Triangle calculator SSA

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Triangle has two solutions with side c=94.28772323097 and with side c=11.66664788334

#1 Obtuse scalene triangle.

Sides: a = 60   b = 50   c = 94.28772323097

Area: T = 1327.95552291
Perimeter: p = 204.287723231
Semiperimeter: s = 102.1443616155

Angle ∠ A = α = 34.2898891559° = 34°17'20″ = 0.59884540546 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 117.7111108441° = 117°42'40″ = 2.05444464085 rad

Height: ha = 44.26551743032
Height: hb = 53.11882091638
Height: hc = 28.16882937672

Median: ma = 69.24662351923
Median: mb = 74.96769332994
Median: mc = 28.76659426379

Inradius: r = 13.00108636769
Circumradius: R = 53.25113617047

Vertex coordinates: A[94.28772323097; 0] B[0; 0] C[52.97768555715; 28.16882937672]
Centroid: CG[49.08880292937; 9.38994312557]
Coordinates of the circumscribed circle: U[47.14436161548; -24.76326125289]
Coordinates of the inscribed circle: I[52.14436161548; 13.00108636769]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.7111108441° = 145°42'40″ = 0.59884540546 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 62.2898891559° = 62°17'20″ = 2.05444464085 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (28° ) ; ; ; ; c**2 -105.954c +1100 =0 ; ; p=1; q=-105.954; r=1100 ; ; D = q**2 - 4pr = 105.954**2 - 4 * 1 * 1100 = 6826.18890499 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 105.95 ± sqrt{ 6826.19 } }{ 2 } ; ; c_{1,2} = 52.97685557 ± 41.3103767381 ; ; c_{1} = 94.2872323081 ; ; c_{2} = 11.6664788319 ; ; ; ; text{ Factored form: } ; ; (c -94.2872323081) (c -11.6664788319) = 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 = 60 ; ; b = 50 ; ; c = 94.29 ; ;

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

p = a+b+c = 60+50+94.29 = 204.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 204.29 }{ 2 } = 102.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.14 * (102.14-60)(102.14-50)(102.14-94.29) } ; ; T = sqrt{ 1763465.09 } = 1327.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1327.96 }{ 60 } = 44.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1327.96 }{ 50 } = 53.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1327.96 }{ 94.29 } = 28.17 ; ;

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{ 50**2+94.29**2-60**2 }{ 2 * 50 * 94.29 } ) = 34° 17'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+94.29**2-50**2 }{ 2 * 60 * 94.29 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 34° 17'20" - 28° = 117° 42'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1327.96 }{ 102.14 } = 13 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 34° 17'20" } = 53.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 94.29**2 - 60**2 } }{ 2 } = 69.246 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 94.29**2+2 * 60**2 - 50**2 } }{ 2 } = 74.967 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 94.29**2 } }{ 2 } = 28.766 ; ;







#2 Obtuse scalene triangle.

Sides: a = 60   b = 50   c = 11.66664788334

Area: T = 164.3122401504
Perimeter: p = 121.6666478833
Semiperimeter: s = 60.83332394167

Angle ∠ A = α = 145.7111108441° = 145°42'40″ = 2.5433138599 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 6.2898891559° = 6°17'20″ = 0.1109761864 rad

Height: ha = 5.47770800501
Height: hb = 6.57224960601
Height: hc = 28.16882937672

Median: ma = 20.44663533224
Median: mb = 35.25769619251
Median: mc = 54.91878779443

Inradius: r = 2.70110299481
Circumradius: R = 53.25113617047

Vertex coordinates: A[11.66664788334; 0] B[0; 0] C[52.97768555715; 28.16882937672]
Centroid: CG[21.5487778135; 9.38994312557]
Coordinates of the circumscribed circle: U[5.83332394167; 52.9310906296]
Coordinates of the inscribed circle: I[10.83332394167; 2.70110299481]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 34.2898891559° = 34°17'20″ = 2.5433138599 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 173.7111108441° = 173°42'40″ = 0.1109761864 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (28° ) ; ; ; ; c**2 -105.954c +1100 =0 ; ; p=1; q=-105.954; r=1100 ; ; D = q**2 - 4pr = 105.954**2 - 4 * 1 * 1100 = 6826.18890499 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 105.95 ± sqrt{ 6826.19 } }{ 2 } ; ; c_{1,2} = 52.97685557 ± 41.3103767381 ; ; c_{1} = 94.2872323081 ; ; c_{2} = 11.6664788319 ; ; ; ; text{ Factored form: } ; ; (c -94.2872323081) (c -11.6664788319) = 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 = 60 ; ; b = 50 ; ; c = 11.67 ; ;

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

p = a+b+c = 60+50+11.67 = 121.67 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 121.67 }{ 2 } = 60.83 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.83 * (60.83-60)(60.83-50)(60.83-11.67) } ; ; T = sqrt{ 26998.57 } = 164.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 164.31 }{ 60 } = 5.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 164.31 }{ 50 } = 6.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 164.31 }{ 11.67 } = 28.17 ; ;

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{ 50**2+11.67**2-60**2 }{ 2 * 50 * 11.67 } ) = 145° 42'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+11.67**2-50**2 }{ 2 * 60 * 11.67 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 145° 42'40" - 28° = 6° 17'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 164.31 }{ 60.83 } = 2.7 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 145° 42'40" } = 53.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 11.67**2 - 60**2 } }{ 2 } = 20.446 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.67**2+2 * 60**2 - 50**2 } }{ 2 } = 35.257 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 11.67**2 } }{ 2 } = 54.918 ; ;
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