Triangle calculator SSA

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Triangle has two solutions with side c=128.1954660419 and with side c=59.16878309783

#1 Acute scalene triangle.

Sides: a = 113   b = 72   c = 128.1954660419

Area: T = 4050.233325686
Perimeter: p = 313.1954660419
Semiperimeter: s = 156.597733021

Angle ∠ A = α = 61.35768472112° = 61°21'25″ = 1.07108790025 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 84.64331527888° = 84°38'35″ = 1.47773017054 rad

Height: ha = 71.68655443692
Height: hb = 112.5066479357
Height: hc = 63.18987980922

Median: ma = 87.27436241942
Median: mb = 115.3499189334
Median: mc = 69.77112853544

Inradius: r = 25.86439994146
Circumradius: R = 64.3788499399

Vertex coordinates: A[128.1954660419; 0] B[0; 0] C[93.68112456987; 63.18987980922]
Centroid: CG[73.95986353726; 21.06329326974]
Coordinates of the circumscribed circle: U[64.09773302096; 6.01102782688]
Coordinates of the inscribed circle: I[84.59773302096; 25.86439994146]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.6433152789° = 118°38'35″ = 1.07108790025 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 95.35768472112° = 95°21'25″ = 1.47773017054 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 = 113 ; ; b = 72 ; ; c = 128.19 ; ;

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

p = a+b+c = 113+72+128.19 = 313.19 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 313.19 }{ 2 } = 156.6 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 156.6 * (156.6-113)(156.6-72)(156.6-128.19) } ; ; T = sqrt{ 16404389.43 } = 4050.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4050.23 }{ 113 } = 71.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4050.23 }{ 72 } = 112.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4050.23 }{ 128.19 } = 63.19 ; ;

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{ 113**2-72**2-128.19**2 }{ 2 * 72 * 128.19 } ) = 61° 21'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 72**2-113**2-128.19**2 }{ 2 * 113 * 128.19 } ) = 34° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 128.19**2-113**2-72**2 }{ 2 * 72 * 113 } ) = 84° 38'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4050.23 }{ 156.6 } = 25.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 113 }{ 2 * sin 61° 21'25" } = 64.38 ; ;





#2 Obtuse scalene triangle.

Sides: a = 113   b = 72   c = 59.16878309783

Area: T = 1869.372206262
Perimeter: p = 244.1687830978
Semiperimeter: s = 122.0843915489

Angle ∠ A = α = 118.6433152789° = 118°38'35″ = 2.07107136511 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 27.35768472112° = 27°21'25″ = 0.47774670568 rad

Height: ha = 33.08662311968
Height: hb = 51.92770017395
Height: hc = 63.18987980922

Median: ma = 33.91440990053
Median: mb = 82.69877394572
Median: mc = 90.00771771823

Inradius: r = 15.31221896126
Circumradius: R = 64.3788499399

Vertex coordinates: A[59.16878309783; 0] B[0; 0] C[93.68112456987; 63.18987980922]
Centroid: CG[50.95496922257; 21.06329326974]
Coordinates of the circumscribed circle: U[29.58439154891; 57.17985198234]
Coordinates of the inscribed circle: I[50.08439154891; 15.31221896126]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61.35768472112° = 61°21'25″ = 2.07107136511 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 152.6433152789° = 152°38'35″ = 0.47774670568 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 113 ; ; b = 72 ; ; beta = 34° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 72**2 = 113**2 + c**2 -2 * 72 * c * cos (34° ) ; ; ; ; c**2 -187.362c +7585 =0 ; ; p=1; q=-187.362491397; r=7585 ; ; D = q**2 - 4pr = 187.362**2 - 4 * 1 * 7585 = 4764.70318266 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 187.36 ± sqrt{ 4764.7 } }{ 2 } ; ; c_{1,2} = 93.6812456987 ± 34.5134147204 ; ; c_{1} = 128.194660419 ; ;
c_{2} = 59.1678309783 ; ; ; ; (c -128.194660419) (c -59.1678309783) = 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 = 113 ; ; b = 72 ; ; c = 59.17 ; ;

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

p = a+b+c = 113+72+59.17 = 244.17 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 244.17 }{ 2 } = 122.08 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 122.08 * (122.08-113)(122.08-72)(122.08-59.17) } ; ; T = sqrt{ 3494551.91 } = 1869.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1869.37 }{ 113 } = 33.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1869.37 }{ 72 } = 51.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1869.37 }{ 59.17 } = 63.19 ; ;

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{ 113**2-72**2-59.17**2 }{ 2 * 72 * 59.17 } ) = 118° 38'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 72**2-113**2-59.17**2 }{ 2 * 113 * 59.17 } ) = 34° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 59.17**2-113**2-72**2 }{ 2 * 72 * 113 } ) = 27° 21'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1869.37 }{ 122.08 } = 15.31 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 113 }{ 2 * sin 118° 38'35" } = 64.38 ; ;




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