Triangle calculator SSA

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Triangle has two solutions with side c=75.14661571851 and with side c=9.42216394626

#1 Acute scalene triangle.

Sides: a = 62   b = 56   c = 75.14661571851

Area: T = 1703.711102681
Perimeter: p = 193.1466157185
Semiperimeter: s = 96.57330785926

Angle ∠ A = α = 54.06879028291° = 54°4'4″ = 0.9443662924 rad
Angle ∠ B = β = 47° = 0.82203047484 rad
Angle ∠ C = γ = 78.93220971709° = 78°55'56″ = 1.37876249811 rad

Height: ha = 54.95884202198
Height: hb = 60.84768223862
Height: hc = 45.34439295004

Median: ma = 58.57702353576
Median: mb = 62.94402293438
Median: mc = 45.58879782956

Inradius: r = 17.64216766623
Circumradius: R = 38.28551689108

Vertex coordinates: A[75.14661571851; 0] B[0; 0] C[42.28438983239; 45.34439295004]
Centroid: CG[39.14333518363; 15.11546431668]
Coordinates of the circumscribed circle: U[37.57330785926; 7.35496886739]
Coordinates of the inscribed circle: I[40.57330785926; 17.64216766623]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.9322097171° = 125°55'56″ = 0.9443662924 rad
∠ B' = β' = 133° = 0.82203047484 rad
∠ C' = γ' = 101.0687902829° = 101°4'4″ = 1.37876249811 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 56 ; ; beta = 47° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 62**2 + c**2 -2 * 62 * c * cos (47° ) ; ; ; ; c**2 -84.568c +708 =0 ; ; p=1; q=-84.568; r=708 ; ; D = q**2 - 4pr = 84.568**2 - 4 * 1 * 708 = 4319.71222986 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 84.57 ± sqrt{ 4319.71 } }{ 2 } ; ; c_{1,2} = 42.28389832 ± 32.8622588612 ; ; c_{1} = 75.1461571812 ; ;
c_{2} = 9.42163945875 ; ; ; ; text{ Factored form: } ; ; (c -75.1461571812) (c -9.42163945875) = 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 = 62 ; ; b = 56 ; ; c = 75.15 ; ;

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

p = a+b+c = 62+56+75.15 = 193.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 193.15 }{ 2 } = 96.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 96.57 * (96.57-62)(96.57-56)(96.57-75.15) } ; ; T = sqrt{ 2902631.26 } = 1703.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1703.71 }{ 62 } = 54.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1703.71 }{ 56 } = 60.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1703.71 }{ 75.15 } = 45.34 ; ;

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{ 56**2+75.15**2-62**2 }{ 2 * 56 * 75.15 } ) = 54° 4'4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+75.15**2-56**2 }{ 2 * 62 * 75.15 } ) = 47° ; ; gamma = 180° - alpha - beta = 180° - 54° 4'4" - 47° = 78° 55'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1703.71 }{ 96.57 } = 17.64 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 54° 4'4" } = 38.29 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 75.15**2 - 62**2 } }{ 2 } = 58.57 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 75.15**2+2 * 62**2 - 56**2 } }{ 2 } = 62.94 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 62**2 - 75.15**2 } }{ 2 } = 45.588 ; ;







#2 Obtuse scalene triangle.

Sides: a = 62   b = 56   c = 9.42216394626

Area: T = 213.6077077786
Perimeter: p = 127.4221639463
Semiperimeter: s = 63.71108197313

Angle ∠ A = α = 125.9322097171° = 125°55'56″ = 2.19879297296 rad
Angle ∠ B = β = 47° = 0.82203047484 rad
Angle ∠ C = γ = 7.06879028291° = 7°4'4″ = 0.12333581756 rad

Height: ha = 6.89105508963
Height: hb = 7.62988242066
Height: hc = 45.34439295004

Median: ma = 25.52222186552
Median: mb = 34.38658058664
Median: mc = 58.88880987761

Inradius: r = 3.35327598403
Circumradius: R = 38.28551689108

Vertex coordinates: A[9.42216394626; 0] B[0; 0] C[42.28438983239; 45.34439295004]
Centroid: CG[17.23551792622; 15.11546431668]
Coordinates of the circumscribed circle: U[4.71108197313; 37.99442408265]
Coordinates of the inscribed circle: I[7.71108197313; 3.35327598403]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 54.06879028291° = 54°4'4″ = 2.19879297296 rad
∠ B' = β' = 133° = 0.82203047484 rad
∠ C' = γ' = 172.9322097171° = 172°55'56″ = 0.12333581756 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 56 ; ; beta = 47° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 62**2 + c**2 -2 * 62 * c * cos (47° ) ; ; ; ; c**2 -84.568c +708 =0 ; ; p=1; q=-84.568; r=708 ; ; D = q**2 - 4pr = 84.568**2 - 4 * 1 * 708 = 4319.71222986 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 84.57 ± sqrt{ 4319.71 } }{ 2 } ; ; c_{1,2} = 42.28389832 ± 32.8622588612 ; ; c_{1} = 75.1461571812 ; ; : Nr. 1
c_{2} = 9.42163945875 ; ; ; ; text{ Factored form: } ; ; (c -75.1461571812) (c -9.42163945875) = 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 = 62 ; ; b = 56 ; ; c = 9.42 ; ;

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

p = a+b+c = 62+56+9.42 = 127.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.42 }{ 2 } = 63.71 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.71 * (63.71-62)(63.71-56)(63.71-9.42) } ; ; T = sqrt{ 45627.98 } = 213.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 213.61 }{ 62 } = 6.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 213.61 }{ 56 } = 7.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 213.61 }{ 9.42 } = 45.34 ; ;

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{ 56**2+9.42**2-62**2 }{ 2 * 56 * 9.42 } ) = 125° 55'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+9.42**2-56**2 }{ 2 * 62 * 9.42 } ) = 47° ; ; gamma = 180° - alpha - beta = 180° - 125° 55'56" - 47° = 7° 4'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 213.61 }{ 63.71 } = 3.35 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 125° 55'56" } = 38.29 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 9.42**2 - 62**2 } }{ 2 } = 25.522 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.42**2+2 * 62**2 - 56**2 } }{ 2 } = 34.386 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 62**2 - 9.42**2 } }{ 2 } = 58.888 ; ;
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