Triangle calculator SSA

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Triangle has two solutions with side c=37.16441666581 and with side c=8.72107256113

#1 Acute scalene triangle.

Sides: a = 29.81   b = 23.76   c = 37.16441666581

Area: T = 353.6855081735
Perimeter: p = 90.73441666581
Semiperimeter: s = 45.36770833291

Angle ∠ A = α = 53.23333611883° = 53°14' = 0.92990974246 rad
Angle ∠ B = β = 39.68° = 39°40'48″ = 0.69325466472 rad
Angle ∠ C = γ = 87.08766388117° = 87°5'12″ = 1.52199485818 rad

Height: ha = 23.72992909584
Height: hb = 29.77114715266
Height: hc = 19.03436613754

Median: ma = 27.39988579451
Median: mb = 31.52441382388
Median: mc = 19.52767260223

Inradius: r = 7.79660727422
Circumradius: R = 18.60661311597

Vertex coordinates: A[37.16441666581; 0] B[0; 0] C[22.94224461347; 19.03436613754]
Centroid: CG[20.03655375976; 6.34545537918]
Coordinates of the circumscribed circle: U[18.58220833291; 0.94656721855]
Coordinates of the inscribed circle: I[21.60770833291; 7.79660727422]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.7676638812° = 126°46' = 0.92990974246 rad
∠ B' = β' = 140.32° = 140°19'12″ = 0.69325466472 rad
∠ C' = γ' = 92.91333611883° = 92°54'48″ = 1.52199485818 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 = 29.81 ; ; b = 23.76 ; ; c = 37.16 ; ;

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

p = a+b+c = 29.81+23.76+37.16 = 90.73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.73 }{ 2 } = 45.37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.37 * (45.37-29.81)(45.37-23.76)(45.37-37.16) } ; ; T = sqrt{ 125093.14 } = 353.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 353.69 }{ 29.81 } = 23.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 353.69 }{ 23.76 } = 29.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 353.69 }{ 37.16 } = 19.03 ; ;

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{ 29.81**2-23.76**2-37.16**2 }{ 2 * 23.76 * 37.16 } ) = 53° 14' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23.76**2-29.81**2-37.16**2 }{ 2 * 29.81 * 37.16 } ) = 39° 40'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 37.16**2-29.81**2-23.76**2 }{ 2 * 23.76 * 29.81 } ) = 87° 5'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 353.69 }{ 45.37 } = 7.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 29.81 }{ 2 * sin 53° 14' } = 18.61 ; ;





#2 Obtuse scalene triangle.

Sides: a = 29.81   b = 23.76   c = 8.72107256113

Area: T = 82.99436691172
Perimeter: p = 62.29107256113
Semiperimeter: s = 31.14553628057

Angle ∠ A = α = 126.7676638812° = 126°46' = 2.2122495229 rad
Angle ∠ B = β = 39.68° = 39°40'48″ = 0.69325466472 rad
Angle ∠ C = γ = 13.55333611883° = 13°33'12″ = 0.23765507774 rad

Height: ha = 5.56881763916
Height: hb = 6.98659990839
Height: hc = 19.03436613754

Median: ma = 9.90663263925
Median: mb = 18.47218482452
Median: mc = 26.66002647769

Inradius: r = 2.66547199339
Circumradius: R = 18.60661311597

Vertex coordinates: A[8.72107256113; 0] B[0; 0] C[22.94224461347; 19.03436613754]
Centroid: CG[10.5544390582; 6.34545537918]
Coordinates of the circumscribed circle: U[4.36603628057; 18.08879891899]
Coordinates of the inscribed circle: I[7.38553628057; 2.66547199339]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 53.23333611883° = 53°14' = 2.2122495229 rad
∠ B' = β' = 140.32° = 140°19'12″ = 0.69325466472 rad
∠ C' = γ' = 166.4476638812° = 166°26'48″ = 0.23765507774 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 29.81 ; ; b = 23.76 ; ; beta = 39° 40'48" ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 23.76**2 = 29.81**2 + c**2 -2 * 23.76 * c * cos (39° 40'48") ; ; ; ; c**2 -45.885c +324.099 =0 ; ; p=1; q=-45.8848922695; r=324.0985 ; ; D = q**2 - 4pr = 45.885**2 - 4 * 1 * 324.099 = 809.029338582 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.88 ± sqrt{ 809.03 } }{ 2 } ; ; c_{1,2} = 22.9424461347 ± 14.2217205234 ; ;
c_{1} = 37.1641666581 ; ; c_{2} = 8.72072561135 ; ; ; ; (c -37.1641666581) (c -8.72072561135) = 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 = 29.81 ; ; b = 23.76 ; ; c = 8.72 ; ;

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

p = a+b+c = 29.81+23.76+8.72 = 62.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.29 }{ 2 } = 31.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.15 * (31.15-29.81)(31.15-23.76)(31.15-8.72) } ; ; T = sqrt{ 6887.95 } = 82.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.99 }{ 29.81 } = 5.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.99 }{ 23.76 } = 6.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.99 }{ 8.72 } = 19.03 ; ;

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{ 29.81**2-23.76**2-8.72**2 }{ 2 * 23.76 * 8.72 } ) = 126° 46' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23.76**2-29.81**2-8.72**2 }{ 2 * 29.81 * 8.72 } ) = 39° 40'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.72**2-29.81**2-23.76**2 }{ 2 * 23.76 * 29.81 } ) = 13° 33'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.99 }{ 31.15 } = 2.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 29.81 }{ 2 * sin 126° 46' } = 18.61 ; ;




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