Triangle calculator SSA

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Triangle has two solutions with side c=15.39220292999 and with side c=4.78110459925

#1 Acute scalene triangle.

Sides: a = 12.8   b = 9.5   c = 15.39220292999

Area: T = 60.64881885889
Perimeter: p = 37.69220292999
Semiperimeter: s = 18.84660146499

Angle ∠ A = α = 56.05497522588° = 56°2'59″ = 0.97882527218 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 85.95502477412° = 85°57'1″ = 1.5500114816 rad

Height: ha = 9.4766279467
Height: hb = 12.76880397029
Height: hc = 7.88804668842

Median: ma = 11.07334946148
Median: mb = 13.33547209564
Median: mc = 8.23550688223

Inradius: r = 3.21880909182
Circumradius: R = 7.7155278916

Vertex coordinates: A[15.39220292999; 0] B[0; 0] C[10.08765376462; 7.88804668842]
Centroid: CG[8.49328556487; 2.62768222947]
Coordinates of the circumscribed circle: U[7.69660146499; 0.54548736187]
Coordinates of the inscribed circle: I[9.34660146499; 3.21880909182]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.9550247741° = 123°57'1″ = 0.97882527218 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 94.05497522588° = 94°2'59″ = 1.5500114816 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 12.8 ; ; b = 9.5 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.5**2 = 12.8**2 + c**2 -2 * 12.8 * c * cos (38° ) ; ; ; ; c**2 -20.173c +73.59 =0 ; ; p=1; q=-20.173; r=73.59 ; ; D = q**2 - 4pr = 20.173**2 - 4 * 1 * 73.59 = 112.59296675 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 20.17 ± sqrt{ 112.59 } }{ 2 } ; ; c_{1,2} = 10.08653765 ± 5.3054916537 ; ; c_{1} = 15.3920293037 ; ; c_{2} = 4.7810459963 ; ; ; ; text{ Factored form: } ; ; (c -15.3920293037) (c -4.7810459963) = 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 = 12.8 ; ; b = 9.5 ; ; c = 15.39 ; ;

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

p = a+b+c = 12.8+9.5+15.39 = 37.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.69 }{ 2 } = 18.85 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.85 * (18.85-12.8)(18.85-9.5)(18.85-15.39) } ; ; T = sqrt{ 3678.2 } = 60.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.65 }{ 12.8 } = 9.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.65 }{ 9.5 } = 12.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.65 }{ 15.39 } = 7.88 ; ;

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{ 9.5**2+15.39**2-12.8**2 }{ 2 * 9.5 * 15.39 } ) = 56° 2'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.8**2+15.39**2-9.5**2 }{ 2 * 12.8 * 15.39 } ) = 38° ; ; gamma = 180° - alpha - beta = 180° - 56° 2'59" - 38° = 85° 57'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.65 }{ 18.85 } = 3.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.8 }{ 2 * sin 56° 2'59" } = 7.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.5**2+2 * 15.39**2 - 12.8**2 } }{ 2 } = 11.073 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.39**2+2 * 12.8**2 - 9.5**2 } }{ 2 } = 13.335 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.5**2+2 * 12.8**2 - 15.39**2 } }{ 2 } = 8.235 ; ;







#2 Obtuse scalene triangle.

Sides: a = 12.8   b = 9.5   c = 4.78110459925

Area: T = 18.83884373076
Perimeter: p = 27.08110459925
Semiperimeter: s = 13.54105229962

Angle ∠ A = α = 123.9550247741° = 123°57'1″ = 2.16333399317 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 18.05497522588° = 18°2'59″ = 0.31550276061 rad

Height: ha = 2.94435058293
Height: hb = 3.96659868016
Height: hc = 7.88804668842

Median: ma = 3.94989492768
Median: mb = 8.41334832496
Median: mc = 11.01550079348

Inradius: r = 1.39112636397
Circumradius: R = 7.7155278916

Vertex coordinates: A[4.78110459925; 0] B[0; 0] C[10.08765376462; 7.88804668842]
Centroid: CG[4.95658612129; 2.62768222947]
Coordinates of the circumscribed circle: U[2.39105229962; 7.33655932655]
Coordinates of the inscribed circle: I[4.04105229962; 1.39112636397]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.05497522588° = 56°2'59″ = 2.16333399317 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 161.9550247741° = 161°57'1″ = 0.31550276061 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.8 ; ; b = 9.5 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.5**2 = 12.8**2 + c**2 -2 * 12.8 * c * cos (38° ) ; ; ; ; c**2 -20.173c +73.59 =0 ; ; p=1; q=-20.173; r=73.59 ; ; D = q**2 - 4pr = 20.173**2 - 4 * 1 * 73.59 = 112.59296675 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 20.17 ± sqrt{ 112.59 } }{ 2 } ; ; c_{1,2} = 10.08653765 ± 5.3054916537 ; ; c_{1} = 15.3920293037 ; ; c_{2} = 4.7810459963 ; ; ; ; text{ Factored form: } ; ; (c -15.3920293037) (c -4.7810459963) = 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 = 12.8 ; ; b = 9.5 ; ; c = 4.78 ; ;

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

p = a+b+c = 12.8+9.5+4.78 = 27.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.08 }{ 2 } = 13.54 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.54 * (13.54-12.8)(13.54-9.5)(13.54-4.78) } ; ; T = sqrt{ 354.89 } = 18.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.84 }{ 12.8 } = 2.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.84 }{ 9.5 } = 3.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.84 }{ 4.78 } = 7.88 ; ;

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{ 9.5**2+4.78**2-12.8**2 }{ 2 * 9.5 * 4.78 } ) = 123° 57'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.8**2+4.78**2-9.5**2 }{ 2 * 12.8 * 4.78 } ) = 38° ; ; gamma = 180° - alpha - beta = 180° - 123° 57'1" - 38° = 18° 2'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.84 }{ 13.54 } = 1.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.8 }{ 2 * sin 123° 57'1" } = 7.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.5**2+2 * 4.78**2 - 12.8**2 } }{ 2 } = 3.949 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.78**2+2 * 12.8**2 - 9.5**2 } }{ 2 } = 8.413 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.5**2+2 * 12.8**2 - 4.78**2 } }{ 2 } = 11.015 ; ;
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