Triangle calculator SSA

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Triangle has two solutions with side c=136.3033383791 and with side c=55.2087726989

#1 Acute scalene triangle.

Sides: a = 125   b = 90   c = 136.3033383791

Area: T = 5475.883289119
Perimeter: p = 351.3033383791
Semiperimeter: s = 175.6521691895

Angle ∠ A = α = 63.22222151534° = 63°13'20″ = 1.10334358148 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 76.77877848466° = 76°46'40″ = 1.3440025138 rad

Height: ha = 87.6144126259
Height: hb = 121.6866286471
Height: hc = 80.34884512108

Median: ma = 97.12439219575
Median: mb = 122.788764684
Median: mc = 84.95879124732

Inradius: r = 31.17546663644
Circumradius: R = 70.00875722087

Vertex coordinates: A[136.3033383791; 0] B[0; 0] C[95.75655553899; 80.34884512108]
Centroid: CG[77.35329797269; 26.78328170703]
Coordinates of the circumscribed circle: U[68.15216918953; 16.01327155211]
Coordinates of the inscribed circle: I[85.65216918953; 31.17546663644]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.7787784847° = 116°46'40″ = 1.10334358148 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 103.2222215153° = 103°13'20″ = 1.3440025138 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 90 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 125**2 + c**2 -2 * 125 * c * cos (40° ) ; ; ; ; c**2 -191.511c +7525 =0 ; ; p=1; q=-191.511; r=7525 ; ; D = q**2 - 4pr = 191.511**2 - 4 * 1 * 7525 = 6576.50555209 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 191.51 ± sqrt{ 6576.51 } }{ 2 } ; ; c_{1,2} = 95.75555539 ± 40.5478284008 ; ; c_{1} = 136.303383791 ; ;
c_{2} = 55.2077269892 ; ; ; ; text{ Factored form: } ; ; (c -136.303383791) (c -55.2077269892) = 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 = 125 ; ; b = 90 ; ; c = 136.3 ; ;

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

p = a+b+c = 125+90+136.3 = 351.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 351.3 }{ 2 } = 175.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.65 * (175.65-125)(175.65-90)(175.65-136.3) } ; ; T = sqrt{ 29985293.44 } = 5475.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5475.88 }{ 125 } = 87.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5475.88 }{ 90 } = 121.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5475.88 }{ 136.3 } = 80.35 ; ;

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{ 90**2+136.3**2-125**2 }{ 2 * 90 * 136.3 } ) = 63° 13'20" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+136.3**2-90**2 }{ 2 * 125 * 136.3 } ) = 40° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 125**2+90**2-136.3**2 }{ 2 * 125 * 90 } ) = 76° 46'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5475.88 }{ 175.65 } = 31.17 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 125 }{ 2 * sin 63° 13'20" } = 70.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 136.3**2 - 125**2 } }{ 2 } = 97.124 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 136.3**2+2 * 125**2 - 90**2 } }{ 2 } = 122.788 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 125**2 - 136.3**2 } }{ 2 } = 84.958 ; ;







#2 Obtuse scalene triangle.

Sides: a = 125   b = 90   c = 55.2087726989

Area: T = 2217.928767922
Perimeter: p = 270.2087726989
Semiperimeter: s = 135.1043863494

Angle ∠ A = α = 116.7787784847° = 116°46'40″ = 2.03881568388 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 23.22222151534° = 23°13'20″ = 0.4055304114 rad

Height: ha = 35.48768428675
Height: hb = 49.28772817604
Height: hc = 80.34884512108

Median: ma = 40.83774406599
Median: mb = 85.50769971385
Median: mc = 105.3599037202

Inradius: r = 16.41664637624
Circumradius: R = 70.00875722087

Vertex coordinates: A[55.2087726989; 0] B[0; 0] C[95.75655553899; 80.34884512108]
Centroid: CG[50.32110941263; 26.78328170703]
Coordinates of the circumscribed circle: U[27.60438634945; 64.33657356897]
Coordinates of the inscribed circle: I[45.10438634945; 16.41664637624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.22222151534° = 63°13'20″ = 2.03881568388 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 156.7787784847° = 156°46'40″ = 0.4055304114 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 90 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 125**2 + c**2 -2 * 125 * c * cos (40° ) ; ; ; ; c**2 -191.511c +7525 =0 ; ; p=1; q=-191.511; r=7525 ; ; D = q**2 - 4pr = 191.511**2 - 4 * 1 * 7525 = 6576.50555209 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 191.51 ± sqrt{ 6576.51 } }{ 2 } ; ; c_{1,2} = 95.75555539 ± 40.5478284008 ; ; c_{1} = 136.303383791 ; ; : Nr. 1
c_{2} = 55.2077269892 ; ; ; ; text{ Factored form: } ; ; (c -136.303383791) (c -55.2077269892) = 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 = 125 ; ; b = 90 ; ; c = 55.21 ; ;

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

p = a+b+c = 125+90+55.21 = 270.21 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 270.21 }{ 2 } = 135.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.1 * (135.1-125)(135.1-90)(135.1-55.21) } ; ; T = sqrt{ 4919203.19 } = 2217.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2217.93 }{ 125 } = 35.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2217.93 }{ 90 } = 49.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2217.93 }{ 55.21 } = 80.35 ; ;

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{ 90**2+55.21**2-125**2 }{ 2 * 90 * 55.21 } ) = 116° 46'40" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+55.21**2-90**2 }{ 2 * 125 * 55.21 } ) = 40° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 125**2+90**2-55.21**2 }{ 2 * 125 * 90 } ) = 23° 13'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2217.93 }{ 135.1 } = 16.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 125 }{ 2 * sin 116° 46'40" } = 70.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 55.21**2 - 125**2 } }{ 2 } = 40.837 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.21**2+2 * 125**2 - 90**2 } }{ 2 } = 85.507 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 125**2 - 55.21**2 } }{ 2 } = 105.359 ; ;
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