Triangle calculator SSA

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Triangle has two solutions with side c=119.8855401388 and with side c=42.74991582851

#1 Acute scalene triangle.

Sides: a = 115   b = 90   c = 119.8855401388

Area: T = 4874.377736648
Perimeter: p = 324.8855401388
Semiperimeter: s = 162.4432700694

Angle ∠ A = α = 64.62553966084° = 64°37'31″ = 1.12879259512 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 70.37546033916° = 70°22'29″ = 1.2288268539 rad

Height: ha = 84.77217802866
Height: hb = 108.3199497033
Height: hc = 81.31772798365

Median: ma = 89.05105740181
Median: mb = 108.5076934032
Median: mc = 84.08795613305

Inradius: r = 30.00767491223
Circumradius: R = 63.64396103068

Vertex coordinates: A[119.8855401388; 0] B[0; 0] C[81.31772798365; 81.31772798365]
Centroid: CG[67.06875604081; 27.10657599455]
Coordinates of the circumscribed circle: U[59.94327006939; 21.37545791425]
Coordinates of the inscribed circle: I[72.44327006939; 30.00767491223]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.3754603392° = 115°22'29″ = 1.12879259512 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 109.6255396608° = 109°37'31″ = 1.2288268539 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 115**2 + c**2 -2 * 115 * c * cos (45° ) ; ; ; ; c**2 -162.635c +5125 =0 ; ; p=1; q=-162.635; r=5125 ; ; D = q**2 - 4pr = 162.635**2 - 4 * 1 * 5125 = 5950 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 162.63 ± sqrt{ 5950 } }{ 2 } ; ; c_{1,2} = 81.31727984 ± 38.5681215514 ; ; c_{1} = 119.885401391 ; ;
c_{2} = 42.7491582886 ; ; ; ; text{ Factored form: } ; ; (c -119.885401391) (c -42.7491582886) = 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 = 115 ; ; b = 90 ; ; c = 119.89 ; ;

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

p = a+b+c = 115+90+119.89 = 324.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 324.89 }{ 2 } = 162.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 162.44 * (162.44-115)(162.44-90)(162.44-119.89) } ; ; T = sqrt{ 23759554.71 } = 4874.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4874.38 }{ 115 } = 84.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4874.38 }{ 90 } = 108.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4874.38 }{ 119.89 } = 81.32 ; ;

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{ 90**2+119.89**2-115**2 }{ 2 * 90 * 119.89 } ) = 64° 37'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 115**2+119.89**2-90**2 }{ 2 * 115 * 119.89 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 64° 37'31" - 45° = 70° 22'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4874.38 }{ 162.44 } = 30.01 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 115 }{ 2 * sin 64° 37'31" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 119.89**2 - 115**2 } }{ 2 } = 89.051 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 119.89**2+2 * 115**2 - 90**2 } }{ 2 } = 108.507 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 115**2 - 119.89**2 } }{ 2 } = 84.08 ; ;







#2 Obtuse scalene triangle.

Sides: a = 115   b = 90   c = 42.74991582851

Area: T = 1738.123263352
Perimeter: p = 247.7499158285
Semiperimeter: s = 123.8754579143

Angle ∠ A = α = 115.3754603392° = 115°22'29″ = 2.01436667024 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 19.62553966084° = 19°37'31″ = 0.34325277878 rad

Height: ha = 30.22882197134
Height: hb = 38.62549474116
Height: hc = 81.31772798365

Median: ma = 40.71223478449
Median: mb = 74.17703799845
Median: mc = 101.0232905158

Inradius: r = 14.03113101005
Circumradius: R = 63.64396103068

Vertex coordinates: A[42.74991582851; 0] B[0; 0] C[81.31772798365; 81.31772798365]
Centroid: CG[41.35554793739; 27.10657599455]
Coordinates of the circumscribed circle: U[21.37545791425; 59.94327006939]
Coordinates of the inscribed circle: I[33.87545791425; 14.03113101005]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.62553966084° = 64°37'31″ = 2.01436667024 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 160.3754603392° = 160°22'29″ = 0.34325277878 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 115**2 + c**2 -2 * 115 * c * cos (45° ) ; ; ; ; c**2 -162.635c +5125 =0 ; ; p=1; q=-162.635; r=5125 ; ; D = q**2 - 4pr = 162.635**2 - 4 * 1 * 5125 = 5950 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 162.63 ± sqrt{ 5950 } }{ 2 } ; ; c_{1,2} = 81.31727984 ± 38.5681215514 ; ; c_{1} = 119.885401391 ; ; : Nr. 1
c_{2} = 42.7491582886 ; ; ; ; text{ Factored form: } ; ; (c -119.885401391) (c -42.7491582886) = 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 = 115 ; ; b = 90 ; ; c = 42.75 ; ;

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

p = a+b+c = 115+90+42.75 = 247.75 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 247.75 }{ 2 } = 123.87 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 123.87 * (123.87-115)(123.87-90)(123.87-42.75) } ; ; T = sqrt{ 3021070.29 } = 1738.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1738.12 }{ 115 } = 30.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1738.12 }{ 90 } = 38.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1738.12 }{ 42.75 } = 81.32 ; ;

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{ 90**2+42.75**2-115**2 }{ 2 * 90 * 42.75 } ) = 115° 22'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 115**2+42.75**2-90**2 }{ 2 * 115 * 42.75 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 115° 22'29" - 45° = 19° 37'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1738.12 }{ 123.87 } = 14.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 115 }{ 2 * sin 115° 22'29" } = 63.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 42.75**2 - 115**2 } }{ 2 } = 40.712 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.75**2+2 * 115**2 - 90**2 } }{ 2 } = 74.17 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 115**2 - 42.75**2 } }{ 2 } = 101.023 ; ;
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