Triangle calculator SSA

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Triangle has two solutions with side c=9.06878427113 and with side c=2.01334336888

#1 Acute scalene triangle.

Sides: a = 8.92   b = 7.83   c = 9.06878427113

Area: T = 31.6954584162
Perimeter: p = 25.81878427113
Semiperimeter: s = 12.90989213557

Angle ∠ A = α = 63.22659572605° = 63°13'33″ = 1.10435011269 rad
Angle ∠ B = β = 51.6° = 51°36' = 0.9010589894 rad
Angle ∠ C = γ = 65.17440427395° = 65°10'27″ = 1.13875016326 rad

Height: ha = 7.10664090049
Height: hb = 8.0965679224
Height: hc = 6.99105456393

Median: ma = 7.20224812196
Median: mb = 8.097746014
Median: mc = 7.0632662893

Inradius: r = 2.45552465143
Circumradius: R = 4.99655757106

Vertex coordinates: A[9.06878427113; 0] B[0; 0] C[5.54106382001; 6.99105456393]
Centroid: CG[4.86994936371; 2.33301818798]
Coordinates of the circumscribed circle: U[4.53439213557; 2.09774588962]
Coordinates of the inscribed circle: I[5.07989213557; 2.45552465143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.774404274° = 116°46'27″ = 1.10435011269 rad
∠ B' = β' = 128.4° = 128°24' = 0.9010589894 rad
∠ C' = γ' = 114.826595726° = 114°49'33″ = 1.13875016326 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 8.92 ; ; b = 7.83 ; ; beta = 51° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.83**2 = 8.92**2 + c**2 -2 * 8.92 * c * cos (51° 36') ; ; ; ; c**2 -11.081c +18.258 =0 ; ; p=1; q=-11.081; r=18.258 ; ; D = q**2 - 4pr = 11.081**2 - 4 * 1 * 18.258 = 49.7646866569 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.08 ± sqrt{ 49.76 } }{ 2 } ; ; c_{1,2} = 5.5406382 ± 3.52720451125 ; ;
c_{1} = 9.06784271125 ; ; c_{2} = 2.01343368875 ; ; ; ; text{ Factored form: } ; ; (c -9.06784271125) (c -2.01343368875) = 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 = 8.92 ; ; b = 7.83 ; ; c = 9.07 ; ;

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

p = a+b+c = 8.92+7.83+9.07 = 25.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.82 }{ 2 } = 12.91 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.91 * (12.91-8.92)(12.91-7.83)(12.91-9.07) } ; ; T = sqrt{ 1004.55 } = 31.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.69 }{ 8.92 } = 7.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.69 }{ 7.83 } = 8.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.69 }{ 9.07 } = 6.99 ; ;

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{ 7.83**2+9.07**2-8.92**2 }{ 2 * 7.83 * 9.07 } ) = 63° 13'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.92**2+9.07**2-7.83**2 }{ 2 * 8.92 * 9.07 } ) = 51° 36' ; ; gamma = 180° - alpha - beta = 180° - 63° 13'33" - 51° 36' = 65° 10'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.69 }{ 12.91 } = 2.46 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.92 }{ 2 * sin 63° 13'33" } = 5 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.83**2+2 * 9.07**2 - 8.92**2 } }{ 2 } = 7.202 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.07**2+2 * 8.92**2 - 7.83**2 } }{ 2 } = 8.097 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.83**2+2 * 8.92**2 - 9.07**2 } }{ 2 } = 7.063 ; ;







#2 Obtuse scalene triangle.

Sides: a = 8.92   b = 7.83   c = 2.01334336888

Area: T = 7.03875000468
Perimeter: p = 18.76334336888
Semiperimeter: s = 9.38217168444

Angle ∠ A = α = 116.774404274° = 116°46'27″ = 2.03880915267 rad
Angle ∠ B = β = 51.6° = 51°36' = 0.9010589894 rad
Angle ∠ C = γ = 11.62659572605° = 11°37'33″ = 0.20329112329 rad

Height: ha = 1.57879148087
Height: hb = 1.79875734475
Height: hc = 6.99105456393

Median: ma = 3.57662840505
Median: mb = 5.1466157072
Median: mc = 8.33221168496

Inradius: r = 0.75501292315
Circumradius: R = 4.99655757106

Vertex coordinates: A[2.01334336888; 0] B[0; 0] C[5.54106382001; 6.99105456393]
Centroid: CG[2.5188023963; 2.33301818798]
Coordinates of the circumscribed circle: U[1.00767168444; 4.89330867431]
Coordinates of the inscribed circle: I[1.55217168444; 0.75501292315]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.22659572605° = 63°13'33″ = 2.03880915267 rad
∠ B' = β' = 128.4° = 128°24' = 0.9010589894 rad
∠ C' = γ' = 168.374404274° = 168°22'27″ = 0.20329112329 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 8.92 ; ; b = 7.83 ; ; beta = 51° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.83**2 = 8.92**2 + c**2 -2 * 8.92 * c * cos (51° 36') ; ; ; ; c**2 -11.081c +18.258 =0 ; ; p=1; q=-11.081; r=18.258 ; ; D = q**2 - 4pr = 11.081**2 - 4 * 1 * 18.258 = 49.7646866569 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.08 ± sqrt{ 49.76 } }{ 2 } ; ; c_{1,2} = 5.5406382 ± 3.52720451125 ; ; : Nr. 1
c_{1} = 9.06784271125 ; ; c_{2} = 2.01343368875 ; ; ; ; text{ Factored form: } ; ; (c -9.06784271125) (c -2.01343368875) = 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 = 8.92 ; ; b = 7.83 ; ; c = 2.01 ; ;

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

p = a+b+c = 8.92+7.83+2.01 = 18.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.76 }{ 2 } = 9.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.38 * (9.38-8.92)(9.38-7.83)(9.38-2.01) } ; ; T = sqrt{ 49.53 } = 7.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.04 }{ 8.92 } = 1.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.04 }{ 7.83 } = 1.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.04 }{ 2.01 } = 6.99 ; ;

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{ 7.83**2+2.01**2-8.92**2 }{ 2 * 7.83 * 2.01 } ) = 116° 46'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.92**2+2.01**2-7.83**2 }{ 2 * 8.92 * 2.01 } ) = 51° 36' ; ; gamma = 180° - alpha - beta = 180° - 116° 46'27" - 51° 36' = 11° 37'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.04 }{ 9.38 } = 0.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.92 }{ 2 * sin 116° 46'27" } = 5 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.83**2+2 * 2.01**2 - 8.92**2 } }{ 2 } = 3.576 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.01**2+2 * 8.92**2 - 7.83**2 } }{ 2 } = 5.146 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.83**2+2 * 8.92**2 - 2.01**2 } }{ 2 } = 8.332 ; ;
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