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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 26.1**2 = 37.8**2 + c**2 -2 * 26.1 * c * cos (35° 54') ; ; ; ; c**2 -61.239c +747.63 =0 ; ; p=1; q=-61.2391480177; 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} = 30.6195740089 ± 13.7814481272 ; ;
c_{1} = 44.4010221361 ; ; c_{2} = 16.8381258816 ; ; ; ; (c -44.4010221361) (c -16.8381258816) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37.8**2-26.1**2-16.84**2 }{ 2 * 26.1 * 16.84 } ) = 121° 52'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26.1**2-37.8**2-16.84**2 }{ 2 * 37.8 * 16.84 } ) = 35° 54' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.84**2-37.8**2-26.1**2 }{ 2 * 26.1 * 37.8 } ) = 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 ; ;




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