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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 90**2 = 130**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -235.64c +8800 =0 ; ; p=1; q=-235.64002463; 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} = 117.820012315 ± 71.2850285954 ; ; c_{1} = 189.10504091 ; ;
c_{2} = 46.5349837193 ; ; ; ; (c -189.10504091) (c -46.5349837193) = 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 = 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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