Triangle calculator SSA

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Triangle has two solutions with side c=75.98658070397 and with side c=22.74110889918

#1 Obtuse scalene triangle.

Sides: a = 57   b = 39   c = 75.98658070397

Area: T = 1082.798775032
Perimeter: p = 171.986580704
Semiperimeter: s = 85.99329035198

Angle ∠ A = α = 46.95109201998° = 46°57'3″ = 0.81994481443 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 103.04990798° = 103°2'57″ = 1.79985457337 rad

Height: ha = 37.99329035198
Height: hb = 55.52880897598
Height: hc = 28.5

Median: ma = 53.24663279085
Median: mb = 64.27441894989
Median: mc = 30.68545120889

Inradius: r = 12.59217105481
Circumradius: R = 39

Vertex coordinates: A[75.98658070397; 0] B[0; 0] C[49.36334480157; 28.5]
Centroid: CG[41.78330850185; 9.5]
Coordinates of the circumscribed circle: U[37.99329035198; -8.80656392234]
Coordinates of the inscribed circle: I[46.99329035198; 12.59217105481]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.04990798° = 133°2'57″ = 0.81994481443 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 76.95109201998° = 76°57'3″ = 1.79985457337 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 57 ; ; b = 39 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 39**2 = 57**2 + c**2 -2 * 57 * c * cos (30° ) ; ; ; ; c**2 -98.727c +1728 =0 ; ; p=1; q=-98.727; r=1728 ; ; D = q**2 - 4pr = 98.727**2 - 4 * 1 * 1728 = 2835 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 98.73 ± sqrt{ 2835 } }{ 2 } = fraction{ 98.73 ± 9 sqrt{ 35 } }{ 2 } ; ; c_{1,2} = 49.36344802 ± 26.6223590239 ; ; c_{1} = 75.9858070439 ; ; c_{2} = 22.7410889961 ; ; ; ; text{ Factored form: } ; ; (c -75.9858070439) (c -22.7410889961) = 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 = 57 ; ; b = 39 ; ; c = 75.99 ; ;

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

p = a+b+c = 57+39+75.99 = 171.99 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 171.99 }{ 2 } = 85.99 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 85.99 * (85.99-57)(85.99-39)(85.99-75.99) } ; ; T = sqrt{ 1172450.97 } = 1082.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1082.8 }{ 57 } = 37.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1082.8 }{ 39 } = 55.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1082.8 }{ 75.99 } = 28.5 ; ;

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{ 39**2+75.99**2-57**2 }{ 2 * 39 * 75.99 } ) = 46° 57'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 57**2+75.99**2-39**2 }{ 2 * 57 * 75.99 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 46° 57'3" - 30° = 103° 2'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1082.8 }{ 85.99 } = 12.59 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 57 }{ 2 * sin 46° 57'3" } = 39 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 75.99**2 - 57**2 } }{ 2 } = 53.246 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 75.99**2+2 * 57**2 - 39**2 } }{ 2 } = 64.274 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 57**2 - 75.99**2 } }{ 2 } = 30.685 ; ;







#2 Obtuse scalene triangle.

Sides: a = 57   b = 39   c = 22.74110889918

Area: T = 324.0610518133
Perimeter: p = 118.7411088992
Semiperimeter: s = 59.37105444959

Angle ∠ A = α = 133.04990798° = 133°2'57″ = 2.32221445093 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 16.95109201998° = 16°57'3″ = 0.29658493687 rad

Height: ha = 11.37105444959
Height: hb = 16.61884881094
Height: hc = 28.5

Median: ma = 14.38215355323
Median: mb = 38.7666332871
Median: mc = 47.49443230067

Inradius: r = 5.45882709471
Circumradius: R = 39

Vertex coordinates: A[22.74110889918; 0] B[0; 0] C[49.36334480157; 28.5]
Centroid: CG[24.03548456692; 9.5]
Coordinates of the circumscribed circle: U[11.37105444959; 37.30656392234]
Coordinates of the inscribed circle: I[20.37105444959; 5.45882709471]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 46.95109201998° = 46°57'3″ = 2.32221445093 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 163.04990798° = 163°2'57″ = 0.29658493687 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 57 ; ; b = 39 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 39**2 = 57**2 + c**2 -2 * 57 * c * cos (30° ) ; ; ; ; c**2 -98.727c +1728 =0 ; ; p=1; q=-98.727; r=1728 ; ; D = q**2 - 4pr = 98.727**2 - 4 * 1 * 1728 = 2835 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 98.73 ± sqrt{ 2835 } }{ 2 } = fraction{ 98.73 ± 9 sqrt{ 35 } }{ 2 } ; ; c_{1,2} = 49.36344802 ± 26.6223590239 ; ; c_{1} = 75.9858070439 ; ; c_{2} = 22.7410889961 ; ; ; ; text{ Factored form: } ; ; (c -75.9858070439) (c -22.7410889961) = 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 = 57 ; ; b = 39 ; ; c = 22.74 ; ;

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

p = a+b+c = 57+39+22.74 = 118.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.74 }{ 2 } = 59.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.37 * (59.37-57)(59.37-39)(59.37-22.74) } ; ; T = sqrt{ 105015.22 } = 324.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 324.06 }{ 57 } = 11.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 324.06 }{ 39 } = 16.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 324.06 }{ 22.74 } = 28.5 ; ;

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{ 39**2+22.74**2-57**2 }{ 2 * 39 * 22.74 } ) = 133° 2'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 57**2+22.74**2-39**2 }{ 2 * 57 * 22.74 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 133° 2'57" - 30° = 16° 57'3" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 324.06 }{ 59.37 } = 5.46 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 57 }{ 2 * sin 133° 2'57" } = 39 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 22.74**2 - 57**2 } }{ 2 } = 14.382 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.74**2+2 * 57**2 - 39**2 } }{ 2 } = 38.766 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 57**2 - 22.74**2 } }{ 2 } = 47.494 ; ;
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