 } = 24 ; ;

6. 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{ 89**2+127.27**2-48**2 }{ 2 * 89 * 127.27 } ) = 15° 38'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 48**2+127.27**2-89**2 }{ 2 * 48 * 127.27 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 15° 38'39" - 30° = 134° 21'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1527.27 }{ 132.14 } = 11.56 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 48 }{ 2 * sin 15° 38'39" } = 89 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 89**2+2 * 127.27**2 - 48**2 } }{ 2 } = 107.162 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 127.27**2+2 * 48**2 - 89**2 } }{ 2 } = 85.269 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 89**2+2 * 48**2 - 127.27**2 } }{ 2 } = 32.603 ; ;
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