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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 50**2 = 60**2 + c**2 -2 * 50 * c * cos (28° ) ; ; ; ; c**2 -105.954c +1100 =0 ; ; p=1; q=-105.953711143; 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} = 52.9768555715 ± 41.3103767381 ; ; c_{1} = 94.2872323097 ; ;
c_{2} = 11.6664788334 ; ; ; ; (c -94.2872323097) (c -11.6664788334) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 60**2-50**2-11.67**2 }{ 2 * 50 * 11.67 } ) = 145° 42'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-60**2-11.67**2 }{ 2 * 60 * 11.67 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.67**2-60**2-50**2 }{ 2 * 50 * 60 } ) = 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 ; ;




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