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?

1. Use Law of Cosines

a = 153 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 153**2 + c**2 -2 * 153 * c * cos (30° ) ; ; ; ; c**2 -265.004c +15309 =0 ; ; p=1; q=-265.004; 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} = 132.50188678 ± 47.4104418878 ; ;
c_{1} = 179.912328668 ; ; c_{2} = 85.0914448922 ; ; ; ; text{ Factored form: } ; ; (c -179.912328668) (c -85.0914448922) = 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 = 179.91 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 90**2+179.91**2-153**2 }{ 2 * 90 * 179.91 } ) = 58° 12'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 153**2+179.91**2-90**2 }{ 2 * 153 * 179.91 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 58° 12'42" - 30° = 91° 47'18" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 179.91**2 - 153**2 } }{ 2 } = 119.925 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 179.91**2+2 * 153**2 - 90**2 } }{ 2 } = 160.822 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 153**2 - 179.91**2 } }{ 2 } = 87.535 ; ;





#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 - 2ac cos beta ; ; 90**2 = 153**2 + c**2 -2 * 153 * c * cos (30° ) ; ; ; ; c**2 -265.004c +15309 =0 ; ; p=1; q=-265.004; 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} = 132.50188678 ± 47.4104418878 ; ; : Nr. 1
c_{1} = 179.912328668 ; ; c_{2} = 85.0914448922 ; ; ; ; text{ Factored form: } ; ; (c -179.912328668) (c -85.0914448922) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+85.09**2-153**2 }{ 2 * 90 * 85.09 } ) = 121° 47'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 153**2+85.09**2-90**2 }{ 2 * 153 * 85.09 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 121° 47'18" - 30° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 85.09**2 - 153**2 } }{ 2 } = 42.638 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 85.09**2+2 * 153**2 - 90**2 } }{ 2 } = 115.325 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 153**2 - 85.09**2 } }{ 2 } = 118.086 ; ;
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