Triangle calculator SSA

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Triangle has two solutions with side c=189.105504091 and with side c=46.53549837193

#1 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 189.105504091

Area: T = 5194.751083893
Perimeter: p = 409.105504091
Semiperimeter: s = 204.5532520455

Angle ∠ A = α = 37.62219467586° = 37°37'19″ = 0.65766268419 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 117.3788053241° = 117°22'41″ = 2.04986334986 rad

Height: ha = 79.91992436759
Height: hb = 115.4398907532
Height: hc = 54.94403740263

Median: ma = 133.0611482965
Median: mb = 155.9021758325
Median: mc = 59.66442344758

Inradius: r = 25.39656823772
Circumradius: R = 106.4799071242

Vertex coordinates: A[189.105504091; 0] B[0; 0] C[117.8220012315; 54.94403740263]
Centroid: CG[102.3088351075; 18.31334580088]
Coordinates of the circumscribed circle: U[94.55325204551; -48.96554315627]
Coordinates of the inscribed circle: I[114.5532520455; 25.39656823772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.3788053241° = 142°22'41″ = 0.65766268419 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 62.62219467586° = 62°37'19″ = 2.04986334986 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (25° ) ; ; ; ; c**2 -235.64c +8800 =0 ; ; p=1; q=-235.64; r=8800 ; ; D = q**2 - 4pr = 235.64**2 - 4 * 1 * 8800 = 20326.2212074 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 235.64 ± sqrt{ 20326.22 } }{ 2 } ; ; c_{1,2} = 117.82001231 ± 71.2850285954 ; ; c_{1} = 189.105040905 ; ;
c_{2} = 46.5349837146 ; ; ; ; (c -189.105040905) (c -46.5349837146) = 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 = 130 ; ; b = 90 ; ; c = 189.11 ; ;

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

p = a+b+c = 130+90+189.11 = 409.11 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 409.11 }{ 2 } = 204.55 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 204.55 * (204.55-130)(204.55-90)(204.55-189.11) } ; ; T = sqrt{ 26985436.28 } = 5194.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5194.75 }{ 130 } = 79.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5194.75 }{ 90 } = 115.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5194.75 }{ 189.11 } = 54.94 ; ;

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{ 130**2-90**2-189.11**2 }{ 2 * 90 * 189.11 } ) = 37° 37'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-189.11**2 }{ 2 * 130 * 189.11 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 189.11**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 117° 22'41" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5194.75 }{ 204.55 } = 25.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 130 }{ 2 * sin 37° 37'19" } = 106.48 ; ;





#2 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 46.53549837193

Area: T = 1278.325470542
Perimeter: p = 266.5354983719
Semiperimeter: s = 133.267749186

Angle ∠ A = α = 142.3788053241° = 142°22'41″ = 2.48549658116 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 12.62219467586° = 12°37'19″ = 0.22202945289 rad

Height: ha = 19.66765339296
Height: hb = 28.40772156761
Height: hc = 54.94403740263

Median: ma = 30.12989288704
Median: mb = 86.64772870601
Median: mc = 109.3555492878

Inradius: r = 9.59221720112
Circumradius: R = 106.4799071242

Vertex coordinates: A[46.53549837193; 0] B[0; 0] C[117.8220012315; 54.94403740263]
Centroid: CG[54.7854998678; 18.31334580088]
Coordinates of the circumscribed circle: U[23.26774918597; 103.9065805589]
Coordinates of the inscribed circle: I[43.26774918597; 9.59221720112]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 37.62219467586° = 37°37'19″ = 2.48549658116 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 167.3788053241° = 167°22'41″ = 0.22202945289 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (25° ) ; ; ; ; c**2 -235.64c +8800 =0 ; ; p=1; q=-235.64; r=8800 ; ; D = q**2 - 4pr = 235.64**2 - 4 * 1 * 8800 = 20326.2212074 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 235.64 ± sqrt{ 20326.22 } }{ 2 } ; ; c_{1,2} = 117.82001231 ± 71.2850285954 ; ; c_{1} = 189.105040905 ; ; : Nr. 1
c_{2} = 46.5349837146 ; ; ; ; (c -189.105040905) (c -46.5349837146) = 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 = 130 ; ; b = 90 ; ; c = 46.53 ; ;

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

p = a+b+c = 130+90+46.53 = 266.53 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 266.53 }{ 2 } = 133.27 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.27 * (133.27-130)(133.27-90)(133.27-46.53) } ; ; T = sqrt{ 1634114.05 } = 1278.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1278.32 }{ 130 } = 19.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1278.32 }{ 90 } = 28.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1278.32 }{ 46.53 } = 54.94 ; ;

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{ 130**2-90**2-46.53**2 }{ 2 * 90 * 46.53 } ) = 142° 22'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-46.53**2 }{ 2 * 130 * 46.53 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.53**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 12° 37'19" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1278.32 }{ 133.27 } = 9.59 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 130 }{ 2 * sin 142° 22'41" } = 106.48 ; ;




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