Triangle calculator SSA

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Triangle has two solutions with side c=9.46992358985 and with side c=6.62111889399

#1 Acute scalene triangle.

Sides: a = 8.74   b = 3.7   c = 9.46992358985

Area: T = 16.16986732488
Perimeter: p = 21.90992358985
Semiperimeter: s = 10.95546179492

Angle ∠ A = α = 67.36442549049° = 67°21'51″ = 1.17657280462 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 89.63657450951° = 89°38'9″ = 1.56444388794 rad

Height: ha = 3.76999252285
Height: hb = 8.74398233777
Height: hc = 3.4154990063

Median: ma = 5.70880044018
Median: mb = 8.92221361932
Median: mc = 4.756627931

Inradius: r = 1.47659687032
Circumradius: R = 4.73547136307

Vertex coordinates: A[9.46992358985; 0] B[0; 0] C[8.04552124192; 3.4154990063]
Centroid: CG[5.83881494392; 1.1388330021]
Coordinates of the circumscribed circle: U[4.73546179492; 0.03301004901]
Coordinates of the inscribed circle: I[7.25546179492; 1.47659687032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.6365745095° = 112°38'9″ = 1.17657280462 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 90.36442549049° = 90°21'51″ = 1.56444388794 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 8.74 ; ; b = 3.7 ; ; beta = 23° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.7**2 = 8.74**2 + c**2 -2 * 8.74 * c * cos (23° ) ; ; ; ; c**2 -16.09c +62.698 =0 ; ; p=1; q=-16.09; r=62.698 ; ; D = q**2 - 4pr = 16.09**2 - 4 * 1 * 62.698 = 8.11137147855 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 16.09 ± sqrt{ 8.11 } }{ 2 } ; ; c_{1,2} = 8.04521242 ± 1.42402347931 ; ; c_{1} = 9.46923589931 ; ;
c_{2} = 6.62118894069 ; ; ; ; text{ Factored form: } ; ; (c -9.46923589931) (c -6.62118894069) = 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 = 8.74 ; ; b = 3.7 ; ; c = 9.47 ; ;

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

p = a+b+c = 8.74+3.7+9.47 = 21.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.91 }{ 2 } = 10.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.95 * (10.95-8.74)(10.95-3.7)(10.95-9.47) } ; ; T = sqrt{ 261.43 } = 16.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.17 }{ 8.74 } = 3.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.17 }{ 3.7 } = 8.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.17 }{ 9.47 } = 3.41 ; ;

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{ 3.7**2+9.47**2-8.74**2 }{ 2 * 3.7 * 9.47 } ) = 67° 21'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.74**2+9.47**2-3.7**2 }{ 2 * 8.74 * 9.47 } ) = 23° ; ; gamma = 180° - alpha - beta = 180° - 67° 21'51" - 23° = 89° 38'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.17 }{ 10.95 } = 1.48 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.74 }{ 2 * sin 67° 21'51" } = 4.73 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 9.47**2 - 8.74**2 } }{ 2 } = 5.708 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.47**2+2 * 8.74**2 - 3.7**2 } }{ 2 } = 8.922 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 8.74**2 - 9.47**2 } }{ 2 } = 4.756 ; ;







#2 Obtuse scalene triangle.

Sides: a = 8.74   b = 3.7   c = 6.62111889399

Area: T = 11.30656472174
Perimeter: p = 19.06111889399
Semiperimeter: s = 9.53105944699

Angle ∠ A = α = 112.6365745095° = 112°38'9″ = 1.96658646073 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 44.36442549049° = 44°21'51″ = 0.77443023183 rad

Height: ha = 2.58771046264
Height: hb = 6.11111606581
Height: hc = 3.4154990063

Median: ma = 3.10993683424
Median: mb = 7.52993672701
Median: mc = 5.8387701967

Inradius: r = 1.18662478519
Circumradius: R = 4.73547136307

Vertex coordinates: A[6.62111889399; 0] B[0; 0] C[8.04552124192; 3.4154990063]
Centroid: CG[4.8898800453; 1.1388330021]
Coordinates of the circumscribed circle: U[3.31105944699; 3.38548895729]
Coordinates of the inscribed circle: I[5.83105944699; 1.18662478519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.36442549049° = 67°21'51″ = 1.96658646073 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 135.6365745095° = 135°38'9″ = 0.77443023183 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.74 ; ; b = 3.7 ; ; beta = 23° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.7**2 = 8.74**2 + c**2 -2 * 8.74 * c * cos (23° ) ; ; ; ; c**2 -16.09c +62.698 =0 ; ; p=1; q=-16.09; r=62.698 ; ; D = q**2 - 4pr = 16.09**2 - 4 * 1 * 62.698 = 8.11137147855 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 16.09 ± sqrt{ 8.11 } }{ 2 } ; ; c_{1,2} = 8.04521242 ± 1.42402347931 ; ; c_{1} = 9.46923589931 ; ; : Nr. 1
c_{2} = 6.62118894069 ; ; ; ; text{ Factored form: } ; ; (c -9.46923589931) (c -6.62118894069) = 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 = 8.74 ; ; b = 3.7 ; ; c = 6.62 ; ;

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

p = a+b+c = 8.74+3.7+6.62 = 19.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.06 }{ 2 } = 9.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.53 * (9.53-8.74)(9.53-3.7)(9.53-6.62) } ; ; T = sqrt{ 127.82 } = 11.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.31 }{ 8.74 } = 2.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.31 }{ 3.7 } = 6.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.31 }{ 6.62 } = 3.41 ; ;

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{ 3.7**2+6.62**2-8.74**2 }{ 2 * 3.7 * 6.62 } ) = 112° 38'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.74**2+6.62**2-3.7**2 }{ 2 * 8.74 * 6.62 } ) = 23° ; ; gamma = 180° - alpha - beta = 180° - 112° 38'9" - 23° = 44° 21'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.31 }{ 9.53 } = 1.19 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.74 }{ 2 * sin 112° 38'9" } = 4.73 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 6.62**2 - 8.74**2 } }{ 2 } = 3.109 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.62**2+2 * 8.74**2 - 3.7**2 } }{ 2 } = 7.529 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 8.74**2 - 6.62**2 } }{ 2 } = 5.838 ; ;
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