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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 27.5**2 = 39.1**2 + c**2 -2 * 27.5 * c * cos (35° 6') ; ; ; ; c**2 -63.979c +772.56 =0 ; ; p=1; q=-63.9793079026; 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} = 31.9896539513 ± 15.8359704447 ; ;
c_{1} = 47.825624396 ; ; c_{2} = 16.1536835066 ; ; ; ; (c -47.825624396) (c -16.1536835066) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 39.1**2-27.5**2-16.15**2 }{ 2 * 27.5 * 16.15 } ) = 125° 9'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.5**2-39.1**2-16.15**2 }{ 2 * 39.1 * 16.15 } ) = 35° 6' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.15**2-39.1**2-27.5**2 }{ 2 * 27.5 * 39.1 } ) = 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 ; ;




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