Triangle calculator SSA

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Triangle has two solutions with side c=40.95986686284 and with side c=21.97333704766

#1 Acute scalene triangle.

Sides: a = 50   b = 40   c = 40.95986686284

Area: T = 795.772159778
Perimeter: p = 130.9598668628
Semiperimeter: s = 65.47993343142

Angle ∠ A = α = 76.2721804145° = 76°16'18″ = 1.33111941088 rad
Angle ∠ B = β = 51° = 0.89901179185 rad
Angle ∠ C = γ = 52.7288195855° = 52°43'42″ = 0.92202806263 rad

Height: ha = 31.83108639112
Height: hb = 39.7898579889
Height: hc = 38.85772980728

Median: ma = 31.84403245571
Median: mb = 41.09550881238
Median: mc = 40.38106496487

Inradius: r = 12.15330190573
Circumradius: R = 25.73551913179

Vertex coordinates: A[40.95986686284; 0] B[0; 0] C[31.46660195525; 38.85772980728]
Centroid: CG[24.1421562727; 12.95224326909]
Coordinates of the circumscribed circle: U[20.47993343142; 15.58551512092]
Coordinates of the inscribed circle: I[25.47993343142; 12.15330190573]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.7288195855° = 103°43'42″ = 1.33111941088 rad
∠ B' = β' = 129° = 0.89901179185 rad
∠ C' = γ' = 127.2721804145° = 127°16'18″ = 0.92202806263 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 50 ; ; b = 40 ; ; beta = 51° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 50**2 + c**2 -2 * 50 * c * cos (51° ) ; ; ; ; c**2 -62.932c +900 =0 ; ; p=1; q=-62.932; r=900 ; ; D = q**2 - 4pr = 62.932**2 - 4 * 1 * 900 = 360.441545911 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.93 ± sqrt{ 360.44 } }{ 2 } ; ; c_{1,2} = 31.46601955 ± 9.49264907588 ; ; c_{1} = 40.9586686259 ; ; c_{2} = 21.9733704741 ; ; ; ; text{ Factored form: } ; ; (c -40.9586686259) (c -21.9733704741) = 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 = 50 ; ; b = 40 ; ; c = 40.96 ; ;

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

p = a+b+c = 50+40+40.96 = 130.96 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 130.96 }{ 2 } = 65.48 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.48 * (65.48-50)(65.48-40)(65.48-40.96) } ; ; T = sqrt{ 633252.44 } = 795.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 795.77 }{ 50 } = 31.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 795.77 }{ 40 } = 39.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 795.77 }{ 40.96 } = 38.86 ; ;

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{ 40**2+40.96**2-50**2 }{ 2 * 40 * 40.96 } ) = 76° 16'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 50**2+40.96**2-40**2 }{ 2 * 50 * 40.96 } ) = 51° ; ; gamma = 180° - alpha - beta = 180° - 76° 16'18" - 51° = 52° 43'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 795.77 }{ 65.48 } = 12.15 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 50 }{ 2 * sin 76° 16'18" } = 25.74 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 40.96**2 - 50**2 } }{ 2 } = 31.84 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40.96**2+2 * 50**2 - 40**2 } }{ 2 } = 41.095 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 50**2 - 40.96**2 } }{ 2 } = 40.381 ; ;







#2 Obtuse scalene triangle.

Sides: a = 50   b = 40   c = 21.97333704766

Area: T = 426.9132903137
Perimeter: p = 111.9733370477
Semiperimeter: s = 55.98766852383

Angle ∠ A = α = 103.7288195855° = 103°43'42″ = 1.81103985448 rad
Angle ∠ B = β = 51° = 0.89901179185 rad
Angle ∠ C = γ = 25.2721804145° = 25°16'18″ = 0.44110761902 rad

Height: ha = 17.07765161255
Height: hb = 21.34656451569
Height: hc = 38.85772980728

Median: ma = 20.40662369155
Median: mb = 33.03765631543
Median: mc = 43.92437150919

Inradius: r = 7.62552577076
Circumradius: R = 25.73551913179

Vertex coordinates: A[21.97333704766; 0] B[0; 0] C[31.46660195525; 38.85772980728]
Centroid: CG[17.81331300097; 12.95224326909]
Coordinates of the circumscribed circle: U[10.98766852383; 23.27221468636]
Coordinates of the inscribed circle: I[15.98766852383; 7.62552577076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.2721804145° = 76°16'18″ = 1.81103985448 rad
∠ B' = β' = 129° = 0.89901179185 rad
∠ C' = γ' = 154.7288195855° = 154°43'42″ = 0.44110761902 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 50 ; ; b = 40 ; ; beta = 51° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 50**2 + c**2 -2 * 50 * c * cos (51° ) ; ; ; ; c**2 -62.932c +900 =0 ; ; p=1; q=-62.932; r=900 ; ; D = q**2 - 4pr = 62.932**2 - 4 * 1 * 900 = 360.441545911 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.93 ± sqrt{ 360.44 } }{ 2 } ; ; c_{1,2} = 31.46601955 ± 9.49264907588 ; ; c_{1} = 40.9586686259 ; ; c_{2} = 21.9733704741 ; ; ; ; text{ Factored form: } ; ; (c -40.9586686259) (c -21.9733704741) = 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 = 50 ; ; b = 40 ; ; c = 21.97 ; ;

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

p = a+b+c = 50+40+21.97 = 111.97 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 111.97 }{ 2 } = 55.99 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.99 * (55.99-50)(55.99-40)(55.99-21.97) } ; ; T = sqrt{ 182254.63 } = 426.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 426.91 }{ 50 } = 17.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 426.91 }{ 40 } = 21.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 426.91 }{ 21.97 } = 38.86 ; ;

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{ 40**2+21.97**2-50**2 }{ 2 * 40 * 21.97 } ) = 103° 43'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 50**2+21.97**2-40**2 }{ 2 * 50 * 21.97 } ) = 51° ; ; gamma = 180° - alpha - beta = 180° - 103° 43'42" - 51° = 25° 16'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 426.91 }{ 55.99 } = 7.63 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 50 }{ 2 * sin 103° 43'42" } = 25.74 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 21.97**2 - 50**2 } }{ 2 } = 20.406 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.97**2+2 * 50**2 - 40**2 } }{ 2 } = 33.037 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 50**2 - 21.97**2 } }{ 2 } = 43.924 ; ;
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