Triangle calculator SSA

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Triangle has two solutions with side c=47.58774152732 and with side c=16.23545442711

#1 Acute scalene triangle.

Sides: a = 39.1   b = 27.5   c = 47.58774152732

Area: T = 537.6010576973
Perimeter: p = 114.1877415273
Semiperimeter: s = 57.09437076366

Angle ∠ A = α = 55.2466134099° = 55°14'46″ = 0.9644226939 rad
Angle ∠ B = β = 35.3° = 35°18' = 0.6166101226 rad
Angle ∠ C = γ = 89.4543865901° = 89°27'14″ = 1.56112644886 rad

Height: ha = 27.49987507403
Height: hb = 39.09882237798
Height: hc = 22.59442331134

Median: ma = 33.58987413606
Median: mb = 41.32334019194
Median: mc = 24.0088112731

Inradius: r = 9.41661090465
Circumradius: R = 23.79547885773

Vertex coordinates: A[47.58774152732; 0] B[0; 0] C[31.91109797721; 22.59442331134]
Centroid: CG[26.49994650151; 7.53114110378]
Coordinates of the circumscribed circle: U[23.79437076366; 0.22768046399]
Coordinates of the inscribed circle: I[29.59437076366; 9.41661090465]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.7543865901° = 124°45'14″ = 0.9644226939 rad
∠ B' = β' = 144.7° = 144°42' = 0.6166101226 rad
∠ C' = γ' = 90.5466134099° = 90°32'46″ = 1.56112644886 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 = 39.1 ; ; b = 27.5 ; ; c = 47.59 ; ;

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

p = a+b+c = 39.1+27.5+47.59 = 114.19 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.19 }{ 2 } = 57.09 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.09 * (57.09-39.1)(57.09-27.5)(57.09-47.59) } ; ; T = sqrt{ 289014.38 } = 537.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 537.6 }{ 39.1 } = 27.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 537.6 }{ 27.5 } = 39.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 537.6 }{ 47.59 } = 22.59 ; ;

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{ 39.1**2-27.5**2-47.59**2 }{ 2 * 27.5 * 47.59 } ) = 55° 14'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.5**2-39.1**2-47.59**2 }{ 2 * 39.1 * 47.59 } ) = 35° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 47.59**2-39.1**2-27.5**2 }{ 2 * 27.5 * 39.1 } ) = 89° 27'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 537.6 }{ 57.09 } = 9.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.1 }{ 2 * sin 55° 14'46" } = 23.79 ; ;





#2 Obtuse scalene triangle.

Sides: a = 39.1   b = 27.5   c = 16.23545442711

Area: T = 183.4043538876
Perimeter: p = 82.83545442711
Semiperimeter: s = 41.41772721356

Angle ∠ A = α = 124.7543865901° = 124°45'14″ = 2.17773657146 rad
Angle ∠ B = β = 35.3° = 35°18' = 0.6166101226 rad
Angle ∠ C = γ = 19.9466134099° = 19°56'46″ = 0.34881257131 rad

Height: ha = 9.38112551855
Height: hb = 13.3388439191
Height: hc = 22.59442331134

Median: ma = 11.30105625455
Median: mb = 26.59217790651
Median: mc = 32.81221912264

Inradius: r = 4.42881897242
Circumradius: R = 23.79547885773

Vertex coordinates: A[16.23545442711; 0] B[0; 0] C[31.91109797721; 22.59442331134]
Centroid: CG[16.04985080144; 7.53114110378]
Coordinates of the circumscribed circle: U[8.11772721356; 22.36774284735]
Coordinates of the inscribed circle: I[13.91772721356; 4.42881897242]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 55.2466134099° = 55°14'46″ = 2.17773657146 rad
∠ B' = β' = 144.7° = 144°42' = 0.6166101226 rad
∠ C' = γ' = 160.0543865901° = 160°3'14″ = 0.34881257131 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 39.1 ; ; b = 27.5 ; ; beta = 35° 18' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 27.5**2 = 39.1**2 + c**2 -2 * 27.5 * c * cos (35° 18') ; ; ; ; c**2 -63.822c +772.56 =0 ; ; p=1; q=-63.8219595443; r=772.56 ; ; D = q**2 - 4pr = 63.822**2 - 4 * 1 * 772.56 = 983.00252007 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.82 ± sqrt{ 983 } }{ 2 } ; ; c_{1,2} = 31.9109797721 ± 15.676435501 ; ;
c_{1} = 47.5874152732 ; ; c_{2} = 16.2345442711 ; ; ; ; (c -47.5874152732) (c -16.2345442711) = 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 = 39.1 ; ; b = 27.5 ; ; c = 16.23 ; ;

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

p = a+b+c = 39.1+27.5+16.23 = 82.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.83 }{ 2 } = 41.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.42 * (41.42-39.1)(41.42-27.5)(41.42-16.23) } ; ; T = sqrt{ 33636.86 } = 183.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 183.4 }{ 39.1 } = 9.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 183.4 }{ 27.5 } = 13.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 183.4 }{ 16.23 } = 22.59 ; ;

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{ 39.1**2-27.5**2-16.23**2 }{ 2 * 27.5 * 16.23 } ) = 124° 45'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.5**2-39.1**2-16.23**2 }{ 2 * 39.1 * 16.23 } ) = 35° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.23**2-39.1**2-27.5**2 }{ 2 * 27.5 * 39.1 } ) = 19° 56'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 183.4 }{ 41.42 } = 4.43 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.1 }{ 2 * sin 124° 45'14" } = 23.79 ; ;




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