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?

1. Use Law of Cosines

a = 161 ; ; b = 122 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 122**2 = 161**2 + c**2 -2 * 161 * c * cos (44° ) ; ; ; ; c**2 -231.627c +11037 =0 ; ; p=1; q=-231.627; 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} = 115.81370785 ± 48.7423319817 ; ; c_{1} = 164.556039832 ; ;
c_{2} = 67.0713758683 ; ; ; ; text{ Factored form: } ; ; (c -164.556039832) (c -67.0713758683) = 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 = 164.56 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 122**2+2 * 164.56**2 - 161**2 } }{ 2 } = 120.42 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 164.56**2+2 * 161**2 - 122**2 } }{ 2 } = 150.927 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 122**2+2 * 161**2 - 164.56**2 } }{ 2 } = 116.76 ; ;







#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 - 2ac cos beta ; ; 122**2 = 161**2 + c**2 -2 * 161 * c * cos (44° ) ; ; ; ; c**2 -231.627c +11037 =0 ; ; p=1; q=-231.627; 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} = 115.81370785 ± 48.7423319817 ; ; c_{1} = 164.556039832 ; ; : Nr. 1
c_{2} = 67.0713758683 ; ; ; ; text{ Factored form: } ; ; (c -164.556039832) (c -67.0713758683) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 122**2+67.07**2-161**2 }{ 2 * 122 * 67.07 } ) = 113° 32'55" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 161**2+67.07**2-122**2 }{ 2 * 161 * 67.07 } ) = 44° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 161**2+122**2-67.07**2 }{ 2 * 161 * 122 } ) = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 122**2+2 * 67.07**2 - 161**2 } }{ 2 } = 56.666 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 67.07**2+2 * 161**2 - 122**2 } }{ 2 } = 107.186 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 122**2+2 * 161**2 - 67.07**2 } }{ 2 } = 138.845 ; ;
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