Triangle calculator SSA

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Triangle has two solutions with side c=110.533016496 and with side c=70.35218356535

#1 Obtuse scalene triangle.

Sides: a = 99   b = 45   c = 110.533016496

Area: T = 2225.355507859
Perimeter: p = 254.533016496
Semiperimeter: s = 127.265508248

Angle ∠ A = α = 63.48553976237° = 63°29'7″ = 1.10880292155 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 92.51546023763° = 92°30'53″ = 1.61546844176 rad

Height: ha = 44.95766682544
Height: hb = 98.90546701596
Height: hc = 40.26769276645

Median: ma = 68.34325832334
Median: mb = 102.4832723827
Median: mc = 53.46774729017

Inradius: r = 17.48659830774
Circumradius: R = 55.31883500504

Vertex coordinates: A[110.533016496; 0] B[0; 0] C[90.44110003066; 40.26769276645]
Centroid: CG[66.99903884221; 13.42223092215]
Coordinates of the circumscribed circle: U[55.26550824799; -2.42770374517]
Coordinates of the inscribed circle: I[82.26550824799; 17.48659830774]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5154602376° = 116°30'53″ = 1.10880292155 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 87.48553976237° = 87°29'7″ = 1.61546844176 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 = 99 ; ; b = 45 ; ; c = 110.53 ; ;

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

p = a+b+c = 99+45+110.53 = 254.53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 254.53 }{ 2 } = 127.27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 127.27 * (127.27-99)(127.27-45)(127.27-110.53) } ; ; T = sqrt{ 4952205.23 } = 2225.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2225.36 }{ 99 } = 44.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2225.36 }{ 45 } = 98.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2225.36 }{ 110.53 } = 40.27 ; ;

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{ 99**2-45**2-110.53**2 }{ 2 * 45 * 110.53 } ) = 63° 29'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-99**2-110.53**2 }{ 2 * 99 * 110.53 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 110.53**2-99**2-45**2 }{ 2 * 45 * 99 } ) = 92° 30'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2225.36 }{ 127.27 } = 17.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 99 }{ 2 * sin 63° 29'7" } = 55.32 ; ;





#2 Obtuse scalene triangle.

Sides: a = 99   b = 45   c = 70.35218356535

Area: T = 1416.426613866
Perimeter: p = 214.3521835653
Semiperimeter: s = 107.1765917827

Angle ∠ A = α = 116.5154602376° = 116°30'53″ = 2.03435634381 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 39.48553976237° = 39°29'7″ = 0.6899150195 rad

Height: ha = 28.61546694679
Height: hb = 62.95222728294
Height: hc = 40.26769276645

Median: ma = 32.20215588117
Median: mb = 82.87990708798
Median: mc = 68.3798759897

Inradius: r = 13.21658993119
Circumradius: R = 55.31883500504

Vertex coordinates: A[70.35218356535; 0] B[0; 0] C[90.44110003066; 40.26769276645]
Centroid: CG[53.59876119867; 13.42223092215]
Coordinates of the circumscribed circle: U[35.17659178267; 42.69439651163]
Coordinates of the inscribed circle: I[62.17659178267; 13.21658993119]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.48553976237° = 63°29'7″ = 2.03435634381 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 140.5154602376° = 140°30'53″ = 0.6899150195 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 99 ; ; b = 45 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 45**2 = 99**2 + c**2 -2 * 45 * c * cos (24° ) ; ; ; ; c**2 -180.882c +7776 =0 ; ; p=1; q=-180.882000613; r=7776 ; ; D = q**2 - 4pr = 180.882**2 - 4 * 1 * 7776 = 1614.29814585 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 180.88 ± sqrt{ 1614.3 } }{ 2 } ; ; c_{1,2} = 90.4410003066 ± 20.0891646532 ; ; c_{1} = 110.53016496 ; ;
c_{2} = 70.3518356535 ; ; ; ; (c -110.53016496) (c -70.3518356535) = 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 = 99 ; ; b = 45 ; ; c = 70.35 ; ;

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

p = a+b+c = 99+45+70.35 = 214.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 214.35 }{ 2 } = 107.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 107.18 * (107.18-99)(107.18-45)(107.18-70.35) } ; ; T = sqrt{ 2006263.01 } = 1416.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1416.43 }{ 99 } = 28.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1416.43 }{ 45 } = 62.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1416.43 }{ 70.35 } = 40.27 ; ;

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{ 99**2-45**2-70.35**2 }{ 2 * 45 * 70.35 } ) = 116° 30'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-99**2-70.35**2 }{ 2 * 99 * 70.35 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 70.35**2-99**2-45**2 }{ 2 * 45 * 99 } ) = 39° 29'7" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1416.43 }{ 107.18 } = 13.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 99 }{ 2 * sin 116° 30'53" } = 55.32 ; ;




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