Triangle calculator SSA

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Triangle has two solutions with side c=51.74440868506 and with side c=11.36992217953

#1 Obtuse scalene triangle.

Sides: a = 37.5   b = 28.6   c = 51.74440868506

Area: T = 524.1422038681
Perimeter: p = 117.8444086851
Semiperimeter: s = 58.92220434253

Angle ∠ A = α = 45.1011398243° = 45°6'5″ = 0.78771678966 rad
Angle ∠ B = β = 32.7° = 32°42' = 0.57107226654 rad
Angle ∠ C = γ = 102.1998601757° = 102°11'55″ = 1.78437020916 rad

Height: ha = 27.9544242063
Height: hb = 36.65332894183
Height: hc = 20.25990120179

Median: ma = 37.36549938044
Median: mb = 42.86444405306
Median: mc = 21.04114440807

Inradius: r = 8.89655169952
Circumradius: R = 26.47697014606

Vertex coordinates: A[51.74440868506; 0] B[0; 0] C[31.55766543229; 20.25990120179]
Centroid: CG[27.76769137245; 6.7533004006]
Coordinates of the circumscribed circle: U[25.87220434253; -5.59330728952]
Coordinates of the inscribed circle: I[30.32220434253; 8.89655169952]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.8998601757° = 134°53'55″ = 0.78771678966 rad
∠ B' = β' = 147.3° = 147°18' = 0.57107226654 rad
∠ C' = γ' = 77.8011398243° = 77°48'5″ = 1.78437020916 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 37.5 ; ; b = 28.6 ; ; beta = 32° 42' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 28.6**2 = 37.5**2 + c**2 -2 * 37.5 * c * cos (32° 42') ; ; ; ; c**2 -63.113c +588.29 =0 ; ; p=1; q=-63.113; r=588.29 ; ; D = q**2 - 4pr = 63.113**2 - 4 * 1 * 588.29 = 1630.12972823 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.11 ± sqrt{ 1630.13 } }{ 2 } ; ; c_{1,2} = 31.55665432 ± 20.1874325276 ; ; c_{1} = 51.7440868476 ; ; c_{2} = 11.3692217924 ; ; ; ; text{ Factored form: } ; ; (c -51.7440868476) (c -11.3692217924) = 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 = 37.5 ; ; b = 28.6 ; ; c = 51.74 ; ;

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

p = a+b+c = 37.5+28.6+51.74 = 117.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 117.84 }{ 2 } = 58.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.92 * (58.92-37.5)(58.92-28.6)(58.92-51.74) } ; ; T = sqrt{ 274724.88 } = 524.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 524.14 }{ 37.5 } = 27.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 524.14 }{ 28.6 } = 36.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 524.14 }{ 51.74 } = 20.26 ; ;

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{ 28.6**2+51.74**2-37.5**2 }{ 2 * 28.6 * 51.74 } ) = 45° 6'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.5**2+51.74**2-28.6**2 }{ 2 * 37.5 * 51.74 } ) = 32° 42' ; ; gamma = 180° - alpha - beta = 180° - 45° 6'5" - 32° 42' = 102° 11'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 524.14 }{ 58.92 } = 8.9 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.5 }{ 2 * sin 45° 6'5" } = 26.47 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.6**2+2 * 51.74**2 - 37.5**2 } }{ 2 } = 37.365 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 51.74**2+2 * 37.5**2 - 28.6**2 } }{ 2 } = 42.864 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.6**2+2 * 37.5**2 - 51.74**2 } }{ 2 } = 21.041 ; ;







#2 Obtuse scalene triangle.

Sides: a = 37.5   b = 28.6   c = 11.36992217953

Area: T = 115.1654600493
Perimeter: p = 77.46992217953
Semiperimeter: s = 38.73546108977

Angle ∠ A = α = 134.8998601757° = 134°53'55″ = 2.3544424757 rad
Angle ∠ B = β = 32.7° = 32°42' = 0.57107226654 rad
Angle ∠ C = γ = 12.4011398243° = 12°24'5″ = 0.21664452312 rad

Height: ha = 6.14221120263
Height: hb = 8.05334685659
Height: hc = 20.25990120179

Median: ma = 11.0477493024
Median: mb = 23.73331962052
Median: mc = 32.86601612738

Inradius: r = 2.97331704495
Circumradius: R = 26.47697014606

Vertex coordinates: A[11.36992217953; 0] B[0; 0] C[31.55766543229; 20.25990120179]
Centroid: CG[14.30986253728; 6.7533004006]
Coordinates of the circumscribed circle: U[5.68546108977; 25.85220849131]
Coordinates of the inscribed circle: I[10.13546108977; 2.97331704495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 45.1011398243° = 45°6'5″ = 2.3544424757 rad
∠ B' = β' = 147.3° = 147°18' = 0.57107226654 rad
∠ C' = γ' = 167.5998601757° = 167°35'55″ = 0.21664452312 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 37.5 ; ; b = 28.6 ; ; beta = 32° 42' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 28.6**2 = 37.5**2 + c**2 -2 * 37.5 * c * cos (32° 42') ; ; ; ; c**2 -63.113c +588.29 =0 ; ; p=1; q=-63.113; r=588.29 ; ; D = q**2 - 4pr = 63.113**2 - 4 * 1 * 588.29 = 1630.12972823 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.11 ± sqrt{ 1630.13 } }{ 2 } ; ; c_{1,2} = 31.55665432 ± 20.1874325276 ; ; c_{1} = 51.7440868476 ; ; c_{2} = 11.3692217924 ; ; ; ; text{ Factored form: } ; ; (c -51.7440868476) (c -11.3692217924) = 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 = 37.5 ; ; b = 28.6 ; ; c = 11.37 ; ;

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

p = a+b+c = 37.5+28.6+11.37 = 77.47 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77.47 }{ 2 } = 38.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.73 * (38.73-37.5)(38.73-28.6)(38.73-11.37) } ; ; T = sqrt{ 13262.89 } = 115.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.16 }{ 37.5 } = 6.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.16 }{ 28.6 } = 8.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.16 }{ 11.37 } = 20.26 ; ;

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{ 28.6**2+11.37**2-37.5**2 }{ 2 * 28.6 * 11.37 } ) = 134° 53'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.5**2+11.37**2-28.6**2 }{ 2 * 37.5 * 11.37 } ) = 32° 42' ; ; gamma = 180° - alpha - beta = 180° - 134° 53'55" - 32° 42' = 12° 24'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.16 }{ 38.73 } = 2.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.5 }{ 2 * sin 134° 53'55" } = 26.47 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.6**2+2 * 11.37**2 - 37.5**2 } }{ 2 } = 11.047 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.37**2+2 * 37.5**2 - 28.6**2 } }{ 2 } = 23.733 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.6**2+2 * 37.5**2 - 11.37**2 } }{ 2 } = 32.86 ; ;
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