Triangle calculator SSA

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Triangle has two solutions with side c=7.51550398384 and with side c=3.83223150135

#1 Acute scalene triangle.

Sides: a = 7.2   b = 4.8   c = 7.51550398384

Area: T = 16.65661938506
Perimeter: p = 19.51550398384
Semiperimeter: s = 9.75875199192

Angle ∠ A = α = 67.44220807656° = 67°26'31″ = 1.17770863638 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 74.55879192344° = 74°33'29″ = 1.30112811741 rad

Height: ha = 4.6276720514
Height: hb = 6.94400807711
Height: hc = 4.43327626223

Median: ma = 5.17766699611
Median: mb = 6.95768607781
Median: mc = 4.82991866869

Inradius: r = 1.70770110016
Circumradius: R = 3.89882461892

Vertex coordinates: A[7.51550398384; 0] B[0; 0] C[5.6743677426; 4.43327626223]
Centroid: CG[4.39662390881; 1.47875875408]
Coordinates of the circumscribed circle: U[3.75875199192; 1.03879631054]
Coordinates of the inscribed circle: I[4.95875199192; 1.70770110016]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5587919234° = 112°33'29″ = 1.17770863638 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 105.4422080766° = 105°26'31″ = 1.30112811741 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 7.2 ; ; b = 4.8 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.8**2 = 7.2**2 + c**2 -2 * 7.2 * c * cos (38° ) ; ; ; ; c**2 -11.347c +28.8 =0 ; ; p=1; q=-11.347; r=28.8 ; ; D = q**2 - 4pr = 11.347**2 - 4 * 1 * 28.8 = 13.5624621358 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.35 ± sqrt{ 13.56 } }{ 2 } ; ;
c_{1,2} = 5.67367743 ± 1.84136241244 ; ; c_{1} = 7.51503983841 ; ; c_{2} = 3.83231501353 ; ; ; ; text{ Factored form: } ; ; (c -7.51503983841) (c -3.83231501353) = 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 = 7.2 ; ; b = 4.8 ; ; c = 7.52 ; ;

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

p = a+b+c = 7.2+4.8+7.52 = 19.52 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.52 }{ 2 } = 9.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.76 * (9.76-7.2)(9.76-4.8)(9.76-7.52) } ; ; T = sqrt{ 277.43 } = 16.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.66 }{ 7.2 } = 4.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.66 }{ 4.8 } = 6.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.66 }{ 7.52 } = 4.43 ; ;

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{ 4.8**2+7.52**2-7.2**2 }{ 2 * 4.8 * 7.52 } ) = 67° 26'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.2**2+7.52**2-4.8**2 }{ 2 * 7.2 * 7.52 } ) = 38° ; ;
 gamma = 180° - alpha - beta = 180° - 67° 26'31" - 38° = 74° 33'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.66 }{ 9.76 } = 1.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.2 }{ 2 * sin 67° 26'31" } = 3.9 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 7.52**2 - 7.2**2 } }{ 2 } = 5.177 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.52**2+2 * 7.2**2 - 4.8**2 } }{ 2 } = 6.957 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 7.2**2 - 7.52**2 } }{ 2 } = 4.829 ; ;



#2 Obtuse scalene triangle.

Sides: a = 7.2   b = 4.8   c = 3.83223150135

Area: T = 8.49438713745
Perimeter: p = 15.83223150135
Semiperimeter: s = 7.91661575068

Angle ∠ A = α = 112.5587919234° = 112°33'29″ = 1.96545062898 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 29.44220807656° = 29°26'31″ = 0.5143861248 rad

Height: ha = 2.35994087151
Height: hb = 3.53991130727
Height: hc = 4.43327626223

Median: ma = 2.43296747069
Median: mb = 5.24443607028
Median: mc = 5.8111053296

Inradius: r = 1.07329790769
Circumradius: R = 3.89882461892

Vertex coordinates: A[3.83223150135; 0] B[0; 0] C[5.6743677426; 4.43327626223]
Centroid: CG[3.16986641465; 1.47875875408]
Coordinates of the circumscribed circle: U[1.91661575068; 3.3954799517]
Coordinates of the inscribed circle: I[3.11661575068; 1.07329790769]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.44220807656° = 67°26'31″ = 1.96545062898 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 150.5587919234° = 150°33'29″ = 0.5143861248 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 7.2 ; ; b = 4.8 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.8**2 = 7.2**2 + c**2 -2 * 7.2 * c * cos (38° ) ; ; ; ; c**2 -11.347c +28.8 =0 ; ; p=1; q=-11.347; r=28.8 ; ; D = q**2 - 4pr = 11.347**2 - 4 * 1 * 28.8 = 13.5624621358 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.35 ± sqrt{ 13.56 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 5.67367743 ± 1.84136241244 ; ; c_{1} = 7.51503983841 ; ; c_{2} = 3.83231501353 ; ; ; ; text{ Factored form: } ; ; (c -7.51503983841) (c -3.83231501353) = 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 = 7.2 ; ; b = 4.8 ; ; c = 3.83 ; ;

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

p = a+b+c = 7.2+4.8+3.83 = 15.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.83 }{ 2 } = 7.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.92 * (7.92-7.2)(7.92-4.8)(7.92-3.83) } ; ; T = sqrt{ 72.15 } = 8.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.49 }{ 7.2 } = 2.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.49 }{ 4.8 } = 3.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.49 }{ 3.83 } = 4.43 ; ;

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{ 4.8**2+3.83**2-7.2**2 }{ 2 * 4.8 * 3.83 } ) = 112° 33'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.2**2+3.83**2-4.8**2 }{ 2 * 7.2 * 3.83 } ) = 38° ; ;
 gamma = 180° - alpha - beta = 180° - 112° 33'29" - 38° = 29° 26'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.49 }{ 7.92 } = 1.07 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.2 }{ 2 * sin 112° 33'29" } = 3.9 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 3.83**2 - 7.2**2 } }{ 2 } = 2.43 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.83**2+2 * 7.2**2 - 4.8**2 } }{ 2 } = 5.244 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 7.2**2 - 3.83**2 } }{ 2 } = 5.811 ; ;
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