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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 90**2 = 135**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -244.703c +10125 =0 ; ; p=1; q=-244.7031025; 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} = 52.7462203243 ; ; ; ; (c -191.956882176) (c -52.7462203243) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 135**2-90**2-52.75**2 }{ 2 * 90 * 52.75 } ) = 140° 39'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-135**2-52.75**2 }{ 2 * 135 * 52.75 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 52.75**2-135**2-90**2 }{ 2 * 90 * 135 } ) = 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 ; ;




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