Triangle calculator SSA

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Triangle has two solutions with side c=29.51772945206 and with side c=9.079916543

#1 Obtuse scalene triangle.

Sides: a = 20.18   b = 11.8   c = 29.51772945206

Area: T = 87.07769191326
Perimeter: p = 61.49772945206
Semiperimeter: s = 30.74986472603

Angle ∠ A = α = 300.0003420187° = 30°1″ = 0.5243604745 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 1332.999657981° = 132°59'59″ = 2.32112819358 rad

Height: ha = 8.63300217178
Height: hb = 14.7598799853
Height: hc = 5.99000610013

Median: ma = 20.08659960646
Median: mb = 24.58553927752
Median: mc = 7.44443623666

Inradius: r = 2.83218943073
Circumradius: R = 20.1879791357

Vertex coordinates: A[29.51772945206; 0] B[0; 0] C[19.29882299753; 5.99000610013]
Centroid: CG[16.27218414987; 1.96766870004]
Coordinates of the circumscribed circle: U[14.75986472603; -13.76224965126]
Coordinates of the inscribed circle: I[18.94986472603; 2.83218943073]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 1509.999657981° = 149°59'59″ = 0.5243604745 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 477.0003420187° = 47°1″ = 2.32112819358 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 20.18 ; ; b = 11.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.8**2 = 20.18**2 + c**2 -2 * 20.18 * c * cos (17° ) ; ; ; ; c**2 -38.596c +267.992 =0 ; ; p=1; q=-38.596; r=267.992 ; ; D = q**2 - 4pr = 38.596**2 - 4 * 1 * 267.992 = 417.717120724 ; ; D>0 ; ;
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 38.6 ± sqrt{ 417.72 } }{ 2 } ; ; c_{1,2} = 19.29822998 ± 10.2190645453 ; ; c_{1} = 29.5172945206 ; ; c_{2} = 9.07916543004 ; ; ; ; text{ Factored form: } ; ; (c -29.5172945206) (c -9.07916543004) = 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 = 20.18 ; ; b = 11.8 ; ; c = 29.52 ; ;

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

p = a+b+c = 20.18+11.8+29.52 = 61.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61.5 }{ 2 } = 30.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.75 * (30.75-20.18)(30.75-11.8)(30.75-29.52) } ; ; T = sqrt{ 7582.39 } = 87.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 87.08 }{ 20.18 } = 8.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 87.08 }{ 11.8 } = 14.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 87.08 }{ 29.52 } = 5.9 ; ;

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{ 11.8**2+29.52**2-20.18**2 }{ 2 * 11.8 * 29.52 } ) = 30° 1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.18**2+29.52**2-11.8**2 }{ 2 * 20.18 * 29.52 } ) = 17° ; ;
 gamma = 180° - alpha - beta = 180° - 30° 1" - 17° = 132° 59'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 87.08 }{ 30.75 } = 2.83 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20.18 }{ 2 * sin 30° 1" } = 20.18 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 29.52**2 - 20.18**2 } }{ 2 } = 20.086 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.52**2+2 * 20.18**2 - 11.8**2 } }{ 2 } = 24.585 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 20.18**2 - 29.52**2 } }{ 2 } = 7.444 ; ;



#2 Obtuse scalene triangle.

Sides: a = 20.18   b = 11.8   c = 9.079916543

Area: T = 26.78438149391
Perimeter: p = 41.059916543
Semiperimeter: s = 20.5329582715

Angle ∠ A = α = 1509.999657981° = 149°59'59″ = 2.61879879086 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 133.0003420187° = 13°1″ = 0.22768987721 rad

Height: ha = 2.65444910742
Height: hb = 4.54396296507
Height: hc = 5.99000610013

Median: ma = 3.0054583574
Median: mb = 14.49221296728
Median: mc = 15.89442879291

Inradius: r = 1.30546448781
Circumradius: R = 20.1879791357

Vertex coordinates: A[9.079916543; 0] B[0; 0] C[19.29882299753; 5.99000610013]
Centroid: CG[9.45991318018; 1.96766870004]
Coordinates of the circumscribed circle: U[4.5439582715; 19.66325575139]
Coordinates of the inscribed circle: I[8.7329582715; 1.30546448781]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 300.0003420187° = 30°1″ = 2.61879879086 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 1676.999657981° = 166°59'59″ = 0.22768987721 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 20.18 ; ; b = 11.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.8**2 = 20.18**2 + c**2 -2 * 20.18 * c * cos (17° ) ; ; ; ; c**2 -38.596c +267.992 =0 ; ; p=1; q=-38.596; r=267.992 ; ; D = q**2 - 4pr = 38.596**2 - 4 * 1 * 267.992 = 417.717120724 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 38.6 ± sqrt{ 417.72 } }{ 2 } ; ; c_{1,2} = 19.29822998 ± 10.2190645453 ; ; c_{1} = 29.5172945206 ; ; c_{2} = 9.07916543004 ; ; ; ; text{ Factored form: } ; ; (c -29.5172945206) (c -9.07916543004) = 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 = 20.18 ; ; b = 11.8 ; ; c = 9.08 ; ;

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

p = a+b+c = 20.18+11.8+9.08 = 41.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41.06 }{ 2 } = 20.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.53 * (20.53-20.18)(20.53-11.8)(20.53-9.08) } ; ; T = sqrt{ 717.37 } = 26.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.78 }{ 20.18 } = 2.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.78 }{ 11.8 } = 4.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.78 }{ 9.08 } = 5.9 ; ;

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{ 11.8**2+9.08**2-20.18**2 }{ 2 * 11.8 * 9.08 } ) = 149° 59'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.18**2+9.08**2-11.8**2 }{ 2 * 20.18 * 9.08 } ) = 17° ; ;
 gamma = 180° - alpha - beta = 180° - 149° 59'59" - 17° = 13° 1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.78 }{ 20.53 } = 1.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20.18 }{ 2 * sin 149° 59'59" } = 20.18 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 9.08**2 - 20.18**2 } }{ 2 } = 3.005 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.08**2+2 * 20.18**2 - 11.8**2 } }{ 2 } = 14.492 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 20.18**2 - 9.08**2 } }{ 2 } = 15.894 ; ;
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