Triangle calculator SSA

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Triangle has two solutions with side c=161.4365688114 and with side c=11.7699392643

#1 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 161.4365688114

Area: T = 4035.892220285
Perimeter: p = 351.4365688114
Semiperimeter: s = 175.7187844057

Angle ∠ A = α = 33.74989885959° = 33°44'56″ = 0.58990309702 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 116.2511011404° = 116°15'4″ = 2.02989629078 rad

Height: ha = 80.7187844057
Height: hb = 89.68664933966
Height: hc = 50

Median: ma = 120.7510737879
Median: mb = 126.5143796475
Median: mc = 50.34551055297

Inradius: r = 22.9688027092
Circumradius: R = 90

Vertex coordinates: A[161.4365688114; 0] B[0; 0] C[86.60325403784; 50]
Centroid: CG[82.67994094975; 16.66766666667]
Coordinates of the circumscribed circle: U[80.7187844057; -39.80774069841]
Coordinates of the inscribed circle: I[85.7187844057; 22.9688027092]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.2511011404° = 146°15'4″ = 0.58990309702 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 63.74989885959° = 63°44'56″ = 2.02989629078 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -173.205c +1900 =0 ; ; p=1; q=-173.205; r=1900 ; ; D = q**2 - 4pr = 173.205**2 - 4 * 1 * 1900 = 22400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 173.21 ± sqrt{ 22400 } }{ 2 } ; ; c_{1,2} = 86.60254038 ± 74.8331477355 ; ; c_{1} = 161.435688115 ; ;
c_{2} = 11.7693926445 ; ; ; ; (c -161.435688115) (c -11.7693926445) = 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 = 100 ; ; b = 90 ; ; c = 161.44 ; ;

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

p = a+b+c = 100+90+161.44 = 351.44 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 351.44 }{ 2 } = 175.72 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.72 * (175.72-100)(175.72-90)(175.72-161.44) } ; ; T = sqrt{ 16288425.87 } = 4035.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4035.89 }{ 100 } = 80.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4035.89 }{ 90 } = 89.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4035.89 }{ 161.44 } = 50 ; ;

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{ 100**2-90**2-161.44**2 }{ 2 * 90 * 161.44 } ) = 33° 44'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-161.44**2 }{ 2 * 100 * 161.44 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 161.44**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 116° 15'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4035.89 }{ 175.72 } = 22.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 33° 44'56" } = 90 ; ;





#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 11.7699392643

Area: T = 294.2354816074
Perimeter: p = 201.7699392643
Semiperimeter: s = 100.8854696322

Angle ∠ A = α = 146.2511011404° = 146°15'4″ = 2.55325616834 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 3.74989885959° = 3°44'56″ = 0.06554321946 rad

Height: ha = 5.88546963215
Height: hb = 6.53985514683
Height: hc = 50

Median: ma = 40.2440021143
Median: mb = 55.17548067653
Median: mc = 94.94993041007

Inradius: r = 2.91765455892
Circumradius: R = 90

Vertex coordinates: A[11.7699392643; 0] B[0; 0] C[86.60325403784; 50]
Centroid: CG[32.79106443405; 16.66766666667]
Coordinates of the circumscribed circle: U[5.88546963215; 89.80774069841]
Coordinates of the inscribed circle: I[10.88546963215; 2.91765455892]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.74989885959° = 33°44'56″ = 2.55325616834 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 176.2511011404° = 176°15'4″ = 0.06554321946 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (30° ) ; ; ; ; c**2 -173.205c +1900 =0 ; ; p=1; q=-173.205; r=1900 ; ; D = q**2 - 4pr = 173.205**2 - 4 * 1 * 1900 = 22400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 173.21 ± sqrt{ 22400 } }{ 2 } ; ; c_{1,2} = 86.60254038 ± 74.8331477355 ; ; c_{1} = 161.435688115 ; ; : Nr. 1
c_{2} = 11.7693926445 ; ; ; ; (c -161.435688115) (c -11.7693926445) = 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 = 100 ; ; b = 90 ; ; c = 11.77 ; ;

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

p = a+b+c = 100+90+11.77 = 201.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.77 }{ 2 } = 100.88 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.88 * (100.88-100)(100.88-90)(100.88-11.77) } ; ; T = sqrt{ 86574.13 } = 294.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 294.23 }{ 100 } = 5.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 294.23 }{ 90 } = 6.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 294.23 }{ 11.77 } = 50 ; ;

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{ 100**2-90**2-11.77**2 }{ 2 * 90 * 11.77 } ) = 146° 15'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-11.77**2 }{ 2 * 100 * 11.77 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.77**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 3° 44'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 294.23 }{ 100.88 } = 2.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 146° 15'4" } = 90 ; ;




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