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?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (25° ) ; ; ; ; c**2 -217.514c +6300 =0 ; ; p=1; q=-217.514; 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} = 108.75693444 ± 74.3509972344 ; ; c_{1} = 183.107931674 ; ;
c_{2} = 34.4059372056 ; ; ; ; text{ Factored form: } ; ; (c -183.107931674) (c -34.4059372056) = 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 = 183.11 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 183.11**2 - 120**2 } }{ 2 } = 131.203 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 183.11**2+2 * 120**2 - 90**2 } }{ 2 } = 148.119 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 120**2 - 183.11**2 } }{ 2 } = 53.553 ; ;







#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 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (25° ) ; ; ; ; c**2 -217.514c +6300 =0 ; ; p=1; q=-217.514; 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} = 108.75693444 ± 74.3509972344 ; ; c_{1} = 183.107931674 ; ; : Nr. 1
c_{2} = 34.4059372056 ; ; ; ; text{ Factored form: } ; ; (c -183.107931674) (c -34.4059372056) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+34.41**2-120**2 }{ 2 * 90 * 34.41 } ) = 145° 42'9" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+34.41**2-90**2 }{ 2 * 120 * 34.41 } ) = 25° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 120**2+90**2-34.41**2 }{ 2 * 120 * 90 } ) = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 34.41**2 - 120**2 } }{ 2 } = 32.278 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.41**2+2 * 120**2 - 90**2 } }{ 2 } = 75.94 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 120**2 - 34.41**2 } }{ 2 } = 104.662 ; ;
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