Triangle calculator SSA

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Triangle has two solutions with side c=47.8265624396 and with side c=16.15436835066

#1 Obtuse scalene triangle.

Sides: a = 39.1   b = 27.5   c = 47.8265624396

Area: T = 537.6254710855
Perimeter: p = 114.4265624396
Semiperimeter: s = 57.2132812198

Angle ∠ A = α = 54.84105767961° = 54°50'26″ = 0.95771486288 rad
Angle ∠ B = β = 35.1° = 35°6' = 0.61326105675 rad
Angle ∠ C = γ = 90.05994232039° = 90°3'34″ = 1.57218334574 rad

Height: ha = 27.549998521
Height: hb = 39.10999789712
Height: hc = 22.48327053549

Median: ma = 33.75774832361
Median: mb = 41.46106762419
Median: mc = 23.88994833093

Inradius: r = 9.39769285934
Circumradius: R = 23.91328250588

Vertex coordinates: A[47.8265624396; 0] B[0; 0] C[31.99896539513; 22.48327053549]
Centroid: CG[26.60550927824; 7.49442351183]
Coordinates of the circumscribed circle: U[23.9132812198; -0.02548007172]
Coordinates of the inscribed circle: I[29.7132812198; 9.39769285934]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.1599423204° = 125°9'34″ = 0.95771486288 rad
∠ B' = β' = 144.9° = 144°54' = 0.61326105675 rad
∠ C' = γ' = 89.94105767961° = 89°56'26″ = 1.57218334574 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 39.1 ; ; b = 27.5 ; ; beta = 35° 6' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 27.5**2 = 39.1**2 + c**2 -2 * 39.1 * c * cos (35° 6') ; ; ; ; c**2 -63.979c +772.56 =0 ; ; p=1; q=-63.979; r=772.56 ; ; D = q**2 - 4pr = 63.979**2 - 4 * 1 * 772.56 = 1003.1118397 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.98 ± sqrt{ 1003.11 } }{ 2 } ; ; c_{1,2} = 31.98965395 ± 15.8359704447 ; ;
c_{1} = 47.8256243947 ; ; c_{2} = 16.1536835053 ; ; ; ; text{ Factored form: } ; ; (c -47.8256243947) (c -16.1536835053) = 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 = 39.1 ; ; b = 27.5 ; ; c = 47.83 ; ;

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

p = a+b+c = 39.1+27.5+47.83 = 114.43 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.43 }{ 2 } = 57.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.21 * (57.21-39.1)(57.21-27.5)(57.21-47.83) } ; ; T = sqrt{ 289040.33 } = 537.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 537.62 }{ 39.1 } = 27.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 537.62 }{ 27.5 } = 39.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 537.62 }{ 47.83 } = 22.48 ; ;

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{ 27.5**2+47.83**2-39.1**2 }{ 2 * 27.5 * 47.83 } ) = 54° 50'26" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39.1**2+47.83**2-27.5**2 }{ 2 * 39.1 * 47.83 } ) = 35° 6' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 39.1**2+27.5**2-47.83**2 }{ 2 * 39.1 * 27.5 } ) = 90° 3'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 537.62 }{ 57.21 } = 9.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.1 }{ 2 * sin 54° 50'26" } = 23.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.5**2+2 * 47.83**2 - 39.1**2 } }{ 2 } = 33.757 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 47.83**2+2 * 39.1**2 - 27.5**2 } }{ 2 } = 41.461 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.5**2+2 * 39.1**2 - 47.83**2 } }{ 2 } = 23.889 ; ;







#2 Obtuse scalene triangle.

Sides: a = 39.1   b = 27.5   c = 16.15436835066

Area: T = 181.5899253338
Perimeter: p = 82.75436835066
Semiperimeter: s = 41.37768417533

Angle ∠ A = α = 125.1599423204° = 125°9'34″ = 2.18444440248 rad
Angle ∠ B = β = 35.1° = 35°6' = 0.61326105675 rad
Angle ∠ C = γ = 19.74105767961° = 19°44'26″ = 0.34545380613 rad

Height: ha = 9.28884528562
Height: hb = 13.20664911518
Height: hc = 22.48327053549

Median: ma = 11.24224750574
Median: mb = 26.5677145978
Median: mc = 32.82221667062

Inradius: r = 4.38986687732
Circumradius: R = 23.91328250588

Vertex coordinates: A[16.15436835066; 0] B[0; 0] C[31.99896539513; 22.48327053549]
Centroid: CG[16.04877791526; 7.49442351183]
Coordinates of the circumscribed circle: U[8.07768417533; 22.50875060721]
Coordinates of the inscribed circle: I[13.87768417533; 4.38986687732]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 54.84105767961° = 54°50'26″ = 2.18444440248 rad
∠ B' = β' = 144.9° = 144°54' = 0.61326105675 rad
∠ C' = γ' = 160.2599423204° = 160°15'34″ = 0.34545380613 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 39.1 ; ; b = 27.5 ; ; beta = 35° 6' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 27.5**2 = 39.1**2 + c**2 -2 * 39.1 * c * cos (35° 6') ; ; ; ; c**2 -63.979c +772.56 =0 ; ; p=1; q=-63.979; r=772.56 ; ; D = q**2 - 4pr = 63.979**2 - 4 * 1 * 772.56 = 1003.1118397 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.98 ± sqrt{ 1003.11 } }{ 2 } ; ; c_{1,2} = 31.98965395 ± 15.8359704447 ; ; : Nr. 1
c_{1} = 47.8256243947 ; ; c_{2} = 16.1536835053 ; ; ; ; text{ Factored form: } ; ; (c -47.8256243947) (c -16.1536835053) = 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 = 39.1 ; ; b = 27.5 ; ; c = 16.15 ; ;

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

p = a+b+c = 39.1+27.5+16.15 = 82.75 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.75 }{ 2 } = 41.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.38 * (41.38-39.1)(41.38-27.5)(41.38-16.15) } ; ; T = sqrt{ 32974.66 } = 181.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 181.59 }{ 39.1 } = 9.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 181.59 }{ 27.5 } = 13.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 181.59 }{ 16.15 } = 22.48 ; ;

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{ 27.5**2+16.15**2-39.1**2 }{ 2 * 27.5 * 16.15 } ) = 125° 9'34" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39.1**2+16.15**2-27.5**2 }{ 2 * 39.1 * 16.15 } ) = 35° 6' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 39.1**2+27.5**2-16.15**2 }{ 2 * 39.1 * 27.5 } ) = 19° 44'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 181.59 }{ 41.38 } = 4.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.1 }{ 2 * sin 125° 9'34" } = 23.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.5**2+2 * 16.15**2 - 39.1**2 } }{ 2 } = 11.242 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.15**2+2 * 39.1**2 - 27.5**2 } }{ 2 } = 26.567 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.5**2+2 * 39.1**2 - 16.15**2 } }{ 2 } = 32.822 ; ;
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