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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 116 ; ; b = 100 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 100**2 = 116**2 + c**2 -2 * 100 * c * cos (12° ) ; ; ; ; c**2 -226.93c +3456 =0 ; ; p=1; q=-226.93024337; 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} = 113.465121685 ± 97.0481006461 ; ; c_{1} = 210.513222331 ; ;
c_{2} = 16.417021039 ; ; ; ; (c -210.513222331) (c -16.417021039) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 116**2-100**2-16.42**2 }{ 2 * 100 * 16.42 } ) = 166° 2'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-116**2-16.42**2 }{ 2 * 116 * 16.42 } ) = 12° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.42**2-116**2-100**2 }{ 2 * 100 * 116 } ) = 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 ; ;




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