Triangle calculator SSA

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Triangle has two solutions with side c=507.5087925929 and with side c=62.49223370368

#1 Obtuse scalene triangle.

Sides: a = 336.066   b = 285   c = 507.5087925929

Area: T = 45190.49770472
Perimeter: p = 1128.574392593
Semiperimeter: s = 564.2876962965

Angle ∠ A = α = 38.67326200297° = 38°40'21″ = 0.67549645499 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 109.327737997° = 109°19'39″ = 1.90881227431 rad

Height: ha = 268.9388226701
Height: hb = 317.1266295068
Height: hc = 178.0887847454

Median: ma = 375.7122068413
Median: mb = 406.1365538482
Median: mc = 180.8088197983

Inradius: r = 80.08442479327
Circumradius: R = 268.9098887859

Vertex coordinates: A[507.5087925929; 0] B[0; 0] C[2855.000131483; 178.0887847454]
Centroid: CG[264.1699352471; 59.36326158179]
Coordinates of the circumscribed circle: U[253.7543962965; -898.9995294895]
Coordinates of the inscribed circle: I[279.2876962964; 80.08442479327]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.327737997° = 141°19'39″ = 0.67549645499 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 70.67326200297° = 70°40'21″ = 1.90881227431 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 336.07 ; ; b = 285 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 285**2 = 336.07**2 + c**2 -2 * 336.07 * c * cos (32° ) ; ; ; ; c**2 -570c +31715.356 =0 ; ; p=1; q=-570; r=31715.356 ; ; D = q**2 - 4pr = 570**2 - 4 * 1 * 31715.356 = 198038.874357 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 570 ± sqrt{ 198038.87 } }{ 2 } ; ; c_{1,2} = 285.00013148 ± 222.507794446 ; ; c_{1} = 507.507925926 ; ; c_{2} = 62.4923370339 ; ; ; ; text{ Factored form: } ; ; (c -507.507925926) (c -62.4923370339) = 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 = 336.07 ; ; b = 285 ; ; c = 507.51 ; ;

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

p = a+b+c = 336.07+285+507.51 = 1128.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1128.57 }{ 2 } = 564.29 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 564.29 * (564.29-336.07)(564.29-285)(564.29-507.51) } ; ; T = sqrt{ 2042181023.37 } = 45190.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45190.5 }{ 336.07 } = 268.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45190.5 }{ 285 } = 317.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45190.5 }{ 507.51 } = 178.09 ; ;

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{ 285**2+507.51**2-336.07**2 }{ 2 * 285 * 507.51 } ) = 38° 40'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 336.07**2+507.51**2-285**2 }{ 2 * 336.07 * 507.51 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 38° 40'21" - 32° = 109° 19'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45190.5 }{ 564.29 } = 80.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 336.07 }{ 2 * sin 38° 40'21" } = 268.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 285**2+2 * 507.51**2 - 336.07**2 } }{ 2 } = 375.712 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 507.51**2+2 * 336.07**2 - 285**2 } }{ 2 } = 406.136 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 285**2+2 * 336.07**2 - 507.51**2 } }{ 2 } = 180.808 ; ;







#2 Obtuse scalene triangle.

Sides: a = 336.066   b = 285   c = 62.49223370368

Area: T = 5564.563289262
Perimeter: p = 683.5588337037
Semiperimeter: s = 341.7799168518

Angle ∠ A = α = 141.327737997° = 141°19'39″ = 2.46766281037 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 6.67326200297° = 6°40'21″ = 0.11664591893 rad

Height: ha = 33.11658932628
Height: hb = 39.05495641587
Height: hc = 178.0887847454

Median: ma = 119.7088216114
Median: mb = 195.2354664627
Median: mc = 310.0110250042

Inradius: r = 16.28111645799
Circumradius: R = 268.9098887859

Vertex coordinates: A[62.49223370368; 0] B[0; 0] C[2855.000131483; 178.0887847454]
Centroid: CG[115.831082284; 59.36326158179]
Coordinates of the circumscribed circle: U[31.24661685184; 267.0877376943]
Coordinates of the inscribed circle: I[56.77991685184; 16.28111645799]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 38.67326200297° = 38°40'21″ = 2.46766281037 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 173.327737997° = 173°19'39″ = 0.11664591893 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 336.07 ; ; b = 285 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 285**2 = 336.07**2 + c**2 -2 * 336.07 * c * cos (32° ) ; ; ; ; c**2 -570c +31715.356 =0 ; ; p=1; q=-570; r=31715.356 ; ; D = q**2 - 4pr = 570**2 - 4 * 1 * 31715.356 = 198038.874357 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 570 ± sqrt{ 198038.87 } }{ 2 } ; ; c_{1,2} = 285.00013148 ± 222.507794446 ; ; c_{1} = 507.507925926 ; ; c_{2} = 62.4923370339 ; ; ; ; text{ Factored form: } ; ; (c -507.507925926) (c -62.4923370339) = 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 = 336.07 ; ; b = 285 ; ; c = 62.49 ; ;

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

p = a+b+c = 336.07+285+62.49 = 683.56 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 683.56 }{ 2 } = 341.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 341.78 * (341.78-336.07)(341.78-285)(341.78-62.49) } ; ; T = sqrt{ 30964360.19 } = 5564.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5564.56 }{ 336.07 } = 33.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5564.56 }{ 285 } = 39.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5564.56 }{ 62.49 } = 178.09 ; ;

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{ 285**2+62.49**2-336.07**2 }{ 2 * 285 * 62.49 } ) = 141° 19'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 336.07**2+62.49**2-285**2 }{ 2 * 336.07 * 62.49 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 141° 19'39" - 32° = 6° 40'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5564.56 }{ 341.78 } = 16.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 336.07 }{ 2 * sin 141° 19'39" } = 268.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 285**2+2 * 62.49**2 - 336.07**2 } }{ 2 } = 119.708 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.49**2+2 * 336.07**2 - 285**2 } }{ 2 } = 195.235 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 285**2+2 * 336.07**2 - 62.49**2 } }{ 2 } = 310.01 ; ;
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