Triangle calculator SSA

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Triangle has two solutions with side c=28.0543789651 and with side c=2.03215579716

#1 Obtuse scalene triangle.

Sides: a = 15.73   b = 13.8   c = 28.0543789651

Area: T = 64.51097863024
Perimeter: p = 57.5843789651
Semiperimeter: s = 28.79218948255

Angle ∠ A = α = 19.46768474254° = 19°28'1″ = 0.34397605826 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 143.5333152575° = 143°31'59″ = 2.50551260982 rad

Height: ha = 8.20221343042
Height: hb = 9.34992443917
Height: hc = 4.59990069153

Median: ma = 20.66108163413
Median: mb = 21.67105792929
Median: mc = 4.7109848358

Inradius: r = 2.24105536938
Circumradius: R = 23.66000949768

Vertex coordinates: A[28.0543789651; 0] B[0; 0] C[15.04326738113; 4.59990069153]
Centroid: CG[14.36554878208; 1.53330023051]
Coordinates of the circumscribed circle: U[14.02768948255; -18.97992176991]
Coordinates of the inscribed circle: I[14.99218948255; 2.24105536938]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5333152575° = 160°31'59″ = 0.34397605826 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 36.46768474254° = 36°28'1″ = 2.50551260982 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 15.73 ; ; b = 13.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 13.8**2 = 15.73**2 + c**2 -2 * 15.73 * c * cos (17° ) ; ; ; ; c**2 -30.085c +56.993 =0 ; ; p=1; q=-30.085; r=56.993 ; ; D = q**2 - 4pr = 30.085**2 - 4 * 1 * 56.993 = 677.156541573 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.09 ± sqrt{ 677.16 } }{ 2 } ; ; c_{1,2} = 15.04267381 ± 13.0111158397 ; ; c_{1} = 28.0537896497 ; ;
c_{2} = 2.03155797034 ; ; ; ; text{ Factored form: } ; ; (c -28.0537896497) (c -2.03155797034) = 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 = 15.73 ; ; b = 13.8 ; ; c = 28.05 ; ;

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

p = a+b+c = 15.73+13.8+28.05 = 57.58 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57.58 }{ 2 } = 28.79 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.79 * (28.79-15.73)(28.79-13.8)(28.79-28.05) } ; ; T = sqrt{ 4161.51 } = 64.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 64.51 }{ 15.73 } = 8.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 64.51 }{ 13.8 } = 9.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 64.51 }{ 28.05 } = 4.6 ; ;

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{ 13.8**2+28.05**2-15.73**2 }{ 2 * 13.8 * 28.05 } ) = 19° 28'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.73**2+28.05**2-13.8**2 }{ 2 * 15.73 * 28.05 } ) = 17° ; ; gamma = 180° - alpha - beta = 180° - 19° 28'1" - 17° = 143° 31'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 64.51 }{ 28.79 } = 2.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.73 }{ 2 * sin 19° 28'1" } = 23.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.8**2+2 * 28.05**2 - 15.73**2 } }{ 2 } = 20.661 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.05**2+2 * 15.73**2 - 13.8**2 } }{ 2 } = 21.671 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.8**2+2 * 15.73**2 - 28.05**2 } }{ 2 } = 4.71 ; ;







#2 Obtuse scalene triangle.

Sides: a = 15.73   b = 13.8   c = 2.03215579716

Area: T = 4.67215745802
Perimeter: p = 31.56215579716
Semiperimeter: s = 15.78107789858

Angle ∠ A = α = 160.5333152575° = 160°31'59″ = 2.8021832071 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 2.46768474254° = 2°28'1″ = 0.04330546097 rad

Height: ha = 0.59439700674
Height: hb = 0.67770397942
Height: hc = 4.59990069153

Median: ma = 5.95219231259
Median: mb = 8.84113835963
Median: mc = 14.76215935133

Inradius: r = 0.29660294029
Circumradius: R = 23.66000949768

Vertex coordinates: A[2.03215579716; 0] B[0; 0] C[15.04326738113; 4.59990069153]
Centroid: CG[5.69114105943; 1.53330023051]
Coordinates of the circumscribed circle: U[1.01657789858; 23.57882246144]
Coordinates of the inscribed circle: I[1.98107789858; 0.29660294029]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.46768474254° = 19°28'1″ = 2.8021832071 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 177.5333152575° = 177°31'59″ = 0.04330546097 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 15.73 ; ; b = 13.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 13.8**2 = 15.73**2 + c**2 -2 * 15.73 * c * cos (17° ) ; ; ; ; c**2 -30.085c +56.993 =0 ; ; p=1; q=-30.085; r=56.993 ; ; D = q**2 - 4pr = 30.085**2 - 4 * 1 * 56.993 = 677.156541573 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.09 ± sqrt{ 677.16 } }{ 2 } ; ; c_{1,2} = 15.04267381 ± 13.0111158397 ; ; c_{1} = 28.0537896497 ; ; : Nr. 1
c_{2} = 2.03155797034 ; ; ; ; text{ Factored form: } ; ; (c -28.0537896497) (c -2.03155797034) = 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 = 15.73 ; ; b = 13.8 ; ; c = 2.03 ; ;

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

p = a+b+c = 15.73+13.8+2.03 = 31.56 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.56 }{ 2 } = 15.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.78 * (15.78-15.73)(15.78-13.8)(15.78-2.03) } ; ; T = sqrt{ 21.82 } = 4.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.67 }{ 15.73 } = 0.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.67 }{ 13.8 } = 0.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.67 }{ 2.03 } = 4.6 ; ;

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{ 13.8**2+2.03**2-15.73**2 }{ 2 * 13.8 * 2.03 } ) = 160° 31'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.73**2+2.03**2-13.8**2 }{ 2 * 15.73 * 2.03 } ) = 17° ; ; gamma = 180° - alpha - beta = 180° - 160° 31'59" - 17° = 2° 28'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.67 }{ 15.78 } = 0.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.73 }{ 2 * sin 160° 31'59" } = 23.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.8**2+2 * 2.03**2 - 15.73**2 } }{ 2 } = 5.952 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.03**2+2 * 15.73**2 - 13.8**2 } }{ 2 } = 8.841 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.8**2+2 * 15.73**2 - 2.03**2 } }{ 2 } = 14.762 ; ;
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