Triangle calculator SSA

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Triangle has two solutions with side c=77.78435201238 and with side c=14.14218130505

#1 Acute scalene triangle.

Sides: a = 60   b = 50   c = 77.78435201238

Area: T = 1499.94884892
Perimeter: p = 187.7843520124
Semiperimeter: s = 93.89217600619

Angle ∠ A = α = 50.47548346316° = 50°28'29″ = 0.88109520537 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 89.52551653684° = 89°31'31″ = 1.56325088991 rad

Height: ha = 49.99882829734
Height: hb = 59.9987939568
Height: hc = 38.56772565812

Median: ma = 58.09659379081
Median: mb = 64.80884716794
Median: mc = 39.21100879786

Inradius: r = 15.97552941921
Circumradius: R = 38.89330956715

Vertex coordinates: A[77.78435201238; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[41.24987289036; 12.85657521937]
Coordinates of the circumscribed circle: U[38.89217600619; 0.32223200297]
Coordinates of the inscribed circle: I[43.89217600619; 15.97552941921]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.5255165368° = 129°31'31″ = 0.88109520537 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 90.47548346316° = 90°28'29″ = 1.56325088991 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1100 =0 ; ; p=1; q=-91.925; r=1100 ; ; D = q**2 - 4pr = 91.925**2 - 4 * 1 * 1100 = 4050.2668792 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 91.93 ± sqrt{ 4050.27 } }{ 2 } ; ; c_{1,2} = 45.96266659 ± 31.8208535366 ; ; c_{1} = 77.7835201266 ; ;
c_{2} = 14.1418130534 ; ; ; ; text{ Factored form: } ; ; (c -77.7835201266) (c -14.1418130534) = 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 = 60 ; ; b = 50 ; ; c = 77.78 ; ;

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

p = a+b+c = 60+50+77.78 = 187.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 187.78 }{ 2 } = 93.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 93.89 * (93.89-60)(93.89-50)(93.89-77.78) } ; ; T = sqrt{ 2249845.47 } = 1499.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1499.95 }{ 60 } = 50 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1499.95 }{ 50 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1499.95 }{ 77.78 } = 38.57 ; ;

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{ 50**2+77.78**2-60**2 }{ 2 * 50 * 77.78 } ) = 50° 28'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+77.78**2-50**2 }{ 2 * 60 * 77.78 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 50° 28'29" - 40° = 89° 31'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1499.95 }{ 93.89 } = 15.98 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 50° 28'29" } = 38.89 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 77.78**2 - 60**2 } }{ 2 } = 58.096 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.78**2+2 * 60**2 - 50**2 } }{ 2 } = 64.808 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 77.78**2 } }{ 2 } = 39.21 ; ;





#2 Obtuse scalene triangle.

Sides: a = 60   b = 50   c = 14.14218130505

Area: T = 272.7055466221
Perimeter: p = 124.1421813051
Semiperimeter: s = 62.07109065252

Angle ∠ A = α = 129.5255165368° = 129°31'31″ = 2.26106405999 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 10.47548346316° = 10°28'29″ = 0.18328203529 rad

Height: ha = 9.09901822074
Height: hb = 10.90882186488
Height: hc = 38.56772565812

Median: ma = 21.21330959121
Median: mb = 35.70770782644
Median: mc = 54.77222765723

Inradius: r = 4.39334506758
Circumradius: R = 38.89330956715

Vertex coordinates: A[14.14218130505; 0] B[0; 0] C[45.96326665871; 38.56772565812]
Centroid: CG[20.03548265459; 12.85657521937]
Coordinates of the circumscribed circle: U[7.07109065252; 38.24549365515]
Coordinates of the inscribed circle: I[12.07109065252; 4.39334506758]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 50.47548346316° = 50°28'29″ = 2.26106405999 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 169.5255165368° = 169°31'31″ = 0.18328203529 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (40° ) ; ; ; ; c**2 -91.925c +1100 =0 ; ; p=1; q=-91.925; r=1100 ; ; D = q**2 - 4pr = 91.925**2 - 4 * 1 * 1100 = 4050.2668792 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 91.93 ± sqrt{ 4050.27 } }{ 2 } ; ; c_{1,2} = 45.96266659 ± 31.8208535366 ; ; c_{1} = 77.7835201266 ; ; : Nr. 1
c_{2} = 14.1418130534 ; ; ; ; text{ Factored form: } ; ; (c -77.7835201266) (c -14.1418130534) = 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 = 60 ; ; b = 50 ; ; c = 14.14 ; ;

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

p = a+b+c = 60+50+14.14 = 124.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.14 }{ 2 } = 62.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.07 * (62.07-60)(62.07-50)(62.07-14.14) } ; ; T = sqrt{ 74368.27 } = 272.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 272.71 }{ 60 } = 9.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 272.71 }{ 50 } = 10.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 272.71 }{ 14.14 } = 38.57 ; ;

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{ 50**2+14.14**2-60**2 }{ 2 * 50 * 14.14 } ) = 129° 31'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+14.14**2-50**2 }{ 2 * 60 * 14.14 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 129° 31'31" - 40° = 10° 28'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 272.71 }{ 62.07 } = 4.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 129° 31'31" } = 38.89 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 14.14**2 - 60**2 } }{ 2 } = 21.213 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.14**2+2 * 60**2 - 50**2 } }{ 2 } = 35.707 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 14.14**2 } }{ 2 } = 54.772 ; ;
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