Triangle calculator SSA

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Triangle has two solutions with side c=143.2676546138 and with side c=44.67219780196

#1 Obtuse scalene triangle.

Sides: a = 100   b = 60   c = 143.2676546138

Area: T = 2450.002223219
Perimeter: p = 303.2676546138
Semiperimeter: s = 151.6333273069

Angle ∠ A = α = 34.7532566879° = 34°45'9″ = 0.60765467156 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 125.2477433121° = 125°14'51″ = 2.18659800876 rad

Height: ha = 499.0000446438
Height: hb = 81.66767410729
Height: hc = 34.20220143326

Median: ma = 97.78988113288
Median: mb = 119.8444280719
Median: mc = 40.84994086793

Inradius: r = 16.15774183727
Circumradius: R = 87.71441320049

Vertex coordinates: A[143.2676546138; 0] B[0; 0] C[93.96992620786; 34.20220143326]
Centroid: CG[79.07986027387; 11.40106714442]
Coordinates of the circumscribed circle: U[71.63332730688; -50.621058023]
Coordinates of the inscribed circle: I[91.63332730688; 16.15774183727]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.2477433121° = 145°14'51″ = 0.60765467156 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 54.7532566879° = 54°45'9″ = 2.18659800876 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 60 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 100**2 + c**2 -2 * 100 * c * cos (20° ) ; ; ; ; c**2 -187.939c +6400 =0 ; ; p=1; q=-187.939; r=6400 ; ; D = q**2 - 4pr = 187.939**2 - 4 * 1 * 6400 = 9720.88886238 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 187.94 ± sqrt{ 9720.89 } }{ 2 } ; ; c_{1,2} = 93.96926208 ± 49.297284059 ; ; c_{1} = 143.266546139 ; ;
c_{2} = 44.671978021 ; ; ; ; text{ Factored form: } ; ; (c -143.266546139) (c -44.671978021) = 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 = 60 ; ; c = 143.27 ; ;

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

p = a+b+c = 100+60+143.27 = 303.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 303.27 }{ 2 } = 151.63 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 151.63 * (151.63-100)(151.63-60)(151.63-143.27) } ; ; T = sqrt{ 6002510.94 } = 2450 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2450 }{ 100 } = 49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2450 }{ 60 } = 81.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2450 }{ 143.27 } = 34.2 ; ;

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{ 60**2+143.27**2-100**2 }{ 2 * 60 * 143.27 } ) = 34° 45'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+143.27**2-60**2 }{ 2 * 100 * 143.27 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 34° 45'9" - 20° = 125° 14'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2450 }{ 151.63 } = 16.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 34° 45'9" } = 87.71 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 143.27**2 - 100**2 } }{ 2 } = 97.789 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 143.27**2+2 * 100**2 - 60**2 } }{ 2 } = 119.844 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 100**2 - 143.27**2 } }{ 2 } = 40.849 ; ;







#2 Obtuse scalene triangle.

Sides: a = 100   b = 60   c = 44.67219780196

Area: T = 763.9365816245
Perimeter: p = 204.672197802
Semiperimeter: s = 102.336598901

Angle ∠ A = α = 145.2477433121° = 145°14'51″ = 2.5355045938 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 14.7532566879° = 14°45'9″ = 0.25774808652 rad

Height: ha = 15.27987163249
Height: hb = 25.46545272082
Height: hc = 34.20220143326

Median: ma = 17.25766743636
Median: mb = 71.39988291927
Median: mc = 79.37994910223

Inradius: r = 7.46549771174
Circumradius: R = 87.71441320049

Vertex coordinates: A[44.67219780196; 0] B[0; 0] C[93.96992620786; 34.20220143326]
Centroid: CG[46.21437466994; 11.40106714442]
Coordinates of the circumscribed circle: U[22.33659890098; 84.82325945626]
Coordinates of the inscribed circle: I[42.33659890098; 7.46549771174]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 34.7532566879° = 34°45'9″ = 2.5355045938 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 165.2477433121° = 165°14'51″ = 0.25774808652 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 60 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 100**2 + c**2 -2 * 100 * c * cos (20° ) ; ; ; ; c**2 -187.939c +6400 =0 ; ; p=1; q=-187.939; r=6400 ; ; D = q**2 - 4pr = 187.939**2 - 4 * 1 * 6400 = 9720.88886238 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 187.94 ± sqrt{ 9720.89 } }{ 2 } ; ; c_{1,2} = 93.96926208 ± 49.297284059 ; ; c_{1} = 143.266546139 ; ; : Nr. 1
c_{2} = 44.671978021 ; ; ; ; text{ Factored form: } ; ; (c -143.266546139) (c -44.671978021) = 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 = 60 ; ; c = 44.67 ; ;

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

p = a+b+c = 100+60+44.67 = 204.67 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 204.67 }{ 2 } = 102.34 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.34 * (102.34-100)(102.34-60)(102.34-44.67) } ; ; T = sqrt{ 583597.93 } = 763.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 763.94 }{ 100 } = 15.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 763.94 }{ 60 } = 25.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 763.94 }{ 44.67 } = 34.2 ; ;

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{ 60**2+44.67**2-100**2 }{ 2 * 60 * 44.67 } ) = 145° 14'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+44.67**2-60**2 }{ 2 * 100 * 44.67 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 145° 14'51" - 20° = 14° 45'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 763.94 }{ 102.34 } = 7.46 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 145° 14'51" } = 87.71 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 44.67**2 - 100**2 } }{ 2 } = 17.257 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44.67**2+2 * 100**2 - 60**2 } }{ 2 } = 71.399 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 100**2 - 44.67**2 } }{ 2 } = 79.379 ; ;
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