Triangle calculator SSA

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Triangle has two solutions with side c=187.8365697156 and with side c=108.153303112

#1 Obtuse scalene triangle.

Sides: a = 162   b = 77   c = 187.8365697156

Area: T = 6188.373253379
Perimeter: p = 426.8365697156
Semiperimeter: s = 213.4187848578

Angle ∠ A = α = 58.84106572994° = 58°50'26″ = 1.02769632039 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 97.15993427006° = 97°9'34″ = 1.69657504292 rad

Height: ha = 76.4399660911
Height: hb = 160.737694893
Height: hc = 65.89113361783

Median: ma = 118.511001883
Median: mb = 171.1176552569
Median: mc = 85.2440469957

Inradius: r = 28.99765088442
Circumradius: R = 94.65658434196

Vertex coordinates: A[187.8365697156; 0] B[0; 0] C[147.9944364138; 65.89113361783]
Centroid: CG[111.9433353765; 21.96437787261]
Coordinates of the circumscribed circle: U[93.9187848578; -11.79768814506]
Coordinates of the inscribed circle: I[136.4187848578; 28.99765088442]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.1599342701° = 121°9'34″ = 1.02769632039 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 82.84106572994° = 82°50'26″ = 1.69657504292 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 162 ; ; b = 77 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 77**2 = 162**2 + c**2 -2 * 162 * c * cos (24° ) ; ; ; ; c**2 -295.989c +20315 =0 ; ; p=1; q=-295.989; r=20315 ; ; D = q**2 - 4pr = 295.989**2 - 4 * 1 * 20315 = 6349.32726656 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 295.99 ± sqrt{ 6349.33 } }{ 2 } ; ; c_{1,2} = 147.99436414 ± 39.8413330179 ; ; c_{1} = 187.835697158 ; ; c_{2} = 108.153031122 ; ; ; ; text{ Factored form: } ; ; (c -187.835697158) (c -108.153031122) = 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 = 162 ; ; b = 77 ; ; c = 187.84 ; ;

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

p = a+b+c = 162+77+187.84 = 426.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 426.84 }{ 2 } = 213.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 213.42 * (213.42-162)(213.42-77)(213.42-187.84) } ; ; T = sqrt{ 38295954.62 } = 6188.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6188.37 }{ 162 } = 76.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6188.37 }{ 77 } = 160.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6188.37 }{ 187.84 } = 65.89 ; ;

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{ 77**2+187.84**2-162**2 }{ 2 * 77 * 187.84 } ) = 58° 50'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 162**2+187.84**2-77**2 }{ 2 * 162 * 187.84 } ) = 24° ; ; gamma = 180° - alpha - beta = 180° - 58° 50'26" - 24° = 97° 9'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6188.37 }{ 213.42 } = 29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 162 }{ 2 * sin 58° 50'26" } = 94.66 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 187.84**2 - 162**2 } }{ 2 } = 118.51 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 187.84**2+2 * 162**2 - 77**2 } }{ 2 } = 171.117 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 162**2 - 187.84**2 } }{ 2 } = 85.24 ; ;







#2 Obtuse scalene triangle.

Sides: a = 162   b = 77   c = 108.153303112

Area: T = 3563.174386612
Perimeter: p = 347.153303112
Semiperimeter: s = 173.577651556

Angle ∠ A = α = 121.1599342701° = 121°9'34″ = 2.11546294497 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 34.84106572994° = 34°50'26″ = 0.60880841834 rad

Height: ha = 43.99898008163
Height: hb = 92.55499705486
Height: hc = 65.89113361783

Median: ma = 47.45656537227
Median: mb = 132.2433294992
Median: mc = 114.7276764379

Inradius: r = 20.52879720855
Circumradius: R = 94.65658434196

Vertex coordinates: A[108.153303112; 0] B[0; 0] C[147.9944364138; 65.89113361783]
Centroid: CG[85.38224650861; 21.96437787261]
Coordinates of the circumscribed circle: U[54.07765155601; 77.68882176289]
Coordinates of the inscribed circle: I[96.57765155601; 20.52879720855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.84106572994° = 58°50'26″ = 2.11546294497 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 145.1599342701° = 145°9'34″ = 0.60880841834 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 162 ; ; b = 77 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 77**2 = 162**2 + c**2 -2 * 162 * c * cos (24° ) ; ; ; ; c**2 -295.989c +20315 =0 ; ; p=1; q=-295.989; r=20315 ; ; D = q**2 - 4pr = 295.989**2 - 4 * 1 * 20315 = 6349.32726656 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 295.99 ± sqrt{ 6349.33 } }{ 2 } ; ; c_{1,2} = 147.99436414 ± 39.8413330179 ; ; c_{1} = 187.835697158 ; ; c_{2} = 108.153031122 ; ; ; ; text{ Factored form: } ; ; (c -187.835697158) (c -108.153031122) = 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 = 162 ; ; b = 77 ; ; c = 108.15 ; ;

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

p = a+b+c = 162+77+108.15 = 347.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 347.15 }{ 2 } = 173.58 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 173.58 * (173.58-162)(173.58-77)(173.58-108.15) } ; ; T = sqrt{ 12696208 } = 3563.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3563.17 }{ 162 } = 43.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3563.17 }{ 77 } = 92.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3563.17 }{ 108.15 } = 65.89 ; ;

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{ 77**2+108.15**2-162**2 }{ 2 * 77 * 108.15 } ) = 121° 9'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 162**2+108.15**2-77**2 }{ 2 * 162 * 108.15 } ) = 24° ; ; gamma = 180° - alpha - beta = 180° - 121° 9'34" - 24° = 34° 50'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3563.17 }{ 173.58 } = 20.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 162 }{ 2 * sin 121° 9'34" } = 94.66 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 108.15**2 - 162**2 } }{ 2 } = 47.456 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 108.15**2+2 * 162**2 - 77**2 } }{ 2 } = 132.243 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 162**2 - 108.15**2 } }{ 2 } = 114.727 ; ;
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