Triangle calculator SSA

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Triangle has two solutions with side c=87.38881153687 and with side c=31.09111842936

#1 Acute scalene triangle.

Sides: a = 81   b = 62   c = 87.38881153687

Area: T = 2413.74113305
Perimeter: p = 230.3888115369
Semiperimeter: s = 115.1944057684

Angle ∠ A = α = 62.9998906243° = 62°59'56″ = 1.10995383391 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 74.0011093757° = 74°4″ = 1.29215627361 rad

Height: ha = 59.59985513704
Height: hb = 77.86326235645
Height: hc = 55.24218671651

Median: ma = 64.0321955724
Median: mb = 78.34443766575
Median: mc = 57.38875363043

Inradius: r = 20.95436965623
Circumradius: R = 45.45546547548

Vertex coordinates: A[87.38881153687; 0] B[0; 0] C[59.24396498312; 55.24218671651]
Centroid: CG[48.87659217333; 18.41439557217]
Coordinates of the circumscribed circle: U[43.69440576843; 12.52881667437]
Coordinates of the inscribed circle: I[53.19440576843; 20.95436965623]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.0011093757° = 117°4″ = 1.10995383391 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 105.9998906243° = 105°59'56″ = 1.29215627361 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 81 ; ; b = 62 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 81**2 + c**2 -2 * 81 * c * cos (43° ) ; ; ; ; c**2 -118.479c +2717 =0 ; ; p=1; q=-118.479; r=2717 ; ; D = q**2 - 4pr = 118.479**2 - 4 * 1 * 2717 = 3169.34444847 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 118.48 ± sqrt{ 3169.34 } }{ 2 } ; ;
c_{1,2} = 59.23964983 ± 28.1484655375 ; ; c_{1} = 87.3881153687 ; ; c_{2} = 31.0911842936 ; ; ; ; text{ Factored form: } ; ; (c -87.3881153687) (c -31.0911842936) = 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 = 81 ; ; b = 62 ; ; c = 87.39 ; ;

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

p = a+b+c = 81+62+87.39 = 230.39 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 230.39 }{ 2 } = 115.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 115.19 * (115.19-81)(115.19-62)(115.19-87.39) } ; ; T = sqrt{ 5826147.21 } = 2413.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2413.74 }{ 81 } = 59.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2413.74 }{ 62 } = 77.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2413.74 }{ 87.39 } = 55.24 ; ;

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+87.39**2-81**2 }{ 2 * 62 * 87.39 } ) = 62° 59'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 81**2+87.39**2-62**2 }{ 2 * 81 * 87.39 } ) = 43° ; ;
gamma = 180° - alpha - beta = 180° - 62° 59'56" - 43° = 74° 4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2413.74 }{ 115.19 } = 20.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 81 }{ 2 * sin 62° 59'56" } = 45.45 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 87.39**2 - 81**2 } }{ 2 } = 64.032 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.39**2+2 * 81**2 - 62**2 } }{ 2 } = 78.344 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 81**2 - 87.39**2 } }{ 2 } = 57.388 ; ;



#2 Obtuse scalene triangle.

Sides: a = 81   b = 62   c = 31.09111842936

Area: T = 858.7687536376
Perimeter: p = 174.0911184294
Semiperimeter: s = 87.04655921468

Angle ∠ A = α = 117.0011093757° = 117°4″ = 2.04220543145 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 19.9998906243° = 19°59'56″ = 0.34990467607 rad

Height: ha = 21.20441367007
Height: hb = 27.70221785928
Height: hc = 55.24218671651

Median: ma = 27.66600952708
Median: mb = 52.94217686746
Median: mc = 70.43331922094

Inradius: r = 9.86657211146
Circumradius: R = 45.45546547548

Vertex coordinates: A[31.09111842936; 0] B[0; 0] C[59.24396498312; 55.24218671651]
Centroid: CG[30.11102780416; 18.41439557217]
Coordinates of the circumscribed circle: U[15.54655921468; 42.71437004214]
Coordinates of the inscribed circle: I[25.04655921468; 9.86657211146]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 62.9998906243° = 62°59'56″ = 2.04220543145 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 160.0011093757° = 160°4″ = 0.34990467607 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 81 ; ; b = 62 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 62**2 = 81**2 + c**2 -2 * 81 * c * cos (43° ) ; ; ; ; c**2 -118.479c +2717 =0 ; ; p=1; q=-118.479; r=2717 ; ; D = q**2 - 4pr = 118.479**2 - 4 * 1 * 2717 = 3169.34444847 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 118.48 ± sqrt{ 3169.34 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 59.23964983 ± 28.1484655375 ; ; c_{1} = 87.3881153687 ; ; c_{2} = 31.0911842936 ; ; ; ; text{ Factored form: } ; ; (c -87.3881153687) (c -31.0911842936) = 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 = 81 ; ; b = 62 ; ; c = 31.09 ; ;

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

p = a+b+c = 81+62+31.09 = 174.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 174.09 }{ 2 } = 87.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 87.05 * (87.05-81)(87.05-62)(87.05-31.09) } ; ; T = sqrt{ 737481.68 } = 858.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 858.77 }{ 81 } = 21.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 858.77 }{ 62 } = 27.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 858.77 }{ 31.09 } = 55.24 ; ;

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+31.09**2-81**2 }{ 2 * 62 * 31.09 } ) = 117° 4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 81**2+31.09**2-62**2 }{ 2 * 81 * 31.09 } ) = 43° ; ;
gamma = 180° - alpha - beta = 180° - 117° 4" - 43° = 19° 59'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 858.77 }{ 87.05 } = 9.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 81 }{ 2 * sin 117° 4" } = 45.45 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 31.09**2 - 81**2 } }{ 2 } = 27.66 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.09**2+2 * 81**2 - 62**2 } }{ 2 } = 52.942 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 81**2 - 31.09**2 } }{ 2 } = 70.433 ; ;
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