Triangle calculator SSA

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Triangle has two solutions with side c=210.5133222331 and with side c=16.4177021039

#1 Obtuse scalene triangle.

Sides: a = 116   b = 100   c = 210.5133222331

Area: T = 2538.553327967
Perimeter: p = 426.5133222331
Semiperimeter: s = 213.2576611166

Angle ∠ A = α = 13.95660514012° = 13°57'22″ = 0.24435790475 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 154.0443948599° = 154°2'38″ = 2.68985740958 rad

Height: ha = 43.76881599944
Height: hb = 50.77110655935
Height: hc = 24.11877561349

Median: ma = 154.2532741915
Median: mb = 162.4377398367
Median: mc = 25.47663774099

Inradius: r = 11.90437495053
Circumradius: R = 240.4876717237

Vertex coordinates: A[210.5133222331; 0] B[0; 0] C[113.4655121685; 24.11877561349]
Centroid: CG[107.9932781339; 8.0399252045]
Coordinates of the circumscribed circle: U[105.2576611166; -216.2298830116]
Coordinates of the inscribed circle: I[113.2576611166; 11.90437495053]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.0443948599° = 166°2'38″ = 0.24435790475 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 25.95660514012° = 25°57'22″ = 2.68985740958 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 116 ; ; b = 100 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 100**2 = 116**2 + c**2 -2 * 116 * c * cos (12° ) ; ; ; ; c**2 -226.93c +3456 =0 ; ; p=1; q=-226.93; r=3456 ; ; D = q**2 - 4pr = 226.93**2 - 4 * 1 * 3456 = 37673.3353561 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 226.93 ± sqrt{ 37673.34 } }{ 2 } ; ; c_{1,2} = 113.46512169 ± 97.0481006461 ; ; c_{1} = 210.513222336 ; ;
c_{2} = 16.4170210439 ; ; ; ; text{ Factored form: } ; ; (c -210.513222336) (c -16.4170210439) = 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 = 116 ; ; b = 100 ; ; c = 210.51 ; ;

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

p = a+b+c = 116+100+210.51 = 426.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 426.51 }{ 2 } = 213.26 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 213.26 * (213.26-116)(213.26-100)(213.26-210.51) } ; ; T = sqrt{ 6444252.75 } = 2538.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2538.55 }{ 116 } = 43.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2538.55 }{ 100 } = 50.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2538.55 }{ 210.51 } = 24.12 ; ;

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{ 100**2+210.51**2-116**2 }{ 2 * 100 * 210.51 } ) = 13° 57'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 116**2+210.51**2-100**2 }{ 2 * 116 * 210.51 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 13° 57'22" - 12° = 154° 2'38" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2538.55 }{ 213.26 } = 11.9 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 116 }{ 2 * sin 13° 57'22" } = 240.49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 210.51**2 - 116**2 } }{ 2 } = 154.253 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 210.51**2+2 * 116**2 - 100**2 } }{ 2 } = 162.437 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 116**2 - 210.51**2 } }{ 2 } = 25.476 ; ;







#2 Obtuse scalene triangle.

Sides: a = 116   b = 100   c = 16.4177021039

Area: T = 197.971085494
Perimeter: p = 232.4177021039
Semiperimeter: s = 116.2098510519

Angle ∠ A = α = 166.0443948599° = 166°2'38″ = 2.89880136061 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 1.95660514012° = 1°57'22″ = 0.03441395373 rad

Height: ha = 3.41332906024
Height: hb = 3.95994170988
Height: hc = 24.11877561349

Median: ma = 42.08803908002
Median: mb = 66.05111868924
Median: mc = 107.9844352362

Inradius: r = 1.70435831029
Circumradius: R = 240.4876717237

Vertex coordinates: A[16.4177021039; 0] B[0; 0] C[113.4655121685; 24.11877561349]
Centroid: CG[43.29440475747; 8.0399252045]
Coordinates of the circumscribed circle: U[8.20985105195; 240.3476586251]
Coordinates of the inscribed circle: I[16.20985105195; 1.70435831029]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 13.95660514011° = 13°57'22″ = 2.89880136061 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 178.0443948599° = 178°2'38″ = 0.03441395373 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 116 ; ; b = 100 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 100**2 = 116**2 + c**2 -2 * 116 * c * cos (12° ) ; ; ; ; c**2 -226.93c +3456 =0 ; ; p=1; q=-226.93; r=3456 ; ; D = q**2 - 4pr = 226.93**2 - 4 * 1 * 3456 = 37673.3353561 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 226.93 ± sqrt{ 37673.34 } }{ 2 } ; ; c_{1,2} = 113.46512169 ± 97.0481006461 ; ; c_{1} = 210.513222336 ; ; : Nr. 1
c_{2} = 16.4170210439 ; ; ; ; text{ Factored form: } ; ; (c -210.513222336) (c -16.4170210439) = 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 = 116 ; ; b = 100 ; ; c = 16.42 ; ;

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

p = a+b+c = 116+100+16.42 = 232.42 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 232.42 }{ 2 } = 116.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.21 * (116.21-116)(116.21-100)(116.21-16.42) } ; ; T = sqrt{ 39192.46 } = 197.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 197.97 }{ 116 } = 3.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 197.97 }{ 100 } = 3.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 197.97 }{ 16.42 } = 24.12 ; ;

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{ 100**2+16.42**2-116**2 }{ 2 * 100 * 16.42 } ) = 166° 2'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 116**2+16.42**2-100**2 }{ 2 * 116 * 16.42 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 166° 2'38" - 12° = 1° 57'22" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 197.97 }{ 116.21 } = 1.7 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 116 }{ 2 * sin 166° 2'38" } = 240.49 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 16.42**2 - 116**2 } }{ 2 } = 42.08 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.42**2+2 * 116**2 - 100**2 } }{ 2 } = 66.051 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 116**2 - 16.42**2 } }{ 2 } = 107.984 ; ;
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