Triangle calculator SSA

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Triangle has two solutions with side c=81.63108193372 and with side c=33.41987506894

#1 Acute scalene triangle.

Sides: a = 73   b = 51   c = 81.63108193372

Area: T = 1834.379869928
Perimeter: p = 205.6310819337
Semiperimeter: s = 102.8155409669

Angle ∠ A = α = 61.79224209059° = 61°47'33″ = 1.07884811976 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 80.20875790941° = 80°12'27″ = 1.43998863402 rad

Height: ha = 50.25769506652
Height: hb = 71.93664195796
Height: hc = 44.94332876988

Median: ma = 57.44660210357
Median: mb = 73.11766556458
Median: mc = 47.94989555005

Inradius: r = 17.84114763428
Circumradius: R = 41.41988657598

Vertex coordinates: A[81.63108193372; 0] B[0; 0] C[57.52547850133; 44.94332876988]
Centroid: CG[46.38552014502; 14.98110958996]
Coordinates of the circumscribed circle: U[40.81554096686; 7.04444853903]
Coordinates of the inscribed circle: I[51.81554096686; 17.84114763428]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.2087579094° = 118°12'27″ = 1.07884811976 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 99.79224209059° = 99°47'33″ = 1.43998863402 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 = 73 ; ; b = 51 ; ; c = 81.63 ; ;

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

p = a+b+c = 73+51+81.63 = 205.63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 205.63 }{ 2 } = 102.82 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.82 * (102.82-73)(102.82-51)(102.82-81.63) } ; ; T = sqrt{ 3364945.21 } = 1834.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1834.38 }{ 73 } = 50.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1834.38 }{ 51 } = 71.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1834.38 }{ 81.63 } = 44.94 ; ;

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{ 73**2-51**2-81.63**2 }{ 2 * 51 * 81.63 } ) = 61° 47'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 51**2-73**2-81.63**2 }{ 2 * 73 * 81.63 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 81.63**2-73**2-51**2 }{ 2 * 51 * 73 } ) = 80° 12'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1834.38 }{ 102.82 } = 17.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 61° 47'33" } = 41.42 ; ;





#2 Obtuse scalene triangle.

Sides: a = 73   b = 51   c = 33.41987506894

Area: T = 750.9744263383
Perimeter: p = 157.4198750689
Semiperimeter: s = 78.70993753447

Angle ∠ A = α = 118.2087579094° = 118°12'27″ = 2.0633111456 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 23.79224209059° = 23°47'33″ = 0.41552560818 rad

Height: ha = 20.5754637353
Height: hb = 29.45499711131
Height: hc = 44.94332876988

Median: ma = 22.94989966844
Median: mb = 50.72113608731
Median: mc = 60.71107632598

Inradius: r = 9.54111030782
Circumradius: R = 41.41988657598

Vertex coordinates: A[33.41987506894; 0] B[0; 0] C[57.52547850133; 44.94332876988]
Centroid: CG[30.31545119009; 14.98110958996]
Coordinates of the circumscribed circle: U[16.70993753447; 37.89988023085]
Coordinates of the inscribed circle: I[27.70993753447; 9.54111030782]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61.79224209059° = 61°47'33″ = 2.0633111456 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 156.2087579094° = 156°12'27″ = 0.41552560818 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 73 ; ; b = 51 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 51**2 = 73**2 + c**2 -2 * 51 * c * cos (38° ) ; ; ; ; c**2 -115.05c +2728 =0 ; ; p=1; q=-115.049570027; r=2728 ; ; D = q**2 - 4pr = 115.05**2 - 4 * 1 * 2728 = 2324.4035633 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 115.05 ± sqrt{ 2324.4 } }{ 2 } ; ; c_{1,2} = 57.5247850133 ± 24.1060343239 ; ; c_{1} = 81.6308193372 ; ;
c_{2} = 33.4187506894 ; ; ; ; (c -81.6308193372) (c -33.4187506894) = 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 = 73 ; ; b = 51 ; ; c = 33.42 ; ;

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

p = a+b+c = 73+51+33.42 = 157.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 157.42 }{ 2 } = 78.71 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.71 * (78.71-73)(78.71-51)(78.71-33.42) } ; ; T = sqrt{ 563962.34 } = 750.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 750.97 }{ 73 } = 20.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 750.97 }{ 51 } = 29.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 750.97 }{ 33.42 } = 44.94 ; ;

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{ 73**2-51**2-33.42**2 }{ 2 * 51 * 33.42 } ) = 118° 12'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 51**2-73**2-33.42**2 }{ 2 * 73 * 33.42 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 33.42**2-73**2-51**2 }{ 2 * 51 * 73 } ) = 23° 47'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 750.97 }{ 78.71 } = 9.54 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 118° 12'27" } = 41.42 ; ;




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