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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 16.4**2 = 20.16**2 + c**2 -2 * 16.4 * c * cos (24° ) ; ; ; ; c**2 -36.834c +137.466 =0 ; ; p=1; q=-36.8341528521; r=137.4656 ; ; 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} = 18.4170764261 ± 14.2029258987 ; ;
c_{1} = 32.6200023248 ; ; c_{2} = 4.21415052738 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 20.16**2-16.4**2-4.21**2 }{ 2 * 16.4 * 4.21 } ) = 150° 3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.4**2-20.16**2-4.21**2 }{ 2 * 20.16 * 4.21 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.21**2-20.16**2-16.4**2 }{ 2 * 16.4 * 20.16 } ) = 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 ; ;




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