Triangle calculator SSA

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Triangle has two solutions with side c=191.9576882176 and with side c=52.74662203243

#1 Obtuse scalene triangle.

Sides: a = 135   b = 90   c = 191.9576882176

Area: T = 5475.903266151
Perimeter: p = 416.9576882176
Semiperimeter: s = 208.4788441088

Angle ∠ A = α = 39.34404765045° = 39°20'26″ = 0.68766208443 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 115.6659523496° = 115°39'34″ = 2.01986394963 rad

Height: ha = 81.12444838742
Height: hb = 121.6876725811
Height: hc = 57.0533465335

Median: ma = 133.8566162754
Median: mb = 159.7222328769
Median: mc = 62.85441076331

Inradius: r = 26.26660380274
Circumradius: R = 106.4799071242

Vertex coordinates: A[191.9576882176; 0] B[0; 0] C[122.352155125; 57.0533465335]
Centroid: CG[104.7699477808; 19.01878217783]
Coordinates of the circumscribed circle: U[95.97884410878; -46.10878242697]
Coordinates of the inscribed circle: I[118.4788441088; 26.26660380274]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.6659523496° = 140°39'34″ = 0.68766208443 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 64.34404765045° = 64°20'26″ = 2.01986394963 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 135 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 135**2 + c**2 -2 * 135 * c * cos (25° ) ; ; ; ; c**2 -244.703c +10125 =0 ; ; p=1; q=-244.703; r=10125 ; ; D = q**2 - 4pr = 244.703**2 - 4 * 1 * 10125 = 19379.6083731 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 244.7 ± sqrt{ 19379.61 } }{ 2 } ; ; c_{1,2} = 122.35155125 ± 69.6053309256 ; ; c_{1} = 191.956882176 ; ;
c_{2} = 52.7462203244 ; ; ; ; text{ Factored form: } ; ; (c -191.956882176) (c -52.7462203244) = 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 = 135 ; ; b = 90 ; ; c = 191.96 ; ;

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

p = a+b+c = 135+90+191.96 = 416.96 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 416.96 }{ 2 } = 208.48 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 208.48 * (208.48-135)(208.48-90)(208.48-191.96) } ; ; T = sqrt{ 29985509.96 } = 5475.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5475.9 }{ 135 } = 81.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5475.9 }{ 90 } = 121.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5475.9 }{ 191.96 } = 57.05 ; ;

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{ 90**2+191.96**2-135**2 }{ 2 * 90 * 191.96 } ) = 39° 20'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 135**2+191.96**2-90**2 }{ 2 * 135 * 191.96 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 39° 20'26" - 25° = 115° 39'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5475.9 }{ 208.48 } = 26.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 135 }{ 2 * sin 39° 20'26" } = 106.48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 191.96**2 - 135**2 } }{ 2 } = 133.856 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 191.96**2+2 * 135**2 - 90**2 } }{ 2 } = 159.722 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 135**2 - 191.96**2 } }{ 2 } = 62.854 ; ;







#2 Obtuse scalene triangle.

Sides: a = 135   b = 90   c = 52.74662203243

Area: T = 1504.677732641
Perimeter: p = 277.7466220324
Semiperimeter: s = 138.8733110162

Angle ∠ A = α = 140.6659523496° = 140°39'34″ = 2.45549718093 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 14.34404765045° = 14°20'26″ = 0.25502885313 rad

Height: ha = 22.29215159468
Height: hb = 33.43772739203
Height: hc = 57.0533465335

Median: ma = 29.74661237685
Median: mb = 92.07992152402
Median: mc = 111.6565537527

Inradius: r = 10.83549076697
Circumradius: R = 106.4799071242

Vertex coordinates: A[52.74662203243; 0] B[0; 0] C[122.352155125; 57.0533465335]
Centroid: CG[58.36659238581; 19.01878217783]
Coordinates of the circumscribed circle: U[26.37331101622; 103.1611289605]
Coordinates of the inscribed circle: I[48.87331101622; 10.83549076697]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 39.34404765045° = 39°20'26″ = 2.45549718093 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 165.6659523496° = 165°39'34″ = 0.25502885313 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 135 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 135**2 + c**2 -2 * 135 * c * cos (25° ) ; ; ; ; c**2 -244.703c +10125 =0 ; ; p=1; q=-244.703; r=10125 ; ; D = q**2 - 4pr = 244.703**2 - 4 * 1 * 10125 = 19379.6083731 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 244.7 ± sqrt{ 19379.61 } }{ 2 } ; ; c_{1,2} = 122.35155125 ± 69.6053309256 ; ; c_{1} = 191.956882176 ; ; : Nr. 1
c_{2} = 52.7462203244 ; ; ; ; text{ Factored form: } ; ; (c -191.956882176) (c -52.7462203244) = 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 = 135 ; ; b = 90 ; ; c = 52.75 ; ;

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

p = a+b+c = 135+90+52.75 = 277.75 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 277.75 }{ 2 } = 138.87 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 138.87 * (138.87-135)(138.87-90)(138.87-52.75) } ; ; T = sqrt{ 2264053.86 } = 1504.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1504.68 }{ 135 } = 22.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1504.68 }{ 90 } = 33.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1504.68 }{ 52.75 } = 57.05 ; ;

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{ 90**2+52.75**2-135**2 }{ 2 * 90 * 52.75 } ) = 140° 39'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 135**2+52.75**2-90**2 }{ 2 * 135 * 52.75 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 140° 39'34" - 25° = 14° 20'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1504.68 }{ 138.87 } = 10.83 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 135 }{ 2 * sin 140° 39'34" } = 106.48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 52.75**2 - 135**2 } }{ 2 } = 29.746 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 52.75**2+2 * 135**2 - 90**2 } }{ 2 } = 92.079 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 135**2 - 52.75**2 } }{ 2 } = 111.656 ; ;
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