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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 39**2 = 57**2 + c**2 -2 * 39 * c * cos (30° ) ; ; ; ; c**2 -98.727c +1728 =0 ; ; p=1; q=-98.7268960314; 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} = 49.3634480157 ± 26.6223590239 ; ;
c_{1} = 75.9858070397 ; ; c_{2} = 22.7410889918 ; ; ; ; (c -75.9858070397) (c -22.7410889918) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 57**2-39**2-22.74**2 }{ 2 * 39 * 22.74 } ) = 133° 2'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39**2-57**2-22.74**2 }{ 2 * 57 * 22.74 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.74**2-57**2-39**2 }{ 2 * 39 * 57 } ) = 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 ; ;




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