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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 29.7**2 = 39.7**2 + c**2 -2 * 29.7 * c * cos (36° 36') ; ; ; ; c**2 -63.744c +694 =0 ; ; p=1; q=-63.7437075302; 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} = 31.8718537651 ± 17.9392046207 ; ;
c_{1} = 49.8110583858 ; ; c_{2} = 13.9326491444 ; ; ; ; (c -49.8110583858) (c -13.9326491444) = 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 = 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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