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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 60 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 60**2 = 100**2 + c**2 -2 * 60 * c * cos (20° ) ; ; ; ; c**2 -187.939c +6400 =0 ; ; p=1; q=-187.938524157; 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} = 93.9692620786 ± 49.297284059 ; ; c_{1} = 143.266546138 ; ;
c_{2} = 44.6719780196 ; ; ; ; (c -143.266546138) (c -44.6719780196) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 100**2-60**2-44.67**2 }{ 2 * 60 * 44.67 } ) = 145° 14'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-100**2-44.67**2 }{ 2 * 100 * 44.67 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 44.67**2-100**2-60**2 }{ 2 * 60 * 100 } ) = 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 ; ;




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