Triangle calculator SSA

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Triangle has two solutions with side c=81.63108193372 and with side c=33.41987506894

#1 Acute scalene triangle.

Sides: a = 73   b = 51   c = 81.63108193372

Area: T = 1834.379869928
Perimeter: p = 205.6310819337
Semiperimeter: s = 102.8155409669

Angle ∠ A = α = 61.79224209059° = 61°47'33″ = 1.07884811976 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 80.20875790941° = 80°12'27″ = 1.43998863402 rad

Height: ha = 50.25769506652
Height: hb = 71.93664195796
Height: hc = 44.94332876988

Median: ma = 57.44660210357
Median: mb = 73.11766556458
Median: mc = 47.94989555005

Inradius: r = 17.84114763428
Circumradius: R = 41.41988657598

Vertex coordinates: A[81.63108193372; 0] B[0; 0] C[57.52547850133; 44.94332876988]
Centroid: CG[46.38552014502; 14.98110958996]
Coordinates of the circumscribed circle: U[40.81554096686; 7.04444853903]
Coordinates of the inscribed circle: I[51.81554096686; 17.84114763428]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.2087579094° = 118°12'27″ = 1.07884811976 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 99.79224209059° = 99°47'33″ = 1.43998863402 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 73 ; ; b = 51 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 51**2 = 73**2 + c**2 -2 * 73 * c * cos (38° ) ; ; ; ; c**2 -115.05c +2728 =0 ; ; p=1; q=-115.05; r=2728 ; ; D = q**2 - 4pr = 115.05**2 - 4 * 1 * 2728 = 2324.4035633 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 115.05 ± sqrt{ 2324.4 } }{ 2 } ; ; c_{1,2} = 57.52478501 ± 24.1060343239 ; ; c_{1} = 81.6308193339 ; ;
c_{2} = 33.4187506861 ; ; ; ; text{ Factored form: } ; ; (c -81.6308193339) (c -33.4187506861) = 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 = 73 ; ; b = 51 ; ; c = 81.63 ; ;

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

p = a+b+c = 73+51+81.63 = 205.63 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.63 }{ 2 } = 102.82 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.82 * (102.82-73)(102.82-51)(102.82-81.63) } ; ; T = sqrt{ 3364945.21 } = 1834.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1834.38 }{ 73 } = 50.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1834.38 }{ 51 } = 71.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1834.38 }{ 81.63 } = 44.94 ; ;

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{ 51**2+81.63**2-73**2 }{ 2 * 51 * 81.63 } ) = 61° 47'33" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 73**2+81.63**2-51**2 }{ 2 * 73 * 81.63 } ) = 38° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 73**2+51**2-81.63**2 }{ 2 * 73 * 51 } ) = 80° 12'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1834.38 }{ 102.82 } = 17.84 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 61° 47'33" } = 41.42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 81.63**2 - 73**2 } }{ 2 } = 57.446 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 81.63**2+2 * 73**2 - 51**2 } }{ 2 } = 73.117 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 73**2 - 81.63**2 } }{ 2 } = 47.949 ; ;







#2 Obtuse scalene triangle.

Sides: a = 73   b = 51   c = 33.41987506894

Area: T = 750.9744263383
Perimeter: p = 157.4198750689
Semiperimeter: s = 78.70993753447

Angle ∠ A = α = 118.2087579094° = 118°12'27″ = 2.0633111456 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 23.79224209059° = 23°47'33″ = 0.41552560818 rad

Height: ha = 20.5754637353
Height: hb = 29.45499711131
Height: hc = 44.94332876988

Median: ma = 22.94989966844
Median: mb = 50.72113608731
Median: mc = 60.71107632598

Inradius: r = 9.54111030782
Circumradius: R = 41.41988657598

Vertex coordinates: A[33.41987506894; 0] B[0; 0] C[57.52547850133; 44.94332876988]
Centroid: CG[30.31545119009; 14.98110958996]
Coordinates of the circumscribed circle: U[16.70993753447; 37.89988023085]
Coordinates of the inscribed circle: I[27.70993753447; 9.54111030782]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61.79224209059° = 61°47'33″ = 2.0633111456 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 156.2087579094° = 156°12'27″ = 0.41552560818 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 73 ; ; b = 51 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 51**2 = 73**2 + c**2 -2 * 73 * c * cos (38° ) ; ; ; ; c**2 -115.05c +2728 =0 ; ; p=1; q=-115.05; r=2728 ; ; D = q**2 - 4pr = 115.05**2 - 4 * 1 * 2728 = 2324.4035633 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 115.05 ± sqrt{ 2324.4 } }{ 2 } ; ; c_{1,2} = 57.52478501 ± 24.1060343239 ; ; c_{1} = 81.6308193339 ; ; : Nr. 1
c_{2} = 33.4187506861 ; ; ; ; text{ Factored form: } ; ; (c -81.6308193339) (c -33.4187506861) = 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 = 73 ; ; b = 51 ; ; c = 33.42 ; ;

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

p = a+b+c = 73+51+33.42 = 157.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 157.42 }{ 2 } = 78.71 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.71 * (78.71-73)(78.71-51)(78.71-33.42) } ; ; T = sqrt{ 563962.34 } = 750.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 750.97 }{ 73 } = 20.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 750.97 }{ 51 } = 29.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 750.97 }{ 33.42 } = 44.94 ; ;

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{ 51**2+33.42**2-73**2 }{ 2 * 51 * 33.42 } ) = 118° 12'27" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 73**2+33.42**2-51**2 }{ 2 * 73 * 33.42 } ) = 38° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 73**2+51**2-33.42**2 }{ 2 * 73 * 51 } ) = 23° 47'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 750.97 }{ 78.71 } = 9.54 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 118° 12'27" } = 41.42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 33.42**2 - 73**2 } }{ 2 } = 22.949 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.42**2+2 * 73**2 - 51**2 } }{ 2 } = 50.721 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 73**2 - 33.42**2 } }{ 2 } = 60.711 ; ;
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