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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 50 ; ; b = 40 ; ; beta = 51° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 40**2 = 50**2 + c**2 -2 * 40 * c * cos (51° ) ; ; ; ; c**2 -62.932c +900 =0 ; ; p=1; q=-62.932039105; 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} = 31.4660195525 ± 9.49264907588 ; ; c_{1} = 40.9586686284 ; ;
c_{2} = 21.9733704766 ; ; ; ; (c -40.9586686284) (c -21.9733704766) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 50**2-40**2-21.97**2 }{ 2 * 40 * 21.97 } ) = 103° 43'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-50**2-21.97**2 }{ 2 * 50 * 21.97 } ) = 51° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21.97**2-50**2-40**2 }{ 2 * 40 * 50 } ) = 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 ; ;




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