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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 } ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 6888137 ; ; b = 6371000 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 6371000**2 = 6888137**2 + c**2 -2 * 6371000 * c * cos (28° ) ; ; ; ; c**2 -12163727.967c +6.85679033077E+12 =0 ; ; p=1; q=-12163727.9669; r=6856790330769 ; ; 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 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 6888137**2-6371000**2-592576.31**2 }{ 2 * 6371000 * 592576.31 } ) = 149° 29'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6371000**2-6888137**2-592576.31**2 }{ 2 * 6888137 * 592576.31 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 592576.31**2-6888137**2-6371000**2 }{ 2 * 6371000 * 6888137 } ) = 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 ; ;




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