Triangle calculator SSA

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Triangle has two solutions with side c=128.9421700851 and with side c=4.16546728441

#1 Obtuse scalene triangle.

Sides: a = 91   b = 88   c = 128.9421700851

Area: T = 4001.188029783
Perimeter: p = 307.9421700851
Semiperimeter: s = 153.9710850425

Angle ∠ A = α = 44.85496076577° = 44°50'59″ = 0.78327733219 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 92.15503923423° = 92°9'1″ = 1.60883277534 rad

Height: ha = 87.93880285238
Height: hb = 90.93659158598
Height: hc = 62.06218507657

Median: ma = 100.5722019514
Median: mb = 102.5554771265
Median: mc = 62.09767748393

Inradius: r = 25.98766090678
Circumradius: R = 64.51662841681

Vertex coordinates: A[128.9421700851; 0] B[0; 0] C[66.55331868473; 62.06218507657]
Centroid: CG[65.1654962566; 20.68772835886]
Coordinates of the circumscribed circle: U[64.47108504253; -2.42108197582]
Coordinates of the inscribed circle: I[65.97108504253; 25.98766090678]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.1550392342° = 135°9'1″ = 0.78327733219 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 87.85496076577° = 87°50'59″ = 1.60883277534 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 91 ; ; b = 88 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 88**2 = 91**2 + c**2 -2 * 91 * c * cos (43° ) ; ; ; ; c**2 -133.106c +537 =0 ; ; p=1; q=-133.106; r=537 ; ; D = q**2 - 4pr = 133.106**2 - 4 * 1 * 537 = 15569.3067182 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 133.11 ± sqrt{ 15569.31 } }{ 2 } ; ;
c_{1,2} = 66.55318685 ± 62.3885140033 ; ; c_{1} = 128.941700851 ; ; c_{2} = 4.16467284406 ; ; ; ; text{ Factored form: } ; ; (c -128.941700851) (c -4.16467284406) = 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 = 91 ; ; b = 88 ; ; c = 128.94 ; ;

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

p = a+b+c = 91+88+128.94 = 307.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 307.94 }{ 2 } = 153.97 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 153.97 * (153.97-91)(153.97-88)(153.97-128.94) } ; ; T = sqrt{ 16009443.78 } = 4001.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4001.18 }{ 91 } = 87.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4001.18 }{ 88 } = 90.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4001.18 }{ 128.94 } = 62.06 ; ;

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{ 88**2+128.94**2-91**2 }{ 2 * 88 * 128.94 } ) = 44° 50'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 91**2+128.94**2-88**2 }{ 2 * 91 * 128.94 } ) = 43° ; ;
 gamma = 180° - alpha - beta = 180° - 44° 50'59" - 43° = 92° 9'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4001.18 }{ 153.97 } = 25.99 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 91 }{ 2 * sin 44° 50'59" } = 64.52 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 88**2+2 * 128.94**2 - 91**2 } }{ 2 } = 100.572 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 128.94**2+2 * 91**2 - 88**2 } }{ 2 } = 102.555 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 88**2+2 * 91**2 - 128.94**2 } }{ 2 } = 62.097 ; ;



#2 Obtuse scalene triangle.

Sides: a = 91   b = 88   c = 4.16546728441

Area: T = 129.2343652268
Perimeter: p = 183.1654672844
Semiperimeter: s = 91.5822336422

Angle ∠ A = α = 135.1550392342° = 135°9'1″ = 2.35988193317 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 1.85496076578° = 1°50'59″ = 0.03222817435 rad

Height: ha = 2.84403000498
Height: hb = 2.93771284606
Height: hc = 62.06218507657

Median: ma = 42.54990569807
Median: mb = 47.04443646992
Median: mc = 89.48883449116

Inradius: r = 1.41111198438
Circumradius: R = 64.51662841681

Vertex coordinates: A[4.16546728441; 0] B[0; 0] C[66.55331868473; 62.06218507657]
Centroid: CG[23.57326198971; 20.68772835886]
Coordinates of the circumscribed circle: U[2.0822336422; 64.48326705239]
Coordinates of the inscribed circle: I[3.5822336422; 1.41111198438]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 44.85496076577° = 44°50'59″ = 2.35988193317 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 178.1550392342° = 178°9'1″ = 0.03222817435 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 91 ; ; b = 88 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 88**2 = 91**2 + c**2 -2 * 91 * c * cos (43° ) ; ; ; ; c**2 -133.106c +537 =0 ; ; p=1; q=-133.106; r=537 ; ; D = q**2 - 4pr = 133.106**2 - 4 * 1 * 537 = 15569.3067182 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 133.11 ± sqrt{ 15569.31 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 66.55318685 ± 62.3885140033 ; ; c_{1} = 128.941700851 ; ; c_{2} = 4.16467284406 ; ; ; ; text{ Factored form: } ; ; (c -128.941700851) (c -4.16467284406) = 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 = 91 ; ; b = 88 ; ; c = 4.16 ; ;

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

p = a+b+c = 91+88+4.16 = 183.16 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 183.16 }{ 2 } = 91.58 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 91.58 * (91.58-91)(91.58-88)(91.58-4.16) } ; ; T = sqrt{ 16701.34 } = 129.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.23 }{ 91 } = 2.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.23 }{ 88 } = 2.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.23 }{ 4.16 } = 62.06 ; ;

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{ 88**2+4.16**2-91**2 }{ 2 * 88 * 4.16 } ) = 135° 9'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 91**2+4.16**2-88**2 }{ 2 * 91 * 4.16 } ) = 43° ; ;
 gamma = 180° - alpha - beta = 180° - 135° 9'1" - 43° = 1° 50'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.23 }{ 91.58 } = 1.41 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 91 }{ 2 * sin 135° 9'1" } = 64.52 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 88**2+2 * 4.16**2 - 91**2 } }{ 2 } = 42.549 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.16**2+2 * 91**2 - 88**2 } }{ 2 } = 47.044 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 88**2+2 * 91**2 - 4.16**2 } }{ 2 } = 89.488 ; ;
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