Triangle calculator SSA

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Triangle has two solutions with side c=179.9122328667 and with side c=85.09114448912

#1 Obtuse scalene triangle.

Sides: a = 153   b = 90   c = 179.9122328667

Area: T = 6881.647657151
Perimeter: p = 422.9122328667
Semiperimeter: s = 211.4566164333

Angle ∠ A = α = 58.21216693829° = 58°12'42″ = 1.01659852938 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 91.78883306171° = 91°47'18″ = 1.60220085842 rad

Height: ha = 89.95661643334
Height: hb = 152.9255479367
Height: hc = 76.5

Median: ma = 119.9254863991
Median: mb = 160.8222022756
Median: mc = 87.53550701057

Inradius: r = 32.54440811489
Circumradius: R = 90

Vertex coordinates: A[179.9122328667; 0] B[0; 0] C[132.5021886779; 76.5]
Centroid: CG[104.1388071815; 25.5]
Coordinates of the circumscribed circle: U[89.95661643334; -2.80986470795]
Coordinates of the inscribed circle: I[121.4566164333; 32.54440811489]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.7888330617° = 121°47'18″ = 1.01659852938 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 88.21216693829° = 88°12'42″ = 1.60220085842 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 = 153 ; ; b = 90 ; ; c = 179.91 ; ;

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

p = a+b+c = 153+90+179.91 = 422.91 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 422.91 }{ 2 } = 211.46 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 211.46 * (211.46-153)(211.46-90)(211.46-179.91) } ; ; T = sqrt{ 47357059.54 } = 6881.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6881.65 }{ 153 } = 89.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6881.65 }{ 90 } = 152.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6881.65 }{ 179.91 } = 76.5 ; ;

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{ 153**2-90**2-179.91**2 }{ 2 * 90 * 179.91 } ) = 58° 12'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-153**2-179.91**2 }{ 2 * 153 * 179.91 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 179.91**2-153**2-90**2 }{ 2 * 90 * 153 } ) = 91° 47'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6881.65 }{ 211.46 } = 32.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 153 }{ 2 * sin 58° 12'42" } = 90 ; ;





#2 Obtuse scalene triangle.

Sides: a = 153   b = 90   c = 85.09114448912

Area: T = 3254.748776709
Perimeter: p = 328.0911444891
Semiperimeter: s = 164.0465722446

Angle ∠ A = α = 121.7888330617° = 121°47'18″ = 2.12656073598 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 28.21216693829° = 28°12'42″ = 0.49223865182 rad

Height: ha = 42.54657224456
Height: hb = 72.32877281575
Height: hc = 76.5

Median: ma = 42.63883277913
Median: mb = 115.3254659101
Median: mc = 118.0866246031

Inradius: r = 19.84404915323
Circumradius: R = 90

Vertex coordinates: A[85.09114448912; 0] B[0; 0] C[132.5021886779; 76.5]
Centroid: CG[72.53111105567; 25.5]
Coordinates of the circumscribed circle: U[42.54657224456; 79.30986470795]
Coordinates of the inscribed circle: I[74.04657224456; 19.84404915323]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.21216693829° = 58°12'42″ = 2.12656073598 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 151.7888330617° = 151°47'18″ = 0.49223865182 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 153 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 153**2 + c**2 -2 * 90 * c * cos (30° ) ; ; ; ; c**2 -265.004c +15309 =0 ; ; p=1; q=-265.003773558; r=15309 ; ; D = q**2 - 4pr = 265.004**2 - 4 * 1 * 15309 = 8991 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 265 ± sqrt{ 8991 } }{ 2 } = fraction{ 265 ± 9 sqrt{ 111 } }{ 2 } ; ; c_{1,2} = 132.501886779 ± 47.4104418878 ; ;
c_{1} = 179.912328667 ; ; c_{2} = 85.0914448912 ; ; ; ; (c -179.912328667) (c -85.0914448912) = 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 = 153 ; ; b = 90 ; ; c = 85.09 ; ;

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

p = a+b+c = 153+90+85.09 = 328.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 328.09 }{ 2 } = 164.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 164.05 * (164.05-153)(164.05-90)(164.05-85.09) } ; ; T = sqrt{ 10593383.03 } = 3254.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3254.75 }{ 153 } = 42.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3254.75 }{ 90 } = 72.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3254.75 }{ 85.09 } = 76.5 ; ;

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{ 153**2-90**2-85.09**2 }{ 2 * 90 * 85.09 } ) = 121° 47'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-153**2-85.09**2 }{ 2 * 153 * 85.09 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 85.09**2-153**2-90**2 }{ 2 * 90 * 153 } ) = 28° 12'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3254.75 }{ 164.05 } = 19.84 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 153 }{ 2 * sin 121° 47'18" } = 90 ; ;




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