Triangle calculator SSA

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Triangle has two solutions with side c=44.40110221361 and with side c=16.83881258816

#1 Acute scalene triangle.

Sides: a = 37.8   b = 26.1   c = 44.40110221361

Area: T = 492.0721554638
Perimeter: p = 108.3011022136
Semiperimeter: s = 54.1510511068

Angle ∠ A = α = 58.1287902397° = 58°7'40″ = 1.01545232841 rad
Angle ∠ B = β = 35.9° = 35°54' = 0.62765732015 rad
Angle ∠ C = γ = 85.9722097603° = 85°58'20″ = 1.5500496168 rad

Height: ha = 26.03655319914
Height: hb = 37.70766325393
Height: hc = 22.16548750846

Median: ma = 31.13106984079
Median: mb = 39.1133205997
Median: mc = 23.71099622167

Inradius: r = 9.08771082273
Circumradius: R = 22.2555482971

Vertex coordinates: A[44.40110221361; 0] B[0; 0] C[30.62195740089; 22.16548750846]
Centroid: CG[25.00768653816; 7.38882916949]
Coordinates of the circumscribed circle: U[22.2010511068; 1.56332755964]
Coordinates of the inscribed circle: I[28.0510511068; 9.08771082273]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.8722097603° = 121°52'20″ = 1.01545232841 rad
∠ B' = β' = 144.1° = 144°6' = 0.62765732015 rad
∠ C' = γ' = 94.0287902397° = 94°1'40″ = 1.5500496168 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 37.8 ; ; b = 26.1 ; ; beta = 35° 54' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 26.1**2 = 37.8**2 + c**2 -2 * 37.8 * c * cos (35° 54') ; ; ; ; c**2 -61.239c +747.63 =0 ; ; p=1; q=-61.239; r=747.63 ; ; D = q**2 - 4pr = 61.239**2 - 4 * 1 * 747.63 = 759.713249934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 61.24 ± sqrt{ 759.71 } }{ 2 } ; ; c_{1,2} = 30.61957401 ± 13.7814481272 ; ; c_{1} = 44.4010221372 ; ; c_{2} = 16.8381258828 ; ; ; ; text{ Factored form: } ; ; (c -44.4010221372) (c -16.8381258828) = 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 = 37.8 ; ; b = 26.1 ; ; c = 44.4 ; ;

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

p = a+b+c = 37.8+26.1+44.4 = 108.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108.3 }{ 2 } = 54.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.15 * (54.15-37.8)(54.15-26.1)(54.15-44.4) } ; ; T = sqrt{ 242134.41 } = 492.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 492.07 }{ 37.8 } = 26.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 492.07 }{ 26.1 } = 37.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 492.07 }{ 44.4 } = 22.16 ; ;

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{ 26.1**2+44.4**2-37.8**2 }{ 2 * 26.1 * 44.4 } ) = 58° 7'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.8**2+44.4**2-26.1**2 }{ 2 * 37.8 * 44.4 } ) = 35° 54' ; ; gamma = 180° - alpha - beta = 180° - 58° 7'40" - 35° 54' = 85° 58'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 492.07 }{ 54.15 } = 9.09 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.8 }{ 2 * sin 58° 7'40" } = 22.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.1**2+2 * 44.4**2 - 37.8**2 } }{ 2 } = 31.131 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44.4**2+2 * 37.8**2 - 26.1**2 } }{ 2 } = 39.113 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.1**2+2 * 37.8**2 - 44.4**2 } }{ 2 } = 23.71 ; ;







#2 Obtuse scalene triangle.

Sides: a = 37.8   b = 26.1   c = 16.83881258816

Area: T = 186.6077478413
Perimeter: p = 80.73881258816
Semiperimeter: s = 40.36990629408

Angle ∠ A = α = 121.8722097603° = 121°52'20″ = 2.12770693695 rad
Angle ∠ B = β = 35.9° = 35°54' = 0.62765732015 rad
Angle ∠ C = γ = 22.2287902397° = 22°13'40″ = 0.38879500826 rad

Height: ha = 9.87334115562
Height: hb = 14.29994236331
Height: hc = 22.16548750846

Median: ma = 11.18773250423
Median: mb = 26.18992867716
Median: mc = 31.37110755187

Inradius: r = 4.62325367848
Circumradius: R = 22.2555482971

Vertex coordinates: A[16.83881258816; 0] B[0; 0] C[30.62195740089; 22.16548750846]
Centroid: CG[15.81992332968; 7.38882916949]
Coordinates of the circumscribed circle: U[8.41990629408; 20.60215994882]
Coordinates of the inscribed circle: I[14.26990629408; 4.62325367848]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.1287902397° = 58°7'40″ = 2.12770693695 rad
∠ B' = β' = 144.1° = 144°6' = 0.62765732015 rad
∠ C' = γ' = 157.7722097603° = 157°46'20″ = 0.38879500826 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 37.8 ; ; b = 26.1 ; ; beta = 35° 54' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 26.1**2 = 37.8**2 + c**2 -2 * 37.8 * c * cos (35° 54') ; ; ; ; c**2 -61.239c +747.63 =0 ; ; p=1; q=-61.239; r=747.63 ; ; D = q**2 - 4pr = 61.239**2 - 4 * 1 * 747.63 = 759.713249934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 61.24 ± sqrt{ 759.71 } }{ 2 } ; ; c_{1,2} = 30.61957401 ± 13.7814481272 ; ; c_{1} = 44.4010221372 ; ; c_{2} = 16.8381258828 ; ; ; ; text{ Factored form: } ; ; (c -44.4010221372) (c -16.8381258828) = 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 = 37.8 ; ; b = 26.1 ; ; c = 16.84 ; ;

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

p = a+b+c = 37.8+26.1+16.84 = 80.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80.74 }{ 2 } = 40.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.37 * (40.37-37.8)(40.37-26.1)(40.37-16.84) } ; ; T = sqrt{ 34822.35 } = 186.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 186.61 }{ 37.8 } = 9.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 186.61 }{ 26.1 } = 14.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 186.61 }{ 16.84 } = 22.16 ; ;

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{ 26.1**2+16.84**2-37.8**2 }{ 2 * 26.1 * 16.84 } ) = 121° 52'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.8**2+16.84**2-26.1**2 }{ 2 * 37.8 * 16.84 } ) = 35° 54' ; ; gamma = 180° - alpha - beta = 180° - 121° 52'20" - 35° 54' = 22° 13'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 186.61 }{ 40.37 } = 4.62 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.8 }{ 2 * sin 121° 52'20" } = 22.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.1**2+2 * 16.84**2 - 37.8**2 } }{ 2 } = 11.187 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.84**2+2 * 37.8**2 - 26.1**2 } }{ 2 } = 26.189 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.1**2+2 * 37.8**2 - 16.84**2 } }{ 2 } = 31.371 ; ;
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