Triangle calculator SSA

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Triangle has two solutions with side c=38.96876297655 and with side c=6.62197502017

#1 Obtuse scalene triangle.

Sides: a = 23.5   b = 17.155   c = 38.96876297655

Area: T = 111.4122155518
Perimeter: p = 79.62326297655
Semiperimeter: s = 39.81113148828

Angle ∠ A = α = 19.4710712896° = 19°28'15″ = 0.34398280477 rad
Angle ∠ B = β = 14.083° = 14°4'59″ = 0.24657947186 rad
Angle ∠ C = γ = 146.4466287104° = 146°26'47″ = 2.55659698873 rad

Height: ha = 9.4821885576
Height: hb = 12.98988843507
Height: hc = 5.71881900048

Median: ma = 27.71986326732
Median: mb = 31.01327325226
Median: mc = 6.60770394364

Inradius: r = 2.79985047931
Circumradius: R = 35.25108835541

Vertex coordinates: A[38.96876297655; 0] B[0; 0] C[22.79436899836; 5.71881900048]
Centroid: CG[20.5877106583; 1.90660633349]
Coordinates of the circumscribed circle: U[19.48438148828; -29.37769594915]
Coordinates of the inscribed circle: I[22.65663148828; 2.79985047931]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5299287104° = 160°31'45″ = 0.34398280477 rad
∠ B' = β' = 165.917° = 165°55'1″ = 0.24657947186 rad
∠ C' = γ' = 33.5543712896° = 33°33'13″ = 2.55659698873 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 23.5 ; ; b = 17.16 ; ; beta = 14° 4'59" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 17.16**2 = 23.5**2 + c**2 -2 * 23.5 * c * cos (14° 4'59") ; ; ; ; c**2 -45.587c +257.956 =0 ; ; p=1; q=-45.587; r=257.956 ; ; D = q**2 - 4pr = 45.587**2 - 4 * 1 * 257.956 = 1046.38531228 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.59 ± sqrt{ 1046.39 } }{ 2 } ; ; c_{1,2} = 22.79368998 ± 16.1739397819 ; ;
c_{1} = 38.9676297619 ; ; c_{2} = 6.61975019809 ; ; ; ; text{ Factored form: } ; ; (c -38.9676297619) (c -6.61975019809) = 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 = 23.5 ; ; b = 17.16 ; ; c = 38.97 ; ;

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

p = a+b+c = 23.5+17.16+38.97 = 79.62 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79.62 }{ 2 } = 39.81 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.81 * (39.81-23.5)(39.81-17.16)(39.81-38.97) } ; ; T = sqrt{ 12412.67 } = 111.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.41 }{ 23.5 } = 9.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.41 }{ 17.16 } = 12.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.41 }{ 38.97 } = 5.72 ; ;

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{ 17.16**2+38.97**2-23.5**2 }{ 2 * 17.16 * 38.97 } ) = 19° 28'15" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23.5**2+38.97**2-17.16**2 }{ 2 * 23.5 * 38.97 } ) = 14° 4'59" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 23.5**2+17.16**2-38.97**2 }{ 2 * 23.5 * 17.16 } ) = 146° 26'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.41 }{ 39.81 } = 2.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23.5 }{ 2 * sin 19° 28'15" } = 35.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.16**2+2 * 38.97**2 - 23.5**2 } }{ 2 } = 27.719 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38.97**2+2 * 23.5**2 - 17.16**2 } }{ 2 } = 31.013 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.16**2+2 * 23.5**2 - 38.97**2 } }{ 2 } = 6.607 ; ;







#2 Obtuse scalene triangle.

Sides: a = 23.5   b = 17.155   c = 6.62197502017

Area: T = 18.92664947188
Perimeter: p = 47.27547502017
Semiperimeter: s = 23.63773751008

Angle ∠ A = α = 160.5299287104° = 160°31'45″ = 2.80217646058 rad
Angle ∠ B = β = 14.083° = 14°4'59″ = 0.24657947186 rad
Angle ∠ C = γ = 5.3887712896° = 5°23'16″ = 0.09440333292 rad

Height: ha = 1.6110765508
Height: hb = 2.20765280931
Height: hc = 5.71881900048

Median: ma = 5.56773206183
Median: mb = 14.98220572725
Median: mc = 20.30655839442

Inradius: r = 0.80107020508
Circumradius: R = 35.25108835541

Vertex coordinates: A[6.62197502017; 0] B[0; 0] C[22.79436899836; 5.71881900048]
Centroid: CG[9.80444800618; 1.90660633349]
Coordinates of the circumscribed circle: U[3.31098751008; 35.09551494963]
Coordinates of the inscribed circle: I[6.48223751008; 0.80107020508]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.4710712896° = 19°28'15″ = 2.80217646058 rad
∠ B' = β' = 165.917° = 165°55'1″ = 0.24657947186 rad
∠ C' = γ' = 174.6122287104° = 174°36'44″ = 0.09440333292 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 23.5 ; ; b = 17.16 ; ; beta = 14° 4'59" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 17.16**2 = 23.5**2 + c**2 -2 * 23.5 * c * cos (14° 4'59") ; ; ; ; c**2 -45.587c +257.956 =0 ; ; p=1; q=-45.587; r=257.956 ; ; D = q**2 - 4pr = 45.587**2 - 4 * 1 * 257.956 = 1046.38531228 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.59 ± sqrt{ 1046.39 } }{ 2 } ; ; c_{1,2} = 22.79368998 ± 16.1739397819 ; ; : Nr. 1
c_{1} = 38.9676297619 ; ; c_{2} = 6.61975019809 ; ; ; ; text{ Factored form: } ; ; (c -38.9676297619) (c -6.61975019809) = 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 = 23.5 ; ; b = 17.16 ; ; c = 6.62 ; ;

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

p = a+b+c = 23.5+17.16+6.62 = 47.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.27 }{ 2 } = 23.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.64 * (23.64-23.5)(23.64-17.16)(23.64-6.62) } ; ; T = sqrt{ 358.21 } = 18.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.93 }{ 23.5 } = 1.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.93 }{ 17.16 } = 2.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.93 }{ 6.62 } = 5.72 ; ;

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{ 17.16**2+6.62**2-23.5**2 }{ 2 * 17.16 * 6.62 } ) = 160° 31'45" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23.5**2+6.62**2-17.16**2 }{ 2 * 23.5 * 6.62 } ) = 14° 4'59" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 23.5**2+17.16**2-6.62**2 }{ 2 * 23.5 * 17.16 } ) = 5° 23'16" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.93 }{ 23.64 } = 0.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23.5 }{ 2 * sin 160° 31'45" } = 35.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.16**2+2 * 6.62**2 - 23.5**2 } }{ 2 } = 5.567 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.62**2+2 * 23.5**2 - 17.16**2 } }{ 2 } = 14.982 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.16**2+2 * 23.5**2 - 6.62**2 } }{ 2 } = 20.306 ; ;
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