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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 77**2 = 162**2 + c**2 -2 * 77 * c * cos (24° ) ; ; ; ; c**2 -295.989c +20315 =0 ; ; p=1; q=-295.988728276; 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} = 147.994364138 ± 39.8413330179 ; ;
c_{1} = 187.835697156 ; ; c_{2} = 108.15303112 ; ; ; ; (c -187.835697156) (c -108.15303112) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 162**2-77**2-108.15**2 }{ 2 * 77 * 108.15 } ) = 121° 9'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 77**2-162**2-108.15**2 }{ 2 * 162 * 108.15 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 108.15**2-162**2-77**2 }{ 2 * 77 * 162 } ) = 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 ; ;




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