Triangle calculator SSA

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Triangle has two solutions with side c=164.5566039836 and with side c=67.07113758728

#1 Acute scalene triangle.

Sides: a = 161   b = 122   c = 164.5566039836

Area: T = 9201.974355379
Perimeter: p = 447.5566039836
Semiperimeter: s = 223.7788019918

Angle ∠ A = α = 66.45113682459° = 66°27'5″ = 1.16597951683 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 69.54986317541° = 69°32'55″ = 1.21438526144 rad

Height: ha = 114.3110230482
Height: hb = 150.8522025472
Height: hc = 111.8439997644

Median: ma = 120.4220492954
Median: mb = 150.9276621652
Median: mc = 116.7659699547

Inradius: r = 41.12109892605
Circumradius: R = 87.81329489172

Vertex coordinates: A[164.5566039836; 0] B[0; 0] C[115.8143707854; 111.8439997644]
Centroid: CG[93.45765825636; 37.28799992146]
Coordinates of the circumscribed circle: U[82.27880199181; 30.6832917656]
Coordinates of the inscribed circle: I[101.7788019918; 41.12109892605]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113.5498631754° = 113°32'55″ = 1.16597951683 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 110.4511368246° = 110°27'5″ = 1.21438526144 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 = 161 ; ; b = 122 ; ; c = 164.56 ; ;

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

p = a+b+c = 161+122+164.56 = 447.56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 447.56 }{ 2 } = 223.78 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 223.78 * (223.78-161)(223.78-122)(223.78-164.56) } ; ; T = sqrt{ 84676317.28 } = 9201.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9201.97 }{ 161 } = 114.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9201.97 }{ 122 } = 150.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9201.97 }{ 164.56 } = 111.84 ; ;

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{ 161**2-122**2-164.56**2 }{ 2 * 122 * 164.56 } ) = 66° 27'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 122**2-161**2-164.56**2 }{ 2 * 161 * 164.56 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 164.56**2-161**2-122**2 }{ 2 * 122 * 161 } ) = 69° 32'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9201.97 }{ 223.78 } = 41.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 161 }{ 2 * sin 66° 27'5" } = 87.81 ; ;





#2 Obtuse scalene triangle.

Sides: a = 161   b = 122   c = 67.07113758728

Area: T = 3750.63112598
Perimeter: p = 350.0711375873
Semiperimeter: s = 175.0365687936

Angle ∠ A = α = 113.5498631754° = 113°32'55″ = 1.98217974852 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 22.45113682459° = 22°27'5″ = 0.39218502975 rad

Height: ha = 46.59216926683
Height: hb = 61.48657583573
Height: hc = 111.8439997644

Median: ma = 56.66659927182
Median: mb = 107.186574873
Median: mc = 138.8454724907

Inradius: r = 21.42878088315
Circumradius: R = 87.81329489172

Vertex coordinates: A[67.07113758728; 0] B[0; 0] C[115.8143707854; 111.8439997644]
Centroid: CG[60.96216945758; 37.28799992146]
Coordinates of the circumscribed circle: U[33.53656879364; 81.15770799879]
Coordinates of the inscribed circle: I[53.03656879364; 21.42878088315]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 66.45113682459° = 66°27'5″ = 1.98217974852 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 157.5498631754° = 157°32'55″ = 0.39218502975 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 161 ; ; b = 122 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 122**2 = 161**2 + c**2 -2 * 122 * c * cos (44° ) ; ; ; ; c**2 -231.627c +11037 =0 ; ; p=1; q=-231.627415709; r=11037 ; ; D = q**2 - 4pr = 231.627**2 - 4 * 1 * 11037 = 9503.25970805 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 231.63 ± sqrt{ 9503.26 } }{ 2 } ; ; c_{1,2} = 115.813707855 ± 48.7423319817 ; ;
c_{1} = 164.556039836 ; ; c_{2} = 67.0713758728 ; ; ; ; (c -164.556039836) (c -67.0713758728) = 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 = 161 ; ; b = 122 ; ; c = 67.07 ; ;

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

p = a+b+c = 161+122+67.07 = 350.07 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 350.07 }{ 2 } = 175.04 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.04 * (175.04-161)(175.04-122)(175.04-67.07) } ; ; T = sqrt{ 14067234.85 } = 3750.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3750.63 }{ 161 } = 46.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3750.63 }{ 122 } = 61.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3750.63 }{ 67.07 } = 111.84 ; ;

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{ 161**2-122**2-67.07**2 }{ 2 * 122 * 67.07 } ) = 113° 32'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 122**2-161**2-67.07**2 }{ 2 * 161 * 67.07 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 67.07**2-161**2-122**2 }{ 2 * 122 * 161 } ) = 22° 27'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3750.63 }{ 175.04 } = 21.43 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 161 }{ 2 * sin 113° 32'55" } = 87.81 ; ;




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