Triangle calculator SSA

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Triangle has two solutions with side c=315.4765556671 and with side c=77.74397788241

#1 Obtuse scalene triangle.

Sides: a = 201   b = 126   c = 315.4765556671

Area: T = 6591.901116811
Perimeter: p = 642.4765556671
Semiperimeter: s = 321.2387778336

Angle ∠ A = α = 19.3770087061° = 19°22'12″ = 0.33880717956 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 148.6329912939° = 148°37'48″ = 2.59440813477 rad

Height: ha = 65.59110563991
Height: hb = 104.6333351875
Height: hc = 41.79902498544

Median: ma = 218.1754616829
Median: mb = 256.8932805326
Median: mc = 57.07327017566

Inradius: r = 20.52203173869
Circumradius: R = 303.0133263719

Vertex coordinates: A[315.4765556671; 0] B[0; 0] C[196.6087667747; 41.79902498544]
Centroid: CG[170.694440814; 13.93300832848]
Coordinates of the circumscribed circle: U[157.7387778335; -258.7219599712]
Coordinates of the inscribed circle: I[195.2387778335; 20.52203173869]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.6329912939° = 160°37'48″ = 0.33880717956 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 31.3770087061° = 31°22'12″ = 2.59440813477 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 201 ; ; b = 126 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 126**2 = 201**2 + c**2 -2 * 201 * c * cos (12° ) ; ; ; ; c**2 -393.215c +24525 =0 ; ; p=1; q=-393.215; r=24525 ; ; D = q**2 - 4pr = 393.215**2 - 4 * 1 * 24525 = 56518.3000684 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 393.22 ± sqrt{ 56518.3 } }{ 2 } ; ; c_{1,2} = 196.60766775 ± 118.867888923 ; ; c_{1} = 315.475556673 ; ; c_{2} = 77.7397788266 ; ; ; ; text{ Factored form: } ; ; (c -315.475556673) (c -77.7397788266) = 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 = 201 ; ; b = 126 ; ; c = 315.48 ; ;

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

p = a+b+c = 201+126+315.48 = 642.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 642.48 }{ 2 } = 321.24 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 321.24 * (321.24-201)(321.24-126)(321.24-315.48) } ; ; T = sqrt{ 43453161.01 } = 6591.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6591.9 }{ 201 } = 65.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6591.9 }{ 126 } = 104.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6591.9 }{ 315.48 } = 41.79 ; ;

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{ 126**2+315.48**2-201**2 }{ 2 * 126 * 315.48 } ) = 19° 22'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 201**2+315.48**2-126**2 }{ 2 * 201 * 315.48 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 19° 22'12" - 12° = 148° 37'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6591.9 }{ 321.24 } = 20.52 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 201 }{ 2 * sin 19° 22'12" } = 303.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 126**2+2 * 315.48**2 - 201**2 } }{ 2 } = 218.175 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 315.48**2+2 * 201**2 - 126**2 } }{ 2 } = 256.893 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 126**2+2 * 201**2 - 315.48**2 } }{ 2 } = 57.073 ; ;







#2 Obtuse scalene triangle.

Sides: a = 201   b = 126   c = 77.74397788241

Area: T = 1624.382239034
Perimeter: p = 404.7439778824
Semiperimeter: s = 202.3769889412

Angle ∠ A = α = 160.6329912939° = 160°37'48″ = 2.8043520858 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 7.37700870611° = 7°22'12″ = 0.12986322854 rad

Height: ha = 16.16330088591
Height: hb = 25.78438474657
Height: hc = 41.79902498544

Median: ma = 29.31770019922
Median: mb = 138.7566032683
Median: mc = 163.1879752718

Inradius: r = 8.0276798824
Circumradius: R = 303.0133263719

Vertex coordinates: A[77.74397788241; 0] B[0; 0] C[196.6087667747; 41.79902498544]
Centroid: CG[91.44991488572; 13.93300832848]
Coordinates of the circumscribed circle: U[38.8769889412; 300.5109849567]
Coordinates of the inscribed circle: I[76.3769889412; 8.0276798824]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.3770087061° = 19°22'12″ = 2.8043520858 rad
∠ B' = β' = 168° = 0.20994395102 rad
∠ C' = γ' = 172.6329912939° = 172°37'48″ = 0.12986322854 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 201 ; ; b = 126 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 126**2 = 201**2 + c**2 -2 * 201 * c * cos (12° ) ; ; ; ; c**2 -393.215c +24525 =0 ; ; p=1; q=-393.215; r=24525 ; ; D = q**2 - 4pr = 393.215**2 - 4 * 1 * 24525 = 56518.3000684 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 393.22 ± sqrt{ 56518.3 } }{ 2 } ; ; c_{1,2} = 196.60766775 ± 118.867888923 ; ; c_{1} = 315.475556673 ; ; c_{2} = 77.7397788266 ; ; ; ; text{ Factored form: } ; ; (c -315.475556673) (c -77.7397788266) = 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 = 201 ; ; b = 126 ; ; c = 77.74 ; ;

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

p = a+b+c = 201+126+77.74 = 404.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 404.74 }{ 2 } = 202.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 202.37 * (202.37-201)(202.37-126)(202.37-77.74) } ; ; T = sqrt{ 2638618.15 } = 1624.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1624.38 }{ 201 } = 16.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1624.38 }{ 126 } = 25.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1624.38 }{ 77.74 } = 41.79 ; ;

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{ 126**2+77.74**2-201**2 }{ 2 * 126 * 77.74 } ) = 160° 37'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 201**2+77.74**2-126**2 }{ 2 * 201 * 77.74 } ) = 12° ; ; gamma = 180° - alpha - beta = 180° - 160° 37'48" - 12° = 7° 22'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1624.38 }{ 202.37 } = 8.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 201 }{ 2 * sin 160° 37'48" } = 303.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 126**2+2 * 77.74**2 - 201**2 } }{ 2 } = 29.317 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.74**2+2 * 201**2 - 126**2 } }{ 2 } = 138.756 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 126**2+2 * 201**2 - 77.74**2 } }{ 2 } = 163.18 ; ;
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