Triangle calculator SSA

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Triangle has two solutions with side c=69.82553913618 and with side c=30.63435554772

#1 Obtuse scalene triangle.

Sides: a = 58   b = 35   c = 69.82553913618

Area: T = 1012.468817475
Perimeter: p = 162.8255391362
Semiperimeter: s = 81.41326956809

Angle ∠ A = α = 55.952226763° = 55°57'8″ = 0.97765512941 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 94.048773237° = 94°2'52″ = 1.64114425839 rad

Height: ha = 34.91326956809
Height: hb = 57.85553242712
Height: hc = 29

Median: ma = 47.00331130821
Median: mb = 61.75438876461
Median: mc = 32.79663973676

Inradius: r = 12.43662443262
Circumradius: R = 35

Vertex coordinates: A[69.82553913618; 0] B[0; 0] C[50.22994734195; 29]
Centroid: CG[40.01882882604; 9.66766666667]
Coordinates of the circumscribed circle: U[34.91326956809; -2.47105627485]
Coordinates of the inscribed circle: I[46.41326956809; 12.43662443262]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.048773237° = 124°2'52″ = 0.97765512941 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 85.952226763° = 85°57'8″ = 1.64114425839 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 58 ; ; b = 35 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35**2 = 58**2 + c**2 -2 * 58 * c * cos (30° ) ; ; ; ; c**2 -100.459c +2139 =0 ; ; p=1; q=-100.459; r=2139 ; ; D = q**2 - 4pr = 100.459**2 - 4 * 1 * 2139 = 1536 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 100.46 ± sqrt{ 1536 } }{ 2 } ; ; c_{1,2} = 50.22947342 ± 19.5959179423 ; ; c_{1} = 69.8253913623 ; ;
c_{2} = 30.6335554777 ; ; ; ; (c -69.8253913623) (c -30.6335554777) = 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 = 58 ; ; b = 35 ; ; c = 69.83 ; ;

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

p = a+b+c = 58+35+69.83 = 162.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 162.83 }{ 2 } = 81.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.41 * (81.41-58)(81.41-35)(81.41-69.83) } ; ; T = sqrt{ 1025091.8 } = 1012.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1012.47 }{ 58 } = 34.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1012.47 }{ 35 } = 57.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1012.47 }{ 69.83 } = 29 ; ;

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{ 58**2-35**2-69.83**2 }{ 2 * 35 * 69.83 } ) = 55° 57'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-58**2-69.83**2 }{ 2 * 58 * 69.83 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 69.83**2-58**2-35**2 }{ 2 * 35 * 58 } ) = 94° 2'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1012.47 }{ 81.41 } = 12.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 58 }{ 2 * sin 55° 57'8" } = 35 ; ;





#2 Obtuse scalene triangle.

Sides: a = 58   b = 35   c = 30.63435554772

Area: T = 444.187655442
Perimeter: p = 123.6343555477
Semiperimeter: s = 61.81767777386

Angle ∠ A = α = 124.048773237° = 124°2'52″ = 2.16550413595 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 25.952226763° = 25°57'8″ = 0.45329525185 rad

Height: ha = 15.31767777386
Height: hb = 25.3822088824
Height: hc = 29

Median: ma = 15.51547465525
Median: mb = 42.95329668427
Median: mc = 45.38660806824

Inradius: r = 7.18655339387
Circumradius: R = 35

Vertex coordinates: A[30.63435554772; 0] B[0; 0] C[50.22994734195; 29]
Centroid: CG[26.95443429656; 9.66766666667]
Coordinates of the circumscribed circle: U[15.31767777386; 31.47105627485]
Coordinates of the inscribed circle: I[26.81767777386; 7.18655339387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 55.952226763° = 55°57'8″ = 2.16550413595 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 154.048773237° = 154°2'52″ = 0.45329525185 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 58 ; ; b = 35 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35**2 = 58**2 + c**2 -2 * 58 * c * cos (30° ) ; ; ; ; c**2 -100.459c +2139 =0 ; ; p=1; q=-100.459; r=2139 ; ; D = q**2 - 4pr = 100.459**2 - 4 * 1 * 2139 = 1536 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 100.46 ± sqrt{ 1536 } }{ 2 } ; ; c_{1,2} = 50.22947342 ± 19.5959179423 ; ; c_{1} = 69.8253913623 ; ; : Nr. 1
c_{2} = 30.6335554777 ; ; ; ; (c -69.8253913623) (c -30.6335554777) = 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 = 58 ; ; b = 35 ; ; c = 30.63 ; ;

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

p = a+b+c = 58+35+30.63 = 123.63 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 123.63 }{ 2 } = 61.82 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 61.82 * (61.82-58)(61.82-35)(61.82-30.63) } ; ; T = sqrt{ 197301.7 } = 444.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 444.19 }{ 58 } = 15.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 444.19 }{ 35 } = 25.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 444.19 }{ 30.63 } = 29 ; ;

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{ 58**2-35**2-30.63**2 }{ 2 * 35 * 30.63 } ) = 124° 2'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-58**2-30.63**2 }{ 2 * 58 * 30.63 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30.63**2-58**2-35**2 }{ 2 * 35 * 58 } ) = 25° 57'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 444.19 }{ 61.82 } = 7.19 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 58 }{ 2 * sin 124° 2'52" } = 35 ; ;




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