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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 81**2-62**2-87.39**2 }{ 2 * 62 * 87.39 } ) = 62° 59'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 62**2-81**2-87.39**2 }{ 2 * 81 * 87.39 } ) = 43° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 87.39**2-81**2-62**2 }{ 2 * 62 * 81 } ) = 74° 4" ; ;

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 81 ; ; b = 62 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 62**2 = 81**2 + c**2 -2 * 62 * c * cos (43° ) ; ; ; ; c**2 -118.479c +2717 =0 ; ; p=1; q=-118.479299662; 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} = 59.2396498312 ± 28.1484655375 ; ; c_{1} = 87.3881153687 ; ;
c_{2} = 31.0911842936 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 81**2-62**2-31.09**2 }{ 2 * 62 * 31.09 } ) = 117° 4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 62**2-81**2-31.09**2 }{ 2 * 81 * 31.09 } ) = 43° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 31.09**2-81**2-62**2 }{ 2 * 62 * 81 } ) = 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 ; ;




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