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?

1. Use Law of Cosines

a = 65 ; ; b = 46 ; ; beta = 42° 35' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 46**2 = 65**2 + c**2 -2 * 65 * c * cos (42° 35') ; ; ; ; c**2 -95.718c +2109 =0 ; ; p=1; q=-95.718; 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} = 47.85910683 ± 13.4719748535 ; ; c_{1} = 61.3310816835 ; ; c_{2} = 34.3871319765 ; ; ; ; text{ Factored form: } ; ; (c -61.3310816835) (c -34.3871319765) = 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 = 61.33 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 46**2+61.33**2-65**2 }{ 2 * 46 * 61.33 } ) = 72° 58'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65**2+61.33**2-46**2 }{ 2 * 65 * 61.33 } ) = 42° 35' ; ; gamma = 180° - alpha - beta = 180° - 72° 58'13" - 42° 35' = 64° 26'47" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 61.33**2 - 65**2 } }{ 2 } = 43.388 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 61.33**2+2 * 65**2 - 46**2 } }{ 2 } = 58.858 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 65**2 - 61.33**2 } }{ 2 } = 47.224 ; ;







#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 - 2ac cos beta ; ; 46**2 = 65**2 + c**2 -2 * 65 * c * cos (42° 35') ; ; ; ; c**2 -95.718c +2109 =0 ; ; p=1; q=-95.718; 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} = 47.85910683 ± 13.4719748535 ; ; c_{1} = 61.3310816835 ; ; c_{2} = 34.3871319765 ; ; ; ; text{ Factored form: } ; ; (c -61.3310816835) (c -34.3871319765) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 46**2+34.39**2-65**2 }{ 2 * 46 * 34.39 } ) = 107° 1'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65**2+34.39**2-46**2 }{ 2 * 65 * 34.39 } ) = 42° 35' ; ; gamma = 180° - alpha - beta = 180° - 107° 1'47" - 42° 35' = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 34.39**2 - 65**2 } }{ 2 } = 24.351 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.39**2+2 * 65**2 - 46**2 } }{ 2 } = 46.634 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 65**2 - 34.39**2 } }{ 2 } = 53.618 ; ;
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