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?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (30° ) ; ; ; ; c**2 -207.846c +6300 =0 ; ; p=1; q=-207.846; 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} = 103.92304845 ± 67.082039325 ; ; c_{1} = 171.005087775 ; ; c_{2} = 36.841009125 ; ; ; ; text{ Factored form: } ; ; (c -171.005087775) (c -36.841009125) = 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 = 171.01 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 90**2+171.01**2-120**2 }{ 2 * 90 * 171.01 } ) = 41° 48'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+171.01**2-90**2 }{ 2 * 120 * 171.01 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 41° 48'37" - 30° = 108° 11'23" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 171.01**2 - 120**2 } }{ 2 } = 122.766 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 171.01**2+2 * 120**2 - 90**2 } }{ 2 } = 140.7 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 120**2 - 171.01**2 } }{ 2 } = 62.764 ; ;







#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 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (30° ) ; ; ; ; c**2 -207.846c +6300 =0 ; ; p=1; q=-207.846; 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} = 103.92304845 ± 67.082039325 ; ; c_{1} = 171.005087775 ; ; c_{2} = 36.841009125 ; ; ; ; text{ Factored form: } ; ; (c -171.005087775) (c -36.841009125) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+36.84**2-120**2 }{ 2 * 90 * 36.84 } ) = 138° 11'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+36.84**2-90**2 }{ 2 * 120 * 36.84 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 138° 11'23" - 30° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 36.84**2 - 120**2 } }{ 2 } = 33.595 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 36.84**2+2 * 120**2 - 90**2 } }{ 2 } = 76.509 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 120**2 - 36.84**2 } }{ 2 } = 104.454 ; ;
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