Triangle calculator SSA

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Triangle has two solutions with side c=10.26109558648 and with side c=2.15325187605

#1 Obtuse scalene triangle.

Sides: a = 6.37   b = 4.3   c = 10.26109558648

Area: T = 7.35216578966
Perimeter: p = 20.93109558648
Semiperimeter: s = 10.46554779324

Angle ∠ A = α = 19.46656365902° = 19°27'56″ = 0.34397394495 rad
Angle ∠ B = β = 13° = 0.22768928028 rad
Angle ∠ C = γ = 147.534436341° = 147°32'4″ = 2.57549604013 rad

Height: ha = 2.30882128404
Height: hb = 3.41993757659
Height: hc = 1.43329382162

Median: ma = 7.19333568402
Median: mb = 8.26549596266
Median: mc = 1.79221066333

Inradius: r = 0.70224674787
Circumradius: R = 9.55876346876

Vertex coordinates: A[10.26109558648; 0] B[0; 0] C[6.20767373127; 1.43329382162]
Centroid: CG[5.48992310592; 0.47876460721]
Coordinates of the circumscribed circle: U[5.13304779324; -8.06439058158]
Coordinates of the inscribed circle: I[6.16554779324; 0.70224674787]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.534436341° = 160°32'4″ = 0.34397394495 rad
∠ B' = β' = 167° = 0.22768928028 rad
∠ C' = γ' = 32.46656365902° = 32°27'56″ = 2.57549604013 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6.37 ; ; b = 4.3 ; ; beta = 13° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.3**2 = 6.37**2 + c**2 -2 * 6.37 * c * cos (13° ) ; ; ; ; c**2 -12.413c +22.087 =0 ; ; p=1; q=-12.413; r=22.087 ; ; D = q**2 - 4pr = 12.413**2 - 4 * 1 * 22.087 = 65.7467522746 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.41 ± sqrt{ 65.75 } }{ 2 } ; ; c_{1,2} = 6.20673731 ± 4.05421855216 ; ; c_{1} = 10.2609558622 ; ;
c_{2} = 2.15251875784 ; ; ; ; text{ Factored form: } ; ; (c -10.2609558622) (c -2.15251875784) = 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 = 6.37 ; ; b = 4.3 ; ; c = 10.26 ; ;

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

p = a+b+c = 6.37+4.3+10.26 = 20.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.93 }{ 2 } = 10.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.47 * (10.47-6.37)(10.47-4.3)(10.47-10.26) } ; ; T = sqrt{ 54.05 } = 7.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.35 }{ 6.37 } = 2.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.35 }{ 4.3 } = 3.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.35 }{ 10.26 } = 1.43 ; ;

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{ 4.3**2+10.26**2-6.37**2 }{ 2 * 4.3 * 10.26 } ) = 19° 27'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.37**2+10.26**2-4.3**2 }{ 2 * 6.37 * 10.26 } ) = 13° ; ; gamma = 180° - alpha - beta = 180° - 19° 27'56" - 13° = 147° 32'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.35 }{ 10.47 } = 0.7 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.37 }{ 2 * sin 19° 27'56" } = 9.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.3**2+2 * 10.26**2 - 6.37**2 } }{ 2 } = 7.193 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.26**2+2 * 6.37**2 - 4.3**2 } }{ 2 } = 8.265 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.3**2+2 * 6.37**2 - 10.26**2 } }{ 2 } = 1.792 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.37   b = 4.3   c = 2.15325187605

Area: T = 1.54222131965
Perimeter: p = 12.82325187605
Semiperimeter: s = 6.41112593803

Angle ∠ A = α = 160.534436341° = 160°32'4″ = 2.80218532041 rad
Angle ∠ B = β = 13° = 0.22768928028 rad
Angle ∠ C = γ = 6.46656365902° = 6°27'56″ = 0.11328466467 rad

Height: ha = 0.48442113647
Height: hb = 0.71773084635
Height: hc = 1.43329382162

Median: ma = 1.1910564365
Median: mb = 4.24105917638
Median: mc = 5.32768298026

Inradius: r = 0.24105476218
Circumradius: R = 9.55876346876

Vertex coordinates: A[2.15325187605; 0] B[0; 0] C[6.20767373127; 1.43329382162]
Centroid: CG[2.78664186911; 0.47876460721]
Coordinates of the circumscribed circle: U[1.07662593803; 9.4976844032]
Coordinates of the inscribed circle: I[2.11112593803; 0.24105476218]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.46656365902° = 19°27'56″ = 2.80218532041 rad
∠ B' = β' = 167° = 0.22768928028 rad
∠ C' = γ' = 173.534436341° = 173°32'4″ = 0.11328466467 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6.37 ; ; b = 4.3 ; ; beta = 13° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.3**2 = 6.37**2 + c**2 -2 * 6.37 * c * cos (13° ) ; ; ; ; c**2 -12.413c +22.087 =0 ; ; p=1; q=-12.413; r=22.087 ; ; D = q**2 - 4pr = 12.413**2 - 4 * 1 * 22.087 = 65.7467522746 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.41 ± sqrt{ 65.75 } }{ 2 } ; ; c_{1,2} = 6.20673731 ± 4.05421855216 ; ; c_{1} = 10.2609558622 ; ; : Nr. 1
c_{2} = 2.15251875784 ; ; ; ; text{ Factored form: } ; ; (c -10.2609558622) (c -2.15251875784) = 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 = 6.37 ; ; b = 4.3 ; ; c = 2.15 ; ;

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

p = a+b+c = 6.37+4.3+2.15 = 12.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.82 }{ 2 } = 6.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.41 * (6.41-6.37)(6.41-4.3)(6.41-2.15) } ; ; T = sqrt{ 2.38 } = 1.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.54 }{ 6.37 } = 0.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.54 }{ 4.3 } = 0.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.54 }{ 2.15 } = 1.43 ; ;

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{ 4.3**2+2.15**2-6.37**2 }{ 2 * 4.3 * 2.15 } ) = 160° 32'4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.37**2+2.15**2-4.3**2 }{ 2 * 6.37 * 2.15 } ) = 13° ; ; gamma = 180° - alpha - beta = 180° - 160° 32'4" - 13° = 6° 27'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.54 }{ 6.41 } = 0.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.37 }{ 2 * sin 160° 32'4" } = 9.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.3**2+2 * 2.15**2 - 6.37**2 } }{ 2 } = 1.191 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.15**2+2 * 6.37**2 - 4.3**2 } }{ 2 } = 4.241 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.3**2+2 * 6.37**2 - 2.15**2 } }{ 2 } = 5.327 ; ;
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