Triangle calculator SSA

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Triangle has two solutions with side c=183.1087931679 and with side c=34.406593721

#1 Obtuse scalene triangle.

Sides: a = 120   b = 90   c = 183.1087931679

Area: T = 4643.085534782
Perimeter: p = 393.1087931679
Semiperimeter: s = 196.5543965839

Angle ∠ A = α = 34.29875702283° = 34°17'51″ = 0.59986055259 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 120.7022429772° = 120°42'9″ = 2.10766548147 rad

Height: ha = 77.3854755797
Height: hb = 103.1879674396
Height: hc = 50.71441914089

Median: ma = 131.2033114757
Median: mb = 148.1199064681
Median: mc = 53.553251011

Inradius: r = 23.622244551
Circumradius: R = 106.4799071242

Vertex coordinates: A[183.1087931679; 0] B[0; 0] C[108.7576934444; 50.71441914089]
Centroid: CG[97.28882887077; 16.90547304696]
Coordinates of the circumscribed circle: U[91.55439658394; -54.36660183535]
Coordinates of the inscribed circle: I[106.5543965839; 23.622244551]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.7022429772° = 145°42'9″ = 0.59986055259 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 59.29875702283° = 59°17'51″ = 2.10766548147 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 = 120 ; ; b = 90 ; ; c = 183.11 ; ;

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

p = a+b+c = 120+90+183.11 = 393.11 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 393.11 }{ 2 } = 196.55 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 196.55 * (196.55-120)(196.55-90)(196.55-183.11) } ; ; T = sqrt{ 21558241.55 } = 4643.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4643.09 }{ 120 } = 77.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4643.09 }{ 90 } = 103.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4643.09 }{ 183.11 } = 50.71 ; ;

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{ 120**2-90**2-183.11**2 }{ 2 * 90 * 183.11 } ) = 34° 17'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-183.11**2 }{ 2 * 120 * 183.11 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 183.11**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 120° 42'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4643.09 }{ 196.55 } = 23.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 34° 17'51" } = 106.48 ; ;





#2 Obtuse scalene triangle.

Sides: a = 120   b = 90   c = 34.406593721

Area: T = 872.4354642636
Perimeter: p = 244.406593721
Semiperimeter: s = 122.2032968605

Angle ∠ A = α = 145.7022429772° = 145°42'9″ = 2.54329871277 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 9.29875702283° = 9°17'51″ = 0.16222732129 rad

Height: ha = 14.54105773773
Height: hb = 19.3877436503
Height: hc = 50.71441914089

Median: ma = 32.27882319474
Median: mb = 75.94400043301
Median: mc = 104.6621635145

Inradius: r = 7.1399226261
Circumradius: R = 106.4799071242

Vertex coordinates: A[34.406593721; 0] B[0; 0] C[108.7576934444; 50.71441914089]
Centroid: CG[47.72109572181; 16.90547304696]
Coordinates of the circumscribed circle: U[17.2032968605; 105.0880209762]
Coordinates of the inscribed circle: I[32.2032968605; 7.1399226261]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 34.29875702283° = 34°17'51″ = 2.54329871277 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 170.7022429772° = 170°42'9″ = 0.16222732129 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 120**2 + c**2 -2 * 90 * c * cos (25° ) ; ; ; ; c**2 -217.514c +6300 =0 ; ; p=1; q=-217.513868889; r=6300 ; ; D = q**2 - 4pr = 217.514**2 - 4 * 1 * 6300 = 22112.283159 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 217.51 ± sqrt{ 22112.28 } }{ 2 } ; ; c_{1,2} = 108.756934444 ± 74.3509972344 ; ; c_{1} = 183.107931679 ; ;
c_{2} = 34.40593721 ; ; ; ; (c -183.107931679) (c -34.40593721) = 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 = 120 ; ; b = 90 ; ; c = 34.41 ; ;

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

p = a+b+c = 120+90+34.41 = 244.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 244.41 }{ 2 } = 122.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 122.2 * (122.2-120)(122.2-90)(122.2-34.41) } ; ; T = sqrt{ 761142.21 } = 872.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 872.43 }{ 120 } = 14.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 872.43 }{ 90 } = 19.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 872.43 }{ 34.41 } = 50.71 ; ;

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{ 120**2-90**2-34.41**2 }{ 2 * 90 * 34.41 } ) = 145° 42'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-34.41**2 }{ 2 * 120 * 34.41 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 34.41**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 9° 17'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 872.43 }{ 122.2 } = 7.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 145° 42'9" } = 106.48 ; ;




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