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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 201 ; ; b = 126 ; ; beta = 12° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 126**2 = 201**2 + c**2 -2 * 126 * c * cos (12° ) ; ; ; ; c**2 -393.215c +24525 =0 ; ; p=1; q=-393.215335495; 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} = 196.607667747 ± 118.867888923 ; ;
c_{1} = 315.475556671 ; ; c_{2} = 77.7397788241 ; ; ; ; (c -315.475556671) (c -77.7397788241) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 201**2-126**2-77.74**2 }{ 2 * 126 * 77.74 } ) = 160° 37'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 126**2-201**2-77.74**2 }{ 2 * 201 * 77.74 } ) = 12° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 77.74**2-201**2-126**2 }{ 2 * 126 * 201 } ) = 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 ; ;




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