Triangle calculator SSA

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Triangle has two solutions with side c=32.62200023248 and with side c=4.21441505274

#1 Obtuse scalene triangle.

Sides: a = 20.16   b = 16.4   c = 32.62200023248

Area: T = 133.7398922446
Perimeter: p = 69.18800023248
Semiperimeter: s = 34.59900011624

Angle ∠ A = α = 29.99992364436° = 29°59'57″ = 0.5243585449 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 126.0010763556° = 126°3″ = 2.19991281841 rad

Height: ha = 13.26877502427
Height: hb = 16.31096246886
Height: hc = 8.21998107244

Median: ma = 23.76877486488
Median: mb = 25.84657941614
Median: mc = 8.46662070659

Inradius: r = 3.86664041039
Circumradius: R = 20.16604653517

Vertex coordinates: A[32.62200023248; 0] B[0; 0] C[18.41770764261; 8.21998107244]
Centroid: CG[17.01223595836; 2.73332702415]
Coordinates of the circumscribed circle: U[16.31100011624; -11.85502415706]
Coordinates of the inscribed circle: I[18.19900011624; 3.86664041039]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0010763556° = 150°3″ = 0.5243585449 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 53.99992364436° = 53°59'57″ = 2.19991281841 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 20.16 ; ; b = 16.4 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.4**2 = 20.16**2 + c**2 -2 * 20.16 * c * cos (24° ) ; ; ; ; c**2 -36.834c +137.466 =0 ; ; p=1; q=-36.834; r=137.466 ; ; D = q**2 - 4pr = 36.834**2 - 4 * 1 * 137.466 = 806.892416336 ; ; D>0 ; ;
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.83 ± sqrt{ 806.89 } }{ 2 } ; ; c_{1,2} = 18.41707643 ± 14.2029258987 ; ; c_{1} = 32.6200023248 ; ; c_{2} = 4.21415052738 ; ; ; ; text{ Factored form: } ; ; (c -32.6200023248) (c -4.21415052738) = 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 = 20.16 ; ; b = 16.4 ; ; c = 32.62 ; ;

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

p = a+b+c = 20.16+16.4+32.62 = 69.18 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69.18 }{ 2 } = 34.59 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.59 * (34.59-20.16)(34.59-16.4)(34.59-32.62) } ; ; T = sqrt{ 17886.1 } = 133.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.74 }{ 20.16 } = 13.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.74 }{ 16.4 } = 16.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.74 }{ 32.62 } = 8.2 ; ;

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{ 16.4**2+32.62**2-20.16**2 }{ 2 * 16.4 * 32.62 } ) = 29° 59'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.16**2+32.62**2-16.4**2 }{ 2 * 20.16 * 32.62 } ) = 24° ; ;
 gamma = 180° - alpha - beta = 180° - 29° 59'57" - 24° = 126° 3" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.74 }{ 34.59 } = 3.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20.16 }{ 2 * sin 29° 59'57" } = 20.16 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.4**2+2 * 32.62**2 - 20.16**2 } }{ 2 } = 23.768 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.62**2+2 * 20.16**2 - 16.4**2 } }{ 2 } = 25.846 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.4**2+2 * 20.16**2 - 32.62**2 } }{ 2 } = 8.466 ; ;



#2 Obtuse scalene triangle.

Sides: a = 20.16   b = 16.4   c = 4.21441505274

Area: T = 17.27876183443
Perimeter: p = 40.77441505274
Semiperimeter: s = 20.38770752637

Angle ∠ A = α = 150.0010763556° = 150°3″ = 2.61880072046 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 5.99992364437° = 5°59'57″ = 0.10547064285 rad

Height: ha = 1.71440494389
Height: hb = 2.10770266274
Height: hc = 8.21998107244

Median: ma = 6.46216663744
Median: mb = 12.03554614508
Median: mc = 18.25552193587

Inradius: r = 0.84774790092
Circumradius: R = 20.16604653517

Vertex coordinates: A[4.21441505274; 0] B[0; 0] C[18.41770764261; 8.21998107244]
Centroid: CG[7.54437423178; 2.73332702415]
Coordinates of the circumscribed circle: U[2.10770752637; 20.0550052295]
Coordinates of the inscribed circle: I[3.98770752637; 0.84774790092]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99992364436° = 29°59'57″ = 2.61880072046 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 174.0010763556° = 174°3″ = 0.10547064285 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 20.16 ; ; b = 16.4 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.4**2 = 20.16**2 + c**2 -2 * 20.16 * c * cos (24° ) ; ; ; ; c**2 -36.834c +137.466 =0 ; ; p=1; q=-36.834; r=137.466 ; ; D = q**2 - 4pr = 36.834**2 - 4 * 1 * 137.466 = 806.892416336 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.83 ± sqrt{ 806.89 } }{ 2 } ; ; c_{1,2} = 18.41707643 ± 14.2029258987 ; ; c_{1} = 32.6200023248 ; ; c_{2} = 4.21415052738 ; ; ; ; text{ Factored form: } ; ; (c -32.6200023248) (c -4.21415052738) = 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 = 20.16 ; ; b = 16.4 ; ; c = 4.21 ; ;

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

p = a+b+c = 20.16+16.4+4.21 = 40.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.77 }{ 2 } = 20.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.39 * (20.39-20.16)(20.39-16.4)(20.39-4.21) } ; ; T = sqrt{ 298.52 } = 17.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.28 }{ 20.16 } = 1.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.28 }{ 16.4 } = 2.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.28 }{ 4.21 } = 8.2 ; ;

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{ 16.4**2+4.21**2-20.16**2 }{ 2 * 16.4 * 4.21 } ) = 150° 3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.16**2+4.21**2-16.4**2 }{ 2 * 20.16 * 4.21 } ) = 24° ; ;
 gamma = 180° - alpha - beta = 180° - 150° 3" - 24° = 5° 59'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.28 }{ 20.39 } = 0.85 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20.16 }{ 2 * sin 150° 3" } = 20.16 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.4**2+2 * 4.21**2 - 20.16**2 } }{ 2 } = 6.462 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.21**2+2 * 20.16**2 - 16.4**2 } }{ 2 } = 12.035 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.4**2+2 * 20.16**2 - 4.21**2 } }{ 2 } = 18.255 ; ;
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