Triangle calculator SSA

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Triangle has two solutions with side c=49.81110583858 and with side c=13.93326491444

#1 Obtuse scalene triangle.

Sides: a = 39.7   b = 29.7   c = 49.81110583858

Area: T = 589.5177052351
Perimeter: p = 119.2111058386
Semiperimeter: s = 59.60655291929

Angle ∠ A = α = 52.84221036648° = 52°50'32″ = 0.92222686926 rad
Angle ∠ B = β = 36.6° = 36°36' = 0.63987905062 rad
Angle ∠ C = γ = 90.55878963352° = 90°33'28″ = 1.58105334547 rad

Height: ha = 29.6998592058
Height: hb = 39.69881180034
Height: hc = 23.67701275361

Median: ma = 35.88330498809
Median: mb = 42.52216799851
Median: mc = 24.67439663537

Inradius: r = 9.899030817
Circumradius: R = 24.90767099068

Vertex coordinates: A[49.81110583858; 0] B[0; 0] C[31.87218537651; 23.67701275361]
Centroid: CG[27.22876373836; 7.8990042512]
Coordinates of the circumscribed circle: U[24.90655291929; -0.24325159885]
Coordinates of the inscribed circle: I[29.90655291929; 9.899030817]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.1587896335° = 127°9'28″ = 0.92222686926 rad
∠ B' = β' = 143.4° = 143°24' = 0.63987905062 rad
∠ C' = γ' = 89.44221036648° = 89°26'32″ = 1.58105334547 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 39.7 ; ; b = 29.7 ; ; beta = 36° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 29.7**2 = 39.7**2 + c**2 -2 * 39.7 * c * cos (36° 36') ; ; ; ; c**2 -63.744c +694 =0 ; ; p=1; q=-63.744; r=694 ; ; D = q**2 - 4pr = 63.744**2 - 4 * 1 * 694 = 1287.26024969 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.74 ± sqrt{ 1287.26 } }{ 2 } ; ; c_{1,2} = 31.87185377 ± 17.9392046207 ; ; c_{1} = 49.8110583907 ; ;
c_{2} = 13.9326491493 ; ; ; ; (c -49.8110583907) (c -13.9326491493) = 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 = 39.7 ; ; b = 29.7 ; ; c = 49.81 ; ;

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

p = a+b+c = 39.7+29.7+49.81 = 119.21 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 119.21 }{ 2 } = 59.61 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.61 * (59.61-39.7)(59.61-29.7)(59.61-49.81) } ; ; T = sqrt{ 347530.36 } = 589.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 589.52 }{ 39.7 } = 29.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 589.52 }{ 29.7 } = 39.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 589.52 }{ 49.81 } = 23.67 ; ;

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{ 39.7**2-29.7**2-49.81**2 }{ 2 * 29.7 * 49.81 } ) = 52° 50'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29.7**2-39.7**2-49.81**2 }{ 2 * 39.7 * 49.81 } ) = 36° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 49.81**2-39.7**2-29.7**2 }{ 2 * 29.7 * 39.7 } ) = 90° 33'28" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 589.52 }{ 59.61 } = 9.89 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.7 }{ 2 * sin 52° 50'32" } = 24.91 ; ;





#2 Obtuse scalene triangle.

Sides: a = 39.7   b = 29.7   c = 13.93326491444

Area: T = 164.8943791082
Perimeter: p = 83.33326491444
Semiperimeter: s = 41.66663245722

Angle ∠ A = α = 127.1587896335° = 127°9'28″ = 2.2199323961 rad
Angle ∠ B = β = 36.6° = 36°36' = 0.63987905062 rad
Angle ∠ C = γ = 16.24221036648° = 16°14'32″ = 0.28334781864 rad

Height: ha = 8.30769919941
Height: hb = 11.10439589954
Height: hc = 23.67701275361

Median: ma = 12.00334101859
Median: mb = 25.77994851789
Median: mc = 34.3599282908

Inradius: r = 3.9577483478
Circumradius: R = 24.90767099068

Vertex coordinates: A[13.93326491444; 0] B[0; 0] C[31.87218537651; 23.67701275361]
Centroid: CG[15.26881676365; 7.8990042512]
Coordinates of the circumscribed circle: U[6.96663245722; 23.91326435247]
Coordinates of the inscribed circle: I[11.96663245722; 3.9577483478]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 52.84221036648° = 52°50'32″ = 2.2199323961 rad
∠ B' = β' = 143.4° = 143°24' = 0.63987905062 rad
∠ C' = γ' = 163.7587896335° = 163°45'28″ = 0.28334781864 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 39.7 ; ; b = 29.7 ; ; beta = 36° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 29.7**2 = 39.7**2 + c**2 -2 * 39.7 * c * cos (36° 36') ; ; ; ; c**2 -63.744c +694 =0 ; ; p=1; q=-63.744; r=694 ; ; D = q**2 - 4pr = 63.744**2 - 4 * 1 * 694 = 1287.26024969 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.74 ± sqrt{ 1287.26 } }{ 2 } ; ; c_{1,2} = 31.87185377 ± 17.9392046207 ; ; c_{1} = 49.8110583907 ; ; : Nr. 1
c_{2} = 13.9326491493 ; ; ; ; (c -49.8110583907) (c -13.9326491493) = 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 = 39.7 ; ; b = 29.7 ; ; c = 13.93 ; ;

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

p = a+b+c = 39.7+29.7+13.93 = 83.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 83.33 }{ 2 } = 41.67 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.67 * (41.67-39.7)(41.67-29.7)(41.67-13.93) } ; ; T = sqrt{ 27189.96 } = 164.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 164.89 }{ 39.7 } = 8.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 164.89 }{ 29.7 } = 11.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 164.89 }{ 13.93 } = 23.67 ; ;

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{ 39.7**2-29.7**2-13.93**2 }{ 2 * 29.7 * 13.93 } ) = 127° 9'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29.7**2-39.7**2-13.93**2 }{ 2 * 39.7 * 13.93 } ) = 36° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.93**2-39.7**2-29.7**2 }{ 2 * 29.7 * 39.7 } ) = 16° 14'32" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 164.89 }{ 41.67 } = 3.96 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.7 }{ 2 * sin 127° 9'28" } = 24.91 ; ;




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