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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 90**2 = 100**2 + c**2 -2 * 90 * c * cos (30° ) ; ; ; ; c**2 -173.205c +1900 =0 ; ; p=1; q=-173.205080757; 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} = 86.6025403784 ± 74.8331477355 ; ; c_{1} = 161.435688114 ; ;
c_{2} = 11.769392643 ; ; ; ; (c -161.435688114) (c -11.769392643) = 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 = 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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