Triangle calculator SSA

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Triangle has two solutions with side c=171.0055087779 and with side c=36.84110091291

#1 Obtuse scalene triangle.

Sides: a = 120   b = 90   c = 171.0055087779

Area: T = 5130.153263337
Perimeter: p = 381.0055087779
Semiperimeter: s = 190.503254389

Angle ∠ A = α = 41.81103148958° = 41°48'37″ = 0.73297276562 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 108.1989685104° = 108°11'23″ = 1.88882662218 rad

Height: ha = 85.50325438896
Height: hb = 114.0033391853
Height: hc = 60

Median: ma = 122.766550828
Median: mb = 140.7699573643
Median: mc = 62.76439624977

Inradius: r = 26.93295754725
Circumradius: R = 90

Vertex coordinates: A[171.0055087779; 0] B[0; 0] C[103.9233048454; 60]
Centroid: CG[91.64327120778; 20]
Coordinates of the circumscribed circle: U[85.50325438896; -28.09547501931]
Coordinates of the inscribed circle: I[100.503254389; 26.93295754725]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.1989685104° = 138°11'23″ = 0.73297276562 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 71.81103148958° = 71°48'37″ = 1.88882662218 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 = 120 ; ; b = 90 ; ; c = 171.01 ; ;

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

p = a+b+c = 120+90+171.01 = 381.01 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 381.01 }{ 2 } = 190.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 190.5 * (190.5-120)(190.5-90)(190.5-171.01) } ; ; T = sqrt{ 26318466.04 } = 5130.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5130.15 }{ 120 } = 85.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5130.15 }{ 90 } = 114 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5130.15 }{ 171.01 } = 60 ; ;

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{ 120**2-90**2-171.01**2 }{ 2 * 90 * 171.01 } ) = 41° 48'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-171.01**2 }{ 2 * 120 * 171.01 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 171.01**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 108° 11'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5130.15 }{ 190.5 } = 26.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 41° 48'37" } = 90 ; ;





#2 Obtuse scalene triangle.

Sides: a = 120   b = 90   c = 36.84110091291

Area: T = 1105.233027387
Perimeter: p = 246.8411009129
Semiperimeter: s = 123.4210504565

Angle ∠ A = α = 138.1989685104° = 138°11'23″ = 2.41218649974 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 11.81103148958° = 11°48'37″ = 0.20661288806 rad

Height: ha = 18.42105045646
Height: hb = 24.56106727528
Height: hc = 60

Median: ma = 33.59550885819
Median: mb = 76.50990189247
Median: mc = 104.4544224479

Inradius: r = 8.95549972087
Circumradius: R = 90

Vertex coordinates: A[36.84110091291; 0] B[0; 0] C[103.9233048454; 60]
Centroid: CG[46.92113525278; 20]
Coordinates of the circumscribed circle: U[18.42105045646; 88.09547501931]
Coordinates of the inscribed circle: I[33.42105045646; 8.95549972087]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.81103148958° = 41°48'37″ = 2.41218649974 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 168.1989685104° = 168°11'23″ = 0.20661288806 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 120**2 + c**2 -2 * 90 * c * cos (30° ) ; ; ; ; c**2 -207.846c +6300 =0 ; ; p=1; q=-207.846096908; r=6300 ; ; D = q**2 - 4pr = 207.846**2 - 4 * 1 * 6300 = 18000 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 207.85 ± sqrt{ 18000 } }{ 2 } = fraction{ 207.85 ± 60 sqrt{ 5 } }{ 2 } ; ; c_{1,2} = 103.923048454 ± 67.082039325 ; ;
c_{1} = 171.005087779 ; ; c_{2} = 36.8410091291 ; ; ; ; (c -171.005087779) (c -36.8410091291) = 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 = 120 ; ; b = 90 ; ; c = 36.84 ; ;

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

p = a+b+c = 120+90+36.84 = 246.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 246.84 }{ 2 } = 123.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 123.42 * (123.42-120)(123.42-90)(123.42-36.84) } ; ; T = sqrt{ 1221533.96 } = 1105.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1105.23 }{ 120 } = 18.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1105.23 }{ 90 } = 24.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1105.23 }{ 36.84 } = 60 ; ;

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{ 120**2-90**2-36.84**2 }{ 2 * 90 * 36.84 } ) = 138° 11'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-36.84**2 }{ 2 * 120 * 36.84 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 36.84**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 11° 48'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1105.23 }{ 123.42 } = 8.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 138° 11'23" } = 90 ; ;




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