Triangle calculator SSA

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Triangle has two solutions with side c=93.37992172023 and with side c=20.93661360972

#1 Obtuse scalene triangle.

Sides: a = 66   b = 49   c = 93.37992172023

Area: T = 1540.757708384
Perimeter: p = 208.3799217202
Semiperimeter: s = 104.1989608601

Angle ∠ A = α = 42.33554018762° = 42°20'7″ = 0.73988921529 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 107.6654598124° = 107°39'53″ = 1.87991017251 rad

Height: ha = 46.69896086012
Height: hb = 62.88880442383
Height: hc = 33

Median: ma = 66.86880723713
Median: mb = 77.05657532093
Median: mc = 34.62105206297

Inradius: r = 14.78880110553
Circumradius: R = 49

Vertex coordinates: A[93.37992172023; 0] B[0; 0] C[57.15876766498; 33]
Centroid: CG[50.17989646174; 11]
Coordinates of the circumscribed circle: U[46.69896086012; -14.86987742827]
Coordinates of the inscribed circle: I[55.19896086012; 14.78880110553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6654598124° = 137°39'53″ = 0.73988921529 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 72.33554018762° = 72°20'7″ = 1.87991017251 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 66 ; ; b = 49 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 49**2 = 66**2 + c**2 -2 * 66 * c * cos (30° ) ; ; ; ; c**2 -114.315c +1955 =0 ; ; p=1; q=-114.315; r=1955 ; ; D = q**2 - 4pr = 114.315**2 - 4 * 1 * 1955 = 5248 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 114.32 ± sqrt{ 5248 } }{ 2 } = fraction{ 114.32 ± 8 sqrt{ 82 } }{ 2 } ; ; c_{1,2} = 57.15767665 ± 36.2215405525 ; ;
c_{1} = 93.3792172025 ; ; c_{2} = 20.9361360975 ; ; ; ; text{ Factored form: } ; ; (c -93.3792172025) (c -20.9361360975) = 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 = 66 ; ; b = 49 ; ; c = 93.38 ; ;

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

p = a+b+c = 66+49+93.38 = 208.38 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 208.38 }{ 2 } = 104.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 104.19 * (104.19-66)(104.19-49)(104.19-93.38) } ; ; T = sqrt{ 2373932.39 } = 1540.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1540.76 }{ 66 } = 46.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1540.76 }{ 49 } = 62.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1540.76 }{ 93.38 } = 33 ; ;

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{ 49**2+93.38**2-66**2 }{ 2 * 49 * 93.38 } ) = 42° 20'7" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 66**2+93.38**2-49**2 }{ 2 * 66 * 93.38 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 42° 20'7" - 30° = 107° 39'53" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1540.76 }{ 104.19 } = 14.79 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 66 }{ 2 * sin 42° 20'7" } = 49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 93.38**2 - 66**2 } }{ 2 } = 66.868 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 93.38**2+2 * 66**2 - 49**2 } }{ 2 } = 77.056 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 66**2 - 93.38**2 } }{ 2 } = 34.621 ; ;







#2 Obtuse scalene triangle.

Sides: a = 66   b = 49   c = 20.93661360972

Area: T = 345.4466245604
Perimeter: p = 135.9366136097
Semiperimeter: s = 67.96880680486

Angle ∠ A = α = 137.6654598124° = 137°39'53″ = 2.40327005007 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 12.33554018762° = 12°20'7″ = 0.21552933773 rad

Height: ha = 10.46880680486
Height: hb = 14.10998467594
Height: hc = 33

Median: ma = 18.18440836266
Median: mb = 42.39899858143
Median: mc = 57.17444659033

Inradius: r = 5.08224785156
Circumradius: R = 49

Vertex coordinates: A[20.93661360972; 0] B[0; 0] C[57.15876766498; 33]
Centroid: CG[26.03112709157; 11]
Coordinates of the circumscribed circle: U[10.46880680486; 47.86987742827]
Coordinates of the inscribed circle: I[18.96880680486; 5.08224785156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.33554018762° = 42°20'7″ = 2.40327005007 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 167.6654598124° = 167°39'53″ = 0.21552933773 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 66 ; ; b = 49 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 49**2 = 66**2 + c**2 -2 * 66 * c * cos (30° ) ; ; ; ; c**2 -114.315c +1955 =0 ; ; p=1; q=-114.315; r=1955 ; ; D = q**2 - 4pr = 114.315**2 - 4 * 1 * 1955 = 5248 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 114.32 ± sqrt{ 5248 } }{ 2 } = fraction{ 114.32 ± 8 sqrt{ 82 } }{ 2 } ; ; c_{1,2} = 57.15767665 ± 36.2215405525 ; ; : Nr. 1
c_{1} = 93.3792172025 ; ; c_{2} = 20.9361360975 ; ; ; ; text{ Factored form: } ; ; (c -93.3792172025) (c -20.9361360975) = 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 = 66 ; ; b = 49 ; ; c = 20.94 ; ;

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

p = a+b+c = 66+49+20.94 = 135.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135.94 }{ 2 } = 67.97 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.97 * (67.97-66)(67.97-49)(67.97-20.94) } ; ; T = sqrt{ 119333.11 } = 345.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 345.45 }{ 66 } = 10.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 345.45 }{ 49 } = 14.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 345.45 }{ 20.94 } = 33 ; ;

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{ 49**2+20.94**2-66**2 }{ 2 * 49 * 20.94 } ) = 137° 39'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 66**2+20.94**2-49**2 }{ 2 * 66 * 20.94 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 137° 39'53" - 30° = 12° 20'7" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 345.45 }{ 67.97 } = 5.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 66 }{ 2 * sin 137° 39'53" } = 49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 20.94**2 - 66**2 } }{ 2 } = 18.184 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.94**2+2 * 66**2 - 49**2 } }{ 2 } = 42.39 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 66**2 - 20.94**2 } }{ 2 } = 57.174 ; ;
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