Triangle calculator SSA

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Triangle has two solutions with side c=83.933308416 and with side c=11.05664267867

#1 Obtuse scalene triangle.

Sides: a = 62   b = 54   c = 83.933308416

Area: T = 1672.486554277
Perimeter: p = 199.933308416
Semiperimeter: s = 99.967654208

Angle ∠ A = α = 47.56326317037° = 47°33'45″ = 0.83301245241 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 92.43773682963° = 92°26'15″ = 1.61333364286 rad

Height: ha = 53.95111465408
Height: hb = 61.94439089913
Height: hc = 39.85328318006

Median: ma = 63.39985907439
Median: mb = 68.66986340938
Median: mc = 40.23444298561

Inradius: r = 16.73304530893
Circumradius: R = 42.00545433252

Vertex coordinates: A[83.933308416; 0] B[0; 0] C[47.49547554734; 39.85328318006]
Centroid: CG[43.80992798778; 13.28442772669]
Coordinates of the circumscribed circle: U[41.967654208; -1.78663386599]
Coordinates of the inscribed circle: I[45.967654208; 16.73304530893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.4377368296° = 132°26'15″ = 0.83301245241 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 87.56326317037° = 87°33'45″ = 1.61333364286 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 = 62 ; ; b = 54 ; ; c = 83.93 ; ;

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

p = a+b+c = 62+54+83.93 = 199.93 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 199.93 }{ 2 } = 99.97 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99.97 * (99.97-62)(99.97-54)(99.97-83.93) } ; ; T = sqrt{ 2797207.89 } = 1672.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1672.49 }{ 62 } = 53.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1672.49 }{ 54 } = 61.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1672.49 }{ 83.93 } = 39.85 ; ;

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{ 62**2-54**2-83.93**2 }{ 2 * 54 * 83.93 } ) = 47° 33'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 54**2-62**2-83.93**2 }{ 2 * 62 * 83.93 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 83.93**2-62**2-54**2 }{ 2 * 54 * 62 } ) = 92° 26'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1672.49 }{ 99.97 } = 16.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 62 }{ 2 * sin 47° 33'45" } = 42 ; ;





#2 Obtuse scalene triangle.

Sides: a = 62   b = 54   c = 11.05664267867

Area: T = 220.3154958523
Perimeter: p = 127.0566426787
Semiperimeter: s = 63.52882133934

Angle ∠ A = α = 132.4377368296° = 132°26'15″ = 2.31114681294 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 7.56326317037° = 7°33'45″ = 0.13219928233 rad

Height: ha = 7.10769341459
Height: hb = 8.16598132786
Height: hc = 39.85328318006

Median: ma = 23.62546118835
Median: mb = 35.41435890111
Median: mc = 57.8744336771

Inradius: r = 3.46879860609
Circumradius: R = 42.00545433252

Vertex coordinates: A[11.05664267867; 0] B[0; 0] C[47.49547554734; 39.85328318006]
Centroid: CG[19.51770607534; 13.28442772669]
Coordinates of the circumscribed circle: U[5.52882133934; 41.63991704605]
Coordinates of the inscribed circle: I[9.52882133934; 3.46879860609]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.56326317037° = 47°33'45″ = 2.31114681294 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 172.4377368296° = 172°26'15″ = 0.13219928233 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 54 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 54**2 = 62**2 + c**2 -2 * 54 * c * cos (40° ) ; ; ; ; c**2 -94.99c +928 =0 ; ; p=1; q=-94.9895109468; r=928 ; ; D = q**2 - 4pr = 94.99**2 - 4 * 1 * 928 = 5311.0071899 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 94.99 ± sqrt{ 5311.01 } }{ 2 } ; ; c_{1,2} = 47.4947554734 ± 36.4383286866 ; ; c_{1} = 83.93308416 ; ;
c_{2} = 11.0564267867 ; ; ; ; (c -83.93308416) (c -11.0564267867) = 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 = 62 ; ; b = 54 ; ; c = 11.06 ; ;

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

p = a+b+c = 62+54+11.06 = 127.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.06 }{ 2 } = 63.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.53 * (63.53-62)(63.53-54)(63.53-11.06) } ; ; T = sqrt{ 48538.68 } = 220.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 220.31 }{ 62 } = 7.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 220.31 }{ 54 } = 8.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 220.31 }{ 11.06 } = 39.85 ; ;

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{ 62**2-54**2-11.06**2 }{ 2 * 54 * 11.06 } ) = 132° 26'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 54**2-62**2-11.06**2 }{ 2 * 62 * 11.06 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.06**2-62**2-54**2 }{ 2 * 54 * 62 } ) = 7° 33'45" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 220.31 }{ 63.53 } = 3.47 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 62 }{ 2 * sin 132° 26'15" } = 42 ; ;




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