Triangle calculator SSA

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Triangle has two solutions with side c=61.33110816824 and with side c=34.38771319753

#1 Acute scalene triangle.

Sides: a = 65   b = 46   c = 61.33110816824

Area: T = 1348.763304217
Perimeter: p = 172.3311081682
Semiperimeter: s = 86.16655408412

Angle ∠ A = α = 72.97702017058° = 72°58'13″ = 1.27435702756 rad
Angle ∠ B = β = 42.58333333333° = 42°35' = 0.74332193731 rad
Angle ∠ C = γ = 64.44664649609° = 64°26'47″ = 1.12548030048 rad

Height: ha = 41.55004012976
Height: hb = 58.64218713988
Height: hc = 43.98330182405

Median: ma = 43.38877954057
Median: mb = 58.85878863889
Median: mc = 47.2244195122

Inradius: r = 15.65331605211
Circumradius: R = 33.99903912875

Vertex coordinates: A[61.33110816824; 0] B[0; 0] C[47.85991068288; 43.98330182405]
Centroid: CG[36.39767295037; 14.66110060802]
Coordinates of the circumscribed circle: U[30.66655408412; 14.66218997676]
Coordinates of the inscribed circle: I[40.16655408412; 15.65331605211]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.0329798294° = 107°1'47″ = 1.27435702756 rad
∠ B' = β' = 137.4176666667° = 137°25' = 0.74332193731 rad
∠ C' = γ' = 115.5543535039° = 115°33'13″ = 1.12548030048 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 = 65 ; ; b = 46 ; ; c = 61.33 ; ;

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

p = a+b+c = 65+46+61.33 = 172.33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 172.33 }{ 2 } = 86.17 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.17 * (86.17-65)(86.17-46)(86.17-61.33) } ; ; T = sqrt{ 1819161.74 } = 1348.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1348.76 }{ 65 } = 41.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1348.76 }{ 46 } = 58.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1348.76 }{ 61.33 } = 43.98 ; ;

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{ 65**2-46**2-61.33**2 }{ 2 * 46 * 61.33 } ) = 72° 58'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 46**2-65**2-61.33**2 }{ 2 * 65 * 61.33 } ) = 42° 35' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 61.33**2-65**2-46**2 }{ 2 * 46 * 65 } ) = 64° 26'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1348.76 }{ 86.17 } = 15.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65 }{ 2 * sin 72° 58'13" } = 33.99 ; ;





#2 Obtuse scalene triangle.

Sides: a = 65   b = 46   c = 34.38771319753

Area: T = 756.2254926455
Perimeter: p = 145.3877131975
Semiperimeter: s = 72.69435659877

Angle ∠ A = α = 107.0329798294° = 107°1'47″ = 1.8688022378 rad
Angle ∠ B = β = 42.58333333333° = 42°35' = 0.74332193731 rad
Angle ∠ C = γ = 30.38768683725° = 30°23'13″ = 0.53303509025 rad

Height: ha = 23.26884592755
Height: hb = 32.87993446285
Height: hc = 43.98330182405

Median: ma = 24.35113330794
Median: mb = 46.63440800568
Median: mc = 53.61879194731

Inradius: r = 10.40329141531
Circumradius: R = 33.99903912875

Vertex coordinates: A[34.38771319753; 0] B[0; 0] C[47.85991068288; 43.98330182405]
Centroid: CG[27.41554129347; 14.66110060802]
Coordinates of the circumscribed circle: U[17.19435659877; 29.32111184729]
Coordinates of the inscribed circle: I[26.69435659877; 10.40329141531]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.97702017058° = 72°58'13″ = 1.8688022378 rad
∠ B' = β' = 137.4176666667° = 137°25' = 0.74332193731 rad
∠ C' = γ' = 149.6133131628° = 149°36'47″ = 0.53303509025 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 65 ; ; b = 46 ; ; beta = 42° 35' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 46**2 = 65**2 + c**2 -2 * 46 * c * cos (42° 35') ; ; ; ; c**2 -95.718c +2109 =0 ; ; p=1; q=-95.7182136577; r=2109 ; ; D = q**2 - 4pr = 95.718**2 - 4 * 1 * 2109 = 725.976425816 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 95.72 ± sqrt{ 725.98 } }{ 2 } ; ; c_{1,2} = 47.8591068288 ± 13.4719748535 ; ; c_{1} = 61.3310816824 ; ;
c_{2} = 34.3871319753 ; ; ; ; (c -61.3310816824) (c -34.3871319753) = 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 = 65 ; ; b = 46 ; ; c = 34.39 ; ;

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

p = a+b+c = 65+46+34.39 = 145.39 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 145.39 }{ 2 } = 72.69 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 72.69 * (72.69-65)(72.69-46)(72.69-34.39) } ; ; T = sqrt{ 571876.14 } = 756.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 756.22 }{ 65 } = 23.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 756.22 }{ 46 } = 32.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 756.22 }{ 34.39 } = 43.98 ; ;

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{ 65**2-46**2-34.39**2 }{ 2 * 46 * 34.39 } ) = 107° 1'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 46**2-65**2-34.39**2 }{ 2 * 65 * 34.39 } ) = 42° 35' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 34.39**2-65**2-46**2 }{ 2 * 46 * 65 } ) = 30° 23'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 756.22 }{ 72.69 } = 10.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65 }{ 2 * sin 107° 1'47" } = 33.99 ; ;




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