Triangle calculator SSA

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Triangle has two solutions with side c=76.93550730769 and with side c=9.203256486

#1 Acute scalene triangle.

Sides: a = 62   b = 56   c = 76.93550730769

Area: T = 1715.616626329
Perimeter: p = 194.9355073077
Semiperimeter: s = 97.46875365385

Angle ∠ A = α = 52.7898819341° = 52°47'20″ = 0.92113387057 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 81.2111180659° = 81°12'40″ = 1.41774024919 rad

Height: ha = 55.34224601062
Height: hb = 61.27220094033
Height: hc = 44.5999067621

Median: ma = 59.72202037394
Median: mb = 64.0121739038
Median: mc = 44.83657963313

Inradius: r = 17.60219249508
Circumradius: R = 38.92545805485

Vertex coordinates: A[76.93550730769; 0] B[0; 0] C[43.06988189685; 44.5999067621]
Centroid: CG[40.00112973485; 14.86663558737]
Coordinates of the circumscribed circle: U[38.46875365385; 5.94774030919]
Coordinates of the inscribed circle: I[41.46875365385; 17.60219249508]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.2111180659° = 127°12'40″ = 0.92113387057 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 98.7898819341° = 98°47'20″ = 1.41774024919 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 56 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 62**2 + c**2 -2 * 62 * c * cos (46° ) ; ; ; ; c**2 -86.138c +708 =0 ; ; p=1; q=-86.138; r=708 ; ; D = q**2 - 4pr = 86.138**2 - 4 * 1 * 708 = 4587.69266935 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 86.14 ± sqrt{ 4587.69 } }{ 2 } ; ; c_{1,2} = 43.06881897 ± 33.8662541084 ; ; c_{1} = 76.9350730784 ; ;
c_{2} = 9.20256486155 ; ; ; ; text{ Factored form: } ; ; (c -76.9350730784) (c -9.20256486155) = 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 = 62 ; ; b = 56 ; ; c = 76.94 ; ;

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

p = a+b+c = 62+56+76.94 = 194.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 194.94 }{ 2 } = 97.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 97.47 * (97.47-62)(97.47-56)(97.47-76.94) } ; ; T = sqrt{ 2943339.16 } = 1715.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1715.62 }{ 62 } = 55.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1715.62 }{ 56 } = 61.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1715.62 }{ 76.94 } = 44.6 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 56**2+76.94**2-62**2 }{ 2 * 56 * 76.94 } ) = 52° 47'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+76.94**2-56**2 }{ 2 * 62 * 76.94 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 52° 47'20" - 46° = 81° 12'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1715.62 }{ 97.47 } = 17.6 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 52° 47'20" } = 38.92 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 76.94**2 - 62**2 } }{ 2 } = 59.72 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 76.94**2+2 * 62**2 - 56**2 } }{ 2 } = 64.012 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 62**2 - 76.94**2 } }{ 2 } = 44.836 ; ;







#2 Obtuse scalene triangle.

Sides: a = 62   b = 56   c = 9.203256486

Area: T = 205.2132906239
Perimeter: p = 127.203256486
Semiperimeter: s = 63.601128243

Angle ∠ A = α = 127.2111180659° = 127°12'40″ = 2.22202539478 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 6.7898819341° = 6°47'20″ = 0.11884872498 rad

Height: ha = 6.6219771169
Height: hb = 7.32990323657
Height: hc = 44.5999067621

Median: ma = 25.48222212533
Median: mb = 34.3566129002
Median: mc = 58.89767588242

Inradius: r = 3.22765529624
Circumradius: R = 38.92545805485

Vertex coordinates: A[9.203256486; 0] B[0; 0] C[43.06988189685; 44.5999067621]
Centroid: CG[17.42437946095; 14.86663558737]
Coordinates of the circumscribed circle: U[4.601128243; 38.65216645291]
Coordinates of the inscribed circle: I[7.601128243; 3.22765529624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 52.7898819341° = 52°47'20″ = 2.22202539478 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 173.2111180659° = 173°12'40″ = 0.11884872498 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 56 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 62**2 + c**2 -2 * 62 * c * cos (46° ) ; ; ; ; c**2 -86.138c +708 =0 ; ; p=1; q=-86.138; r=708 ; ; D = q**2 - 4pr = 86.138**2 - 4 * 1 * 708 = 4587.69266935 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 86.14 ± sqrt{ 4587.69 } }{ 2 } ; ; c_{1,2} = 43.06881897 ± 33.8662541084 ; ; c_{1} = 76.9350730784 ; ; : Nr. 1
c_{2} = 9.20256486155 ; ; ; ; text{ Factored form: } ; ; (c -76.9350730784) (c -9.20256486155) = 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 = 62 ; ; b = 56 ; ; c = 9.2 ; ;

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

p = a+b+c = 62+56+9.2 = 127.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.2 }{ 2 } = 63.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.6 * (63.6-62)(63.6-56)(63.6-9.2) } ; ; T = sqrt{ 42112.34 } = 205.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 205.21 }{ 62 } = 6.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 205.21 }{ 56 } = 7.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 205.21 }{ 9.2 } = 44.6 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 56**2+9.2**2-62**2 }{ 2 * 56 * 9.2 } ) = 127° 12'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+9.2**2-56**2 }{ 2 * 62 * 9.2 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 127° 12'40" - 46° = 6° 47'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 205.21 }{ 63.6 } = 3.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 127° 12'40" } = 38.92 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 9.2**2 - 62**2 } }{ 2 } = 25.482 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.2**2+2 * 62**2 - 56**2 } }{ 2 } = 34.356 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 62**2 - 9.2**2 } }{ 2 } = 58.897 ; ;
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