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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 56**2 = 62**2 + c**2 -2 * 56 * c * cos (47° ) ; ; ; ; c**2 -84.568c +708 =0 ; ; p=1; q=-84.5677966477; 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} = 42.2838983239 ± 32.8622588612 ; ; c_{1} = 75.1461571851 ; ;
c_{2} = 9.42163946263 ; ; ; ; (c -75.1461571851) (c -9.42163946263) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 62**2-56**2-9.42**2 }{ 2 * 56 * 9.42 } ) = 125° 55'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-62**2-9.42**2 }{ 2 * 62 * 9.42 } ) = 47° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.42**2-62**2-56**2 }{ 2 * 56 * 62 } ) = 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 ; ;




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