Triangle calculator SSA

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Triangle has two solutions with side c=11571151.65987 and with side c=592576.3088132

#1 Obtuse scalene triangle.

Sides: a = 6888137   b = 6371000   c = 11571151.65987

Area: T = 1.87093051054E+13
Perimeter: p = 24830288.65987
Semiperimeter: s = 12415144.32994

Angle ∠ A = α = 30.50326880123° = 30°30'10″ = 0.53223723365 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 121.4977311988° = 121°29'50″ = 2.12105281265 rad

Height: ha = 5432326.652246
Height: hb = 5873271.105515
Height: hc = 3233784.442207

Median: ma = 8682107.349914
Median: mb = 8973381.791117
Median: mc = 3247329.439941

Inradius: r = 1506974.435454
Circumradius: R = 6785288.508842

Vertex coordinates: A[11571151.65987; 0] B[0; 0] C[6081863.983343; 3233784.442207]
Centroid: CG[5884338.547739; 1077928.147736]
Coordinates of the circumscribed circle: U[5785575.829937; -3545032.082238]
Coordinates of the inscribed circle: I[6044144.329937; 1506974.435454]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.4977311988° = 149°29'50″ = 0.53223723365 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 58.50326880123° = 58°30'10″ = 2.12105281265 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6888137 ; ; b = 6371000 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6371000**2 = 6888137**2 + c**2 -2 * 6888137 * c * cos (28° ) ; ; ; ; c**2 -12163727.967c +6.85679033077E+12 =0 ; ; p=1; q=-12163727.967; r=6.85679033077E+12 ; ; D = q**2 - 4pr = 12163727.967**2 - 4 * 1 * 6.85679033077E+12 = 1.20529116729E+14 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12163727.97 ± sqrt{ 1.20529116729E+14 } }{ 2 } ; ;
c_{1,2} = 6081863.98343 ± 5489287.6753 ; ; c_{1} = 11571151.6587 ; ; c_{2} = 592576.308132 ; ; ; ; text{ Factored form: } ; ; (c -11571151.6587) (c -592576.308132) = 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 = 6888137 ; ; b = 6371000 ; ; c = 11571151.66 ; ;

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

p = a+b+c = 6888137+6371000+11571151.66 = 24830288.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24830288.66 }{ 2 } = 12415144.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12415144.33 * (12415144.33-6888137)(12415144.33-6371000)(12415144.33-11571151.66) } ; ; T = sqrt{ 3.5 * 10**{ 26 } } = 1.871 * 10**{ 13 } ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.871 * 10**{ 13 } }{ 6888137 } = 5432326.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.871 * 10**{ 13 } }{ 6371000 } = 5873271.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.871 * 10**{ 13 } }{ 11571151.66 } = 3233784.44 ; ;

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{ 6371000**2+11571151.66**2-6888137**2 }{ 2 * 6371000 * 11571151.66 } ) = 30° 30'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6888137**2+11571151.66**2-6371000**2 }{ 2 * 6888137 * 11571151.66 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 30° 30'10" - 28° = 121° 29'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.871 * 10**{ 13 } }{ 12415144.33 } = 1506974.43 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6888137 }{ 2 * sin 30° 30'10" } = 6785288.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6371000**2+2 * 11571151.66**2 - 6888137**2 } }{ 2 } = 8682107.349 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11571151.66**2+2 * 6888137**2 - 6371000**2 } }{ 2 } = 8973381.791 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6371000**2+2 * 6888137**2 - 11571151.66**2 } }{ 2 } = 3247329.439 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6888137   b = 6371000   c = 592576.3088132

Area: T = 958132022989
Perimeter: p = 13851713.30881
Semiperimeter: s = 6925856.654407

Angle ∠ A = α = 149.4977311988° = 149°29'50″ = 2.60992203171 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 2.50326880123° = 2°30'10″ = 0.0443680146 rad

Height: ha = 278197.7255449
Height: hb = 300779.1632765
Height: hc = 3233784.442207

Median: ma = 2934073.279921
Median: mb = 3708285.155029
Median: mc = 6627989.853328

Inradius: r = 138341.3011423
Circumradius: R = 6785288.508842

Vertex coordinates: A[592576.3088132; 0] B[0; 0] C[6081863.983343; 3233784.442207]
Centroid: CG[2224813.431052; 1077928.147736]
Coordinates of the circumscribed circle: U[296288.1544066; 6778816.524446]
Coordinates of the inscribed circle: I[554856.6544066; 138341.3011423]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.50326880123° = 30°30'10″ = 2.60992203171 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 177.4977311988° = 177°29'50″ = 0.0443680146 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6888137 ; ; b = 6371000 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6371000**2 = 6888137**2 + c**2 -2 * 6888137 * c * cos (28° ) ; ; ; ; c**2 -12163727.967c +6.85679033077E+12 =0 ; ; p=1; q=-12163727.967; r=6.85679033077E+12 ; ; D = q**2 - 4pr = 12163727.967**2 - 4 * 1 * 6.85679033077E+12 = 1.20529116729E+14 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12163727.97 ± sqrt{ 1.20529116729E+14 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 6081863.98343 ± 5489287.6753 ; ; c_{1} = 11571151.6587 ; ; c_{2} = 592576.308132 ; ; ; ; text{ Factored form: } ; ; (c -11571151.6587) (c -592576.308132) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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 = 6888137 ; ; b = 6371000 ; ; c = 592576.31 ; ;

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

p = a+b+c = 6888137+6371000+592576.31 = 13851713.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13851713.31 }{ 2 } = 6925856.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6925856.65 * (6925856.65-6888137)(6925856.65-6371000)(6925856.65-592576.31) } ; ; T = sqrt{ 9.18 * 10**{ 23 } } = 958132022989 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 958132022989 }{ 6888137 } = 278197.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 958132022989 }{ 6371000 } = 300779.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 958132022989 }{ 592576.31 } = 3233784.44 ; ;

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{ 6371000**2+592576.31**2-6888137**2 }{ 2 * 6371000 * 592576.31 } ) = 149° 29'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6888137**2+592576.31**2-6371000**2 }{ 2 * 6888137 * 592576.31 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 149° 29'50" - 28° = 2° 30'10" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 958132022989 }{ 6925856.65 } = 138341.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6888137 }{ 2 * sin 149° 29'50" } = 6785288.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6371000**2+2 * 592576.31**2 - 6888137**2 } }{ 2 } = 2934073.279 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 592576.31**2+2 * 6888137**2 - 6371000**2 } }{ 2 } = 3708285.15 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6371000**2+2 * 6888137**2 - 592576.31**2 } }{ 2 } = 6627989.853 ; ;
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