Triangle calculator SSA

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Triangle has two solutions with side c=37.16441666581 and with side c=8.72107256113

#1 Acute scalene triangle.

Sides: a = 29.81   b = 23.76   c = 37.16441666581

Area: T = 353.6855081735
Perimeter: p = 90.73441666581
Semiperimeter: s = 45.36770833291

Angle ∠ A = α = 53.23333611883° = 53°14' = 0.92990974246 rad
Angle ∠ B = β = 39.68° = 39°40'48″ = 0.69325466472 rad
Angle ∠ C = γ = 87.08766388117° = 87°5'12″ = 1.52199485818 rad

Height: ha = 23.72992909584
Height: hb = 29.77114715266
Height: hc = 19.03436613754

Median: ma = 27.39988579451
Median: mb = 31.52441382388
Median: mc = 19.52767260223

Inradius: r = 7.79660727422
Circumradius: R = 18.60661311597

Vertex coordinates: A[37.16441666581; 0] B[0; 0] C[22.94224461347; 19.03436613754]
Centroid: CG[20.03655375976; 6.34545537918]
Coordinates of the circumscribed circle: U[18.58220833291; 0.94656721855]
Coordinates of the inscribed circle: I[21.60770833291; 7.79660727422]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.7676638812° = 126°46' = 0.92990974246 rad
∠ B' = β' = 140.32° = 140°19'12″ = 0.69325466472 rad
∠ C' = γ' = 92.91333611883° = 92°54'48″ = 1.52199485818 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 29.81 ; ; b = 23.76 ; ; beta = 39° 40'48" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 23.76**2 = 29.81**2 + c**2 -2 * 29.81 * c * cos (39° 40'48") ; ; ; ; c**2 -45.885c +324.099 =0 ; ; p=1; q=-45.885; r=324.099 ; ; D = q**2 - 4pr = 45.885**2 - 4 * 1 * 324.099 = 809.029338582 ; ;
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.88 ± sqrt{ 809.03 } }{ 2 } ; ; c_{1,2} = 22.94244613 ± 14.2217205234 ; ; c_{1} = 37.1641666581 ; ; c_{2} = 8.72072561135 ; ; ; ; text{ Factored form: } ; ; (c -37.1641666581) (c -8.72072561135) = 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 = 29.81 ; ; b = 23.76 ; ; c = 37.16 ; ;

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

p = a+b+c = 29.81+23.76+37.16 = 90.73 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.73 }{ 2 } = 45.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.37 * (45.37-29.81)(45.37-23.76)(45.37-37.16) } ; ; T = sqrt{ 125093.14 } = 353.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 353.69 }{ 29.81 } = 23.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 353.69 }{ 23.76 } = 29.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 353.69 }{ 37.16 } = 19.03 ; ;

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{ 23.76**2+37.16**2-29.81**2 }{ 2 * 23.76 * 37.16 } ) = 53° 14' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 29.81**2+37.16**2-23.76**2 }{ 2 * 29.81 * 37.16 } ) = 39° 40'48" ; ;
 gamma = 180° - alpha - beta = 180° - 53° 14' - 39° 40'48" = 87° 5'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 353.69 }{ 45.37 } = 7.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 29.81 }{ 2 * sin 53° 14' } = 18.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.76**2+2 * 37.16**2 - 29.81**2 } }{ 2 } = 27.399 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.16**2+2 * 29.81**2 - 23.76**2 } }{ 2 } = 31.524 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.76**2+2 * 29.81**2 - 37.16**2 } }{ 2 } = 19.527 ; ;



#2 Obtuse scalene triangle.

Sides: a = 29.81   b = 23.76   c = 8.72107256113

Area: T = 82.99436691172
Perimeter: p = 62.29107256113
Semiperimeter: s = 31.14553628057

Angle ∠ A = α = 126.7676638812° = 126°46' = 2.2122495229 rad
Angle ∠ B = β = 39.68° = 39°40'48″ = 0.69325466472 rad
Angle ∠ C = γ = 13.55333611883° = 13°33'12″ = 0.23765507774 rad

Height: ha = 5.56881763916
Height: hb = 6.98659990839
Height: hc = 19.03436613754

Median: ma = 9.90663263925
Median: mb = 18.47218482452
Median: mc = 26.66002647769

Inradius: r = 2.66547199339
Circumradius: R = 18.60661311597

Vertex coordinates: A[8.72107256113; 0] B[0; 0] C[22.94224461347; 19.03436613754]
Centroid: CG[10.5544390582; 6.34545537918]
Coordinates of the circumscribed circle: U[4.36603628057; 18.08879891899]
Coordinates of the inscribed circle: I[7.38553628057; 2.66547199339]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 53.23333611883° = 53°14' = 2.2122495229 rad
∠ B' = β' = 140.32° = 140°19'12″ = 0.69325466472 rad
∠ C' = γ' = 166.4476638812° = 166°26'48″ = 0.23765507774 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 29.81 ; ; b = 23.76 ; ; beta = 39° 40'48" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 23.76**2 = 29.81**2 + c**2 -2 * 29.81 * c * cos (39° 40'48") ; ; ; ; c**2 -45.885c +324.099 =0 ; ; p=1; q=-45.885; r=324.099 ; ; D = q**2 - 4pr = 45.885**2 - 4 * 1 * 324.099 = 809.029338582 ; ; : Nr. 1
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.88 ± sqrt{ 809.03 } }{ 2 } ; ; c_{1,2} = 22.94244613 ± 14.2217205234 ; ; c_{1} = 37.1641666581 ; ; c_{2} = 8.72072561135 ; ; ; ; text{ Factored form: } ; ; (c -37.1641666581) (c -8.72072561135) = 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 = 29.81 ; ; b = 23.76 ; ; c = 8.72 ; ;

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

p = a+b+c = 29.81+23.76+8.72 = 62.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.29 }{ 2 } = 31.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.15 * (31.15-29.81)(31.15-23.76)(31.15-8.72) } ; ; T = sqrt{ 6887.95 } = 82.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.99 }{ 29.81 } = 5.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.99 }{ 23.76 } = 6.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.99 }{ 8.72 } = 19.03 ; ;

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{ 23.76**2+8.72**2-29.81**2 }{ 2 * 23.76 * 8.72 } ) = 126° 46' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 29.81**2+8.72**2-23.76**2 }{ 2 * 29.81 * 8.72 } ) = 39° 40'48" ; ;
 gamma = 180° - alpha - beta = 180° - 126° 46' - 39° 40'48" = 13° 33'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.99 }{ 31.15 } = 2.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 29.81 }{ 2 * sin 126° 46' } = 18.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.76**2+2 * 8.72**2 - 29.81**2 } }{ 2 } = 9.906 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.72**2+2 * 29.81**2 - 23.76**2 } }{ 2 } = 18.472 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.76**2+2 * 29.81**2 - 8.72**2 } }{ 2 } = 26.6 ; ;
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