Triangle calculator SSA

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Triangle has two solutions with side c=161.3632738659 and with side c=38.15500713929

#1 Obtuse scalene triangle.

Sides: a = 100   b = 62   c = 161.3632738659

Area: T = 562.8054782127
Perimeter: p = 323.3632738659
Semiperimeter: s = 161.681136933

Angle ∠ A = α = 6.46600518551° = 6°27'36″ = 0.11327491747 rad
Angle ∠ B = β = 4° = 0.07698131701 rad
Angle ∠ C = γ = 169.5439948145° = 169°32'24″ = 2.95990303088 rad

Height: ha = 11.25660956425
Height: hb = 18.15549929719
Height: hc = 6.97656473744

Median: ma = 111.5399081553
Median: mb = 130.6066151133
Median: mc = 20.31105057326

Inradius: r = 3.4810950121
Circumradius: R = 444.4033197812

Vertex coordinates: A[161.3632738659; 0] B[0; 0] C[99.7566405026; 6.97656473744]
Centroid: CG[87.04397145617; 2.32552157915]
Coordinates of the circumscribed circle: U[80.68113693295; -437.0187984606]
Coordinates of the inscribed circle: I[99.68113693295; 3.4810950121]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.5439948145° = 173°32'24″ = 0.11327491747 rad
∠ B' = β' = 176° = 0.07698131701 rad
∠ C' = γ' = 10.46600518551° = 10°27'36″ = 2.95990303088 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 62 ; ; beta = 4° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 100**2 + c**2 -2 * 100 * c * cos (4° ) ; ; ; ; c**2 -199.513c +6156 =0 ; ; p=1; q=-199.513; r=6156 ; ; D = q**2 - 4pr = 199.513**2 - 4 * 1 * 6156 = 15181.3613748 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 199.51 ± sqrt{ 15181.36 } }{ 2 } ; ; c_{1,2} = 99.75640503 ± 61.6063336331 ; ; c_{1} = 161.362738663 ; ;
c_{2} = 38.1500713969 ; ; ; ; text{ Factored form: } ; ; (c -161.362738663) (c -38.1500713969) = 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 = 100 ; ; b = 62 ; ; c = 161.36 ; ;

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

p = a+b+c = 100+62+161.36 = 323.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 323.36 }{ 2 } = 161.68 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 161.68 * (161.68-100)(161.68-62)(161.68-161.36) } ; ; T = sqrt{ 316749.22 } = 562.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 562.8 }{ 100 } = 11.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 562.8 }{ 62 } = 18.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 562.8 }{ 161.36 } = 6.98 ; ;

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{ 62**2+161.36**2-100**2 }{ 2 * 62 * 161.36 } ) = 6° 27'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+161.36**2-62**2 }{ 2 * 100 * 161.36 } ) = 4° ; ; gamma = 180° - alpha - beta = 180° - 6° 27'36" - 4° = 169° 32'24" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 562.8 }{ 161.68 } = 3.48 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 6° 27'36" } = 444.4 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 161.36**2 - 100**2 } }{ 2 } = 111.539 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 161.36**2+2 * 100**2 - 62**2 } }{ 2 } = 130.606 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 100**2 - 161.36**2 } }{ 2 } = 20.311 ; ;







#2 Obtuse scalene triangle.

Sides: a = 100   b = 62   c = 38.15500713929

Area: T = 133.0610722673
Perimeter: p = 200.1550071393
Semiperimeter: s = 100.0755035696

Angle ∠ A = α = 173.5439948145° = 173°32'24″ = 3.02988434789 rad
Angle ∠ B = β = 4° = 0.07698131701 rad
Angle ∠ C = γ = 2.46600518551° = 2°27'36″ = 0.04329360046 rad

Height: ha = 2.66112144535
Height: hb = 4.29222813765
Height: hc = 6.97656473744

Median: ma = 12.23657661649
Median: mb = 69.04113931902
Median: mc = 80.98223623586

Inradius: r = 1.33296095449
Circumradius: R = 444.4033197812

Vertex coordinates: A[38.15500713929; 0] B[0; 0] C[99.7566405026; 6.97656473744]
Centroid: CG[45.9698825473; 2.32552157915]
Coordinates of the circumscribed circle: U[19.07550356965; 443.994363198]
Coordinates of the inscribed circle: I[38.07550356965; 1.33296095449]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 6.46600518551° = 6°27'36″ = 3.02988434789 rad
∠ B' = β' = 176° = 0.07698131701 rad
∠ C' = γ' = 177.5439948145° = 177°32'24″ = 0.04329360046 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 62 ; ; beta = 4° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 100**2 + c**2 -2 * 100 * c * cos (4° ) ; ; ; ; c**2 -199.513c +6156 =0 ; ; p=1; q=-199.513; r=6156 ; ; D = q**2 - 4pr = 199.513**2 - 4 * 1 * 6156 = 15181.3613748 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 199.51 ± sqrt{ 15181.36 } }{ 2 } ; ; c_{1,2} = 99.75640503 ± 61.6063336331 ; ; c_{1} = 161.362738663 ; ; : Nr. 1
c_{2} = 38.1500713969 ; ; ; ; text{ Factored form: } ; ; (c -161.362738663) (c -38.1500713969) = 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 = 100 ; ; b = 62 ; ; c = 38.15 ; ;

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

p = a+b+c = 100+62+38.15 = 200.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 200.15 }{ 2 } = 100.08 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.08 * (100.08-100)(100.08-62)(100.08-38.15) } ; ; T = sqrt{ 17705.16 } = 133.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.06 }{ 100 } = 2.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.06 }{ 62 } = 4.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.06 }{ 38.15 } = 6.98 ; ;

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{ 62**2+38.15**2-100**2 }{ 2 * 62 * 38.15 } ) = 173° 32'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+38.15**2-62**2 }{ 2 * 100 * 38.15 } ) = 4° ; ; gamma = 180° - alpha - beta = 180° - 173° 32'24" - 4° = 2° 27'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.06 }{ 100.08 } = 1.33 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 173° 32'24" } = 444.4 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 38.15**2 - 100**2 } }{ 2 } = 12.236 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38.15**2+2 * 100**2 - 62**2 } }{ 2 } = 69.041 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 100**2 - 38.15**2 } }{ 2 } = 80.982 ; ;
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