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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 91 ; ; b = 88 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 88**2 = 91**2 + c**2 -2 * 88 * c * cos (43° ) ; ; ; ; c**2 -133.106c +537 =0 ; ; p=1; q=-133.106373695; 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} = 66.5531868473 ± 62.3885140033 ; ; c_{1} = 128.941700851 ; ;
c_{2} = 4.16467284406 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 91**2-88**2-4.16**2 }{ 2 * 88 * 4.16 } ) = 135° 9'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 88**2-91**2-4.16**2 }{ 2 * 91 * 4.16 } ) = 43° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.16**2-91**2-88**2 }{ 2 * 88 * 91 } ) = 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 ; ;




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