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?

1. Use Law of Cosines

a = 113 ; ; b = 72 ; ; beta = 34° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 72**2 = 113**2 + c**2 -2 * 113 * c * cos (34° ) ; ; ; ; c**2 -187.362c +7585 =0 ; ; p=1; q=-187.362; 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} = 93.6812457 ± 34.5134147204 ; ; c_{1} = 128.19466042 ; ; c_{2} = 59.1678309796 ; ; ; ; text{ Factored form: } ; ; (c -128.19466042) (c -59.1678309796) = 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 = 128.19 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 72**2+128.19**2-113**2 }{ 2 * 72 * 128.19 } ) = 61° 21'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 113**2+128.19**2-72**2 }{ 2 * 113 * 128.19 } ) = 34° ; ; gamma = 180° - alpha - beta = 180° - 61° 21'25" - 34° = 84° 38'35" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 128.19**2 - 113**2 } }{ 2 } = 87.274 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 128.19**2+2 * 113**2 - 72**2 } }{ 2 } = 115.349 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 113**2 - 128.19**2 } }{ 2 } = 69.771 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 113 ; ; b = 72 ; ; beta = 34° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 72**2 = 113**2 + c**2 -2 * 113 * c * cos (34° ) ; ; ; ; c**2 -187.362c +7585 =0 ; ; p=1; q=-187.362; 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} = 93.6812457 ± 34.5134147204 ; ; c_{1} = 128.19466042 ; ; c_{2} = 59.1678309796 ; ; ; ; text{ Factored form: } ; ; (c -128.19466042) (c -59.1678309796) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 72**2+59.17**2-113**2 }{ 2 * 72 * 59.17 } ) = 118° 38'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 113**2+59.17**2-72**2 }{ 2 * 113 * 59.17 } ) = 34° ; ; gamma = 180° - alpha - beta = 180° - 118° 38'35" - 34° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 59.17**2 - 113**2 } }{ 2 } = 33.914 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.17**2+2 * 113**2 - 72**2 } }{ 2 } = 82.698 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 113**2 - 59.17**2 } }{ 2 } = 90.007 ; ;
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