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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 115 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 115**2 + c**2 -2 * 90 * c * cos (45° ) ; ; ; ; c**2 -162.635c +5125 =0 ; ; p=1; q=-162.634559673; 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} = 81.3172798365 ± 38.5681215514 ; ; c_{1} = 119.885401388 ; ;
c_{2} = 42.7491582851 ; ; ; ; (c -119.885401388) (c -42.7491582851) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 115**2-90**2-42.75**2 }{ 2 * 90 * 42.75 } ) = 115° 22'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-115**2-42.75**2 }{ 2 * 115 * 42.75 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.75**2-115**2-90**2 }{ 2 * 90 * 115 } ) = 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 ; ;




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