Triangle calculator SSA

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Obtuse scalene triangle.

Sides: a = 75   b = 100   c = 157.6544386372

Area: T = 2956.021974448
Perimeter: p = 332.6544386373
Semiperimeter: s = 166.3277193186

Angle ∠ A = α = 22.0244312837° = 22°1'28″ = 0.38443967745 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 127.9765687163° = 127°58'32″ = 2.23435971035 rad

Height: ha = 78.82771931863
Height: hb = 59.12203948897
Height: hc = 37.5

Median: ma = 126.5754889971
Median: mb = 112.8711399262
Median: mc = 39.9854667241

Inradius: r = 17.77223178505
Circumradius: R = 100

Vertex coordinates: A[157.6544386372; 0] B[0; 0] C[64.95219052838; 37.5]
Centroid: CG[74.20220972188; 12.5]
Coordinates of the circumscribed circle: U[78.82771931863; -61.53327036167]
Coordinates of the inscribed circle: I[66.32771931863; 17.77223178505]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.9765687163° = 157°58'32″ = 0.38443967745 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 52.0244312837° = 52°1'28″ = 2.23435971035 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 75 ; ; b = 100 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 100**2 = 75**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -129.904c -4375 =0 ; ; p=1; q=-129.903810568; r=-4375 ; ; D = q**2 - 4pr = 129.904**2 - 4 * 1 * (-4375) = 34375 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 129.9 ± sqrt{ 34375 } }{ 2 } = fraction{ 129.9 ± 25 sqrt{ 55 } }{ 2 } ; ; c_{1,2} = 64.9519052838 ± 92.7024810887 ; ;
c_{1} = 157.654386373 ; ; c_{2} = -27.7505758049 ; ; ; ; (c -157.654386373) (c +27.7505758049) = 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 = 75 ; ; b = 100 ; ; c = 157.65 ; ;

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

p = a+b+c = 75+100+157.65 = 332.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 332.65 }{ 2 } = 166.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 166.33 * (166.33-75)(166.33-100)(166.33-157.65) } ; ; T = sqrt{ 8738052.73 } = 2956.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2956.02 }{ 75 } = 78.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2956.02 }{ 100 } = 59.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2956.02 }{ 157.65 } = 37.5 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 75**2-100**2-157.65**2 }{ 2 * 100 * 157.65 } ) = 22° 1'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-75**2-157.65**2 }{ 2 * 75 * 157.65 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 157.65**2-75**2-100**2 }{ 2 * 100 * 75 } ) = 127° 58'32" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2956.02 }{ 166.33 } = 17.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 75 }{ 2 * sin 22° 1'28" } = 100 ; ;




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