Triangle calculator SSA

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Triangle has two solutions with side c=142.5044239816 and with side c=30.70108409409

#1 Obtuse scalene triangle.

Sides: a = 100   b = 75   c = 142.5044239816

Area: T = 3562.60659954
Perimeter: p = 317.5044239816
Semiperimeter: s = 158.7522119908

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 = 71.2522119908
Height: hb = 95.0032826544
Height: hc = 50

Median: ma = 102.3054590233
Median: mb = 117.2549644702
Median: mc = 52.30333020814

Inradius: r = 22.44113128937
Circumradius: R = 75

Vertex coordinates: A[142.5044239816; 0] B[0; 0] C[86.60325403784; 50]
Centroid: CG[76.36989267315; 16.66766666667]
Coordinates of the circumscribed circle: U[71.2522119908; -23.41222918276]
Coordinates of the inscribed circle: I[83.7522119908; 22.44113128937]

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 = 100 ; ; b = 75 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 75**2 = 100**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -173.205c +4375 =0 ; ; p=1; q=-173.205; r=4375 ; ; D = q**2 - 4pr = 173.205**2 - 4 * 1 * 4375 = 12500 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 173.21 ± sqrt{ 12500 } }{ 2 } ; ; c_{1,2} = 86.60254038 ± 55.9016994375 ; ; c_{1} = 142.504239817 ; ;
c_{2} = 30.7008409425 ; ; ; ; text{ Factored form: } ; ; (c -142.504239817) (c -30.7008409425) = 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 = 100 ; ; b = 75 ; ; c = 142.5 ; ;

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

p = a+b+c = 100+75+142.5 = 317.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 317.5 }{ 2 } = 158.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 158.75 * (158.75-100)(158.75-75)(158.75-142.5) } ; ; T = sqrt{ 12692161.48 } = 3562.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3562.61 }{ 100 } = 71.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3562.61 }{ 75 } = 95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3562.61 }{ 142.5 } = 50 ; ;

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{ 75**2+142.5**2-100**2 }{ 2 * 75 * 142.5 } ) = 41° 48'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+142.5**2-75**2 }{ 2 * 100 * 142.5 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 41° 48'37" - 30° = 108° 11'23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3562.61 }{ 158.75 } = 22.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 41° 48'37" } = 75 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 142.5**2 - 100**2 } }{ 2 } = 102.305 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 142.5**2+2 * 100**2 - 75**2 } }{ 2 } = 117.25 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 100**2 - 142.5**2 } }{ 2 } = 52.303 ; ;







#2 Obtuse scalene triangle.

Sides: a = 100   b = 75   c = 30.70108409409

Area: T = 767.5211023524
Perimeter: p = 205.7010840941
Semiperimeter: s = 102.855042047

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 = 15.35504204705
Height: hb = 20.4677227294
Height: hc = 50

Median: ma = 27.99659071516
Median: mb = 63.75875157706
Median: mc = 87.04551870661

Inradius: r = 7.46224976739
Circumradius: R = 75

Vertex coordinates: A[30.70108409409; 0] B[0; 0] C[86.60325403784; 50]
Centroid: CG[39.10111271065; 16.66766666667]
Coordinates of the circumscribed circle: U[15.35504204705; 73.41222918276]
Coordinates of the inscribed circle: I[27.85504204705; 7.46224976739]

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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 75 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 75**2 = 100**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -173.205c +4375 =0 ; ; p=1; q=-173.205; r=4375 ; ; D = q**2 - 4pr = 173.205**2 - 4 * 1 * 4375 = 12500 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 173.21 ± sqrt{ 12500 } }{ 2 } ; ; c_{1,2} = 86.60254038 ± 55.9016994375 ; ; c_{1} = 142.504239817 ; ; : Nr. 1
c_{2} = 30.7008409425 ; ; ; ; text{ Factored form: } ; ; (c -142.504239817) (c -30.7008409425) = 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 = 100 ; ; b = 75 ; ; c = 30.7 ; ;

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

p = a+b+c = 100+75+30.7 = 205.7 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.7 }{ 2 } = 102.85 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.85 * (102.85-100)(102.85-75)(102.85-30.7) } ; ; T = sqrt{ 589088.52 } = 767.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 767.52 }{ 100 } = 15.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 767.52 }{ 75 } = 20.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 767.52 }{ 30.7 } = 50 ; ;

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{ 75**2+30.7**2-100**2 }{ 2 * 75 * 30.7 } ) = 138° 11'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+30.7**2-75**2 }{ 2 * 100 * 30.7 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 138° 11'23" - 30° = 11° 48'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 767.52 }{ 102.85 } = 7.46 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 138° 11'23" } = 75 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 30.7**2 - 100**2 } }{ 2 } = 27.996 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30.7**2+2 * 100**2 - 75**2 } }{ 2 } = 63.758 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 100**2 - 30.7**2 } }{ 2 } = 87.045 ; ;
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