Triangle calculator SSA

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Triangle has two solutions with side c=29.05548409391 and with side c=13.78768109772

#1 Acute scalene triangle.

Sides: a = 26.15   b = 16.83   c = 29.05548409391

Area: T = 217.8977125529
Perimeter: p = 72.03548409391
Semiperimeter: s = 36.01774204696

Angle ∠ A = α = 63.0255387041° = 63°1'31″ = 1.11000005162 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 81.9754612959° = 81°58'29″ = 1.43107268992 rad

Height: ha = 16.66551721246
Height: hb = 25.89438948936
Height: hc = 14.99990238106

Median: ma = 19.81881915421
Median: mb = 26.3298519062
Median: mc = 16.50772636891

Inradius: r = 6.05497704358
Circumradius: R = 14.67111047852

Vertex coordinates: A[29.05548409391; 0] B[0; 0] C[21.42108259582; 14.99990238106]
Centroid: CG[16.82552222991; 54.9996746035]
Coordinates of the circumscribed circle: U[14.52774204696; 2.0488260266]
Coordinates of the inscribed circle: I[19.18774204696; 6.05497704358]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.9754612959° = 116°58'29″ = 1.11000005162 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 98.0255387041° = 98°1'31″ = 1.43107268992 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 26.15 ; ; b = 16.83 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.83**2 = 26.15**2 + c**2 -2 * 26.15 * c * cos (35° ) ; ; ; ; c**2 -42.842c +400.574 =0 ; ; p=1; q=-42.842; r=400.574 ; ; D = q**2 - 4pr = 42.842**2 - 4 * 1 * 400.574 = 233.112738919 ; ; D>0 ; ;
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 42.84 ± sqrt{ 233.11 } }{ 2 } ; ; c_{1,2} = 21.42082596 ± 7.63401498097 ; ; c_{1} = 29.0548409391 ; ; c_{2} = 13.7868109772 ; ; ; ; text{ Factored form: } ; ; (c -29.0548409391) (c -13.7868109772) = 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 = 26.15 ; ; b = 16.83 ; ; c = 29.05 ; ;

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

p = a+b+c = 26.15+16.83+29.05 = 72.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72.03 }{ 2 } = 36.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.02 * (36.02-26.15)(36.02-16.83)(36.02-29.05) } ; ; T = sqrt{ 47479.16 } = 217.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 217.9 }{ 26.15 } = 16.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 217.9 }{ 16.83 } = 25.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 217.9 }{ 29.05 } = 15 ; ;

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{ 16.83**2+29.05**2-26.15**2 }{ 2 * 16.83 * 29.05 } ) = 63° 1'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.15**2+29.05**2-16.83**2 }{ 2 * 26.15 * 29.05 } ) = 35° ; ;
 gamma = 180° - alpha - beta = 180° - 63° 1'31" - 35° = 81° 58'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 217.9 }{ 36.02 } = 6.05 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 26.15 }{ 2 * sin 63° 1'31" } = 14.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.83**2+2 * 29.05**2 - 26.15**2 } }{ 2 } = 19.818 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.05**2+2 * 26.15**2 - 16.83**2 } }{ 2 } = 26.329 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.83**2+2 * 26.15**2 - 29.05**2 } }{ 2 } = 16.507 ; ;



#2 Obtuse scalene triangle.

Sides: a = 26.15   b = 16.83   c = 13.78768109772

Area: T = 103.3944353059
Perimeter: p = 56.76768109772
Semiperimeter: s = 28.38334054886

Angle ∠ A = α = 116.9754612959° = 116°58'29″ = 2.04215921374 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 28.0255387041° = 28°1'31″ = 0.4899135278 rad

Height: ha = 7.90877899089
Height: hb = 12.28769106428
Height: hc = 14.99990238106

Median: ma = 8.1065979488
Median: mb = 19.13547093905
Median: mc = 20.88110119671

Inradius: r = 3.6432774758
Circumradius: R = 14.67111047852

Vertex coordinates: A[13.78768109772; 0] B[0; 0] C[21.42108259582; 14.99990238106]
Centroid: CG[11.73658789784; 54.9996746035]
Coordinates of the circumscribed circle: U[6.89334054886; 12.95107635445]
Coordinates of the inscribed circle: I[11.55334054886; 3.6432774758]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.0255387041° = 63°1'31″ = 2.04215921374 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 151.9754612959° = 151°58'29″ = 0.4899135278 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 26.15 ; ; b = 16.83 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.83**2 = 26.15**2 + c**2 -2 * 26.15 * c * cos (35° ) ; ; ; ; c**2 -42.842c +400.574 =0 ; ; p=1; q=-42.842; r=400.574 ; ; D = q**2 - 4pr = 42.842**2 - 4 * 1 * 400.574 = 233.112738919 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 42.84 ± sqrt{ 233.11 } }{ 2 } ; ; c_{1,2} = 21.42082596 ± 7.63401498097 ; ; c_{1} = 29.0548409391 ; ; c_{2} = 13.7868109772 ; ; ; ; text{ Factored form: } ; ; (c -29.0548409391) (c -13.7868109772) = 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 = 26.15 ; ; b = 16.83 ; ; c = 13.79 ; ;

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

p = a+b+c = 26.15+16.83+13.79 = 56.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56.77 }{ 2 } = 28.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.38 * (28.38-26.15)(28.38-16.83)(28.38-13.79) } ; ; T = sqrt{ 10690.39 } = 103.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.39 }{ 26.15 } = 7.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.39 }{ 16.83 } = 12.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.39 }{ 13.79 } = 15 ; ;

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{ 16.83**2+13.79**2-26.15**2 }{ 2 * 16.83 * 13.79 } ) = 116° 58'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.15**2+13.79**2-16.83**2 }{ 2 * 26.15 * 13.79 } ) = 35° ; ;
 gamma = 180° - alpha - beta = 180° - 116° 58'29" - 35° = 28° 1'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.39 }{ 28.38 } = 3.64 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 26.15 }{ 2 * sin 116° 58'29" } = 14.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.83**2+2 * 13.79**2 - 26.15**2 } }{ 2 } = 8.106 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.79**2+2 * 26.15**2 - 16.83**2 } }{ 2 } = 19.135 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.83**2+2 * 26.15**2 - 13.79**2 } }{ 2 } = 20.881 ; ;
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