Triangle calculator SSA

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Triangle has two solutions with side c=60.8211191124 and with side c=42.02548264259

#1 Acute scalene triangle.

Sides: a = 80   b = 62   c = 60.8211191124

Area: T = 1863.669941938
Perimeter: p = 202.8211191124
Semiperimeter: s = 101.4110595562

Angle ∠ A = α = 81.28112911402° = 81°16'53″ = 1.41986261507 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 48.71987088598° = 48°43'7″ = 0.85503018769 rad

Height: ha = 46.59217354844
Height: hb = 60.1188368367
Height: hc = 61.28435554495

Median: ma = 46.60105219377
Median: mb = 63.94222289639
Median: mc = 64.7865767554

Inradius: r = 18.3777462523
Circumradius: R = 40.46876259693

Vertex coordinates: A[60.8211191124; 0] B[0; 0] C[51.42330087749; 61.28435554495]
Centroid: CG[37.41547332996; 20.42878518165]
Coordinates of the circumscribed circle: U[30.4110595562; 26.69987720534]
Coordinates of the inscribed circle: I[39.4110595562; 18.3777462523]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.71987088598° = 98°43'7″ = 1.41986261507 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 131.281129114° = 131°16'53″ = 0.85503018769 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 80 ; ; b = 62 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 80**2 + c**2 -2 * 80 * c * cos (50° ) ; ; ; ; c**2 -102.846c +2556 =0 ; ; p=1; q=-102.846; r=2556 ; ; D = q**2 - 4pr = 102.846**2 - 4 * 1 * 2556 = 353.303325863 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 102.85 ± sqrt{ 353.3 } }{ 2 } ; ; c_{1,2} = 51.42300877 ± 9.39818234904 ; ; c_{1} = 60.821191119 ; ;
c_{2} = 42.024826421 ; ; ; ; text{ Factored form: } ; ; (c -60.821191119) (c -42.024826421) = 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 = 80 ; ; b = 62 ; ; c = 60.82 ; ;

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

p = a+b+c = 80+62+60.82 = 202.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 202.82 }{ 2 } = 101.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 101.41 * (101.41-80)(101.41-62)(101.41-60.82) } ; ; T = sqrt{ 3473263.7 } = 1863.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1863.67 }{ 80 } = 46.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1863.67 }{ 62 } = 60.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1863.67 }{ 60.82 } = 61.28 ; ;

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{ 62**2+60.82**2-80**2 }{ 2 * 62 * 60.82 } ) = 81° 16'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80**2+60.82**2-62**2 }{ 2 * 80 * 60.82 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 81° 16'53" - 50° = 48° 43'7" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1863.67 }{ 101.41 } = 18.38 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 81° 16'53" } = 40.47 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 60.82**2 - 80**2 } }{ 2 } = 46.601 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60.82**2+2 * 80**2 - 62**2 } }{ 2 } = 63.942 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 80**2 - 60.82**2 } }{ 2 } = 64.786 ; ;







#2 Obtuse scalene triangle.

Sides: a = 80   b = 62   c = 42.02548264259

Area: T = 1287.715539026
Perimeter: p = 184.0254826426
Semiperimeter: s = 92.01224132129

Angle ∠ A = α = 98.71987088598° = 98°43'7″ = 1.72329665029 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 31.28112911402° = 31°16'53″ = 0.54659615247 rad

Height: ha = 32.19328847566
Height: hb = 41.53992061375
Height: hc = 61.28435554495

Median: ma = 34.71437295326
Median: mb = 55.87552451275
Median: mc = 68.41440226194

Inradius: r = 13.99550181209
Circumradius: R = 40.46876259693

Vertex coordinates: A[42.02548264259; 0] B[0; 0] C[51.42330087749; 61.28435554495]
Centroid: CG[31.14992784003; 20.42878518165]
Coordinates of the circumscribed circle: U[21.01224132129; 34.58547833962]
Coordinates of the inscribed circle: I[30.01224132129; 13.99550181209]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 81.28112911402° = 81°16'53″ = 1.72329665029 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 148.719870886° = 148°43'7″ = 0.54659615247 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 80 ; ; b = 62 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 80**2 + c**2 -2 * 80 * c * cos (50° ) ; ; ; ; c**2 -102.846c +2556 =0 ; ; p=1; q=-102.846; r=2556 ; ; D = q**2 - 4pr = 102.846**2 - 4 * 1 * 2556 = 353.303325863 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 102.85 ± sqrt{ 353.3 } }{ 2 } ; ; c_{1,2} = 51.42300877 ± 9.39818234904 ; ; c_{1} = 60.821191119 ; ; : Nr. 1
c_{2} = 42.024826421 ; ; ; ; text{ Factored form: } ; ; (c -60.821191119) (c -42.024826421) = 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 = 80 ; ; b = 62 ; ; c = 42.02 ; ;

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

p = a+b+c = 80+62+42.02 = 184.02 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 184.02 }{ 2 } = 92.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 92.01 * (92.01-80)(92.01-62)(92.01-42.02) } ; ; T = sqrt{ 1658210.93 } = 1287.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1287.72 }{ 80 } = 32.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1287.72 }{ 62 } = 41.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1287.72 }{ 42.02 } = 61.28 ; ;

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{ 62**2+42.02**2-80**2 }{ 2 * 62 * 42.02 } ) = 98° 43'7" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80**2+42.02**2-62**2 }{ 2 * 80 * 42.02 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 98° 43'7" - 50° = 31° 16'53" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1287.72 }{ 92.01 } = 14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 98° 43'7" } = 40.47 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 42.02**2 - 80**2 } }{ 2 } = 34.714 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.02**2+2 * 80**2 - 62**2 } }{ 2 } = 55.875 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 80**2 - 42.02**2 } }{ 2 } = 68.414 ; ;
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