Triangle calculator SSA

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Triangle has two solutions with side c=57.81444217843 and with side c=3.03303857514

#1 Obtuse scalene triangle.

Sides: a = 37.7   b = 35.3   c = 57.81444217843

Area: T = 643.6433149567
Perimeter: p = 130.8144421784
Semiperimeter: s = 65.40772108921

Angle ∠ A = α = 39.10662280994° = 39°6'22″ = 0.68325324384 rad
Angle ∠ B = β = 36.2° = 36°12' = 0.63218091892 rad
Angle ∠ C = γ = 104.6943771901° = 104°41'38″ = 1.8277251026 rad

Height: ha = 34.14655251759
Height: hb = 36.46770339698
Height: hc = 22.26658336693

Median: ma = 44.03438072749
Median: mb = 45.50113866066
Median: mc = 22.31773286582

Inradius: r = 9.841055337
Circumradius: R = 29.88545760677

Vertex coordinates: A[57.81444217843; 0] B[0; 0] C[30.42224037678; 22.26658336693]
Centroid: CG[29.4122275184; 7.42219445564]
Coordinates of the circumscribed circle: U[28.90772108921; -7.58803064044]
Coordinates of the inscribed circle: I[30.10772108921; 9.841055337]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.8943771901° = 140°53'38″ = 0.68325324384 rad
∠ B' = β' = 143.8° = 143°48' = 0.63218091892 rad
∠ C' = γ' = 75.30662280994° = 75°18'22″ = 1.8277251026 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 = 37.7 ; ; b = 35.3 ; ; c = 57.81 ; ;

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

p = a+b+c = 37.7+35.3+57.81 = 130.81 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 130.81 }{ 2 } = 65.41 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.41 * (65.41-37.7)(65.41-35.3)(65.41-57.81) } ; ; T = sqrt{ 414276.5 } = 643.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 643.64 }{ 37.7 } = 34.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 643.64 }{ 35.3 } = 36.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 643.64 }{ 57.81 } = 22.27 ; ;

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{ 37.7**2-35.3**2-57.81**2 }{ 2 * 35.3 * 57.81 } ) = 39° 6'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35.3**2-37.7**2-57.81**2 }{ 2 * 37.7 * 57.81 } ) = 36° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 57.81**2-37.7**2-35.3**2 }{ 2 * 35.3 * 37.7 } ) = 104° 41'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 643.64 }{ 65.41 } = 9.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.7 }{ 2 * sin 39° 6'22" } = 29.88 ; ;





#2 Obtuse scalene triangle.

Sides: a = 37.7   b = 35.3   c = 3.03303857514

Area: T = 33.73770325471
Perimeter: p = 76.03303857514
Semiperimeter: s = 38.01551928757

Angle ∠ A = α = 140.8943771901° = 140°53'38″ = 2.45990602152 rad
Angle ∠ B = β = 36.2° = 36°12' = 0.63218091892 rad
Angle ∠ C = γ = 2.90662280994° = 2°54'22″ = 0.05107232491 rad

Height: ha = 1.79897629998
Height: hb = 1.91114466032
Height: hc = 22.26658336693

Median: ma = 16.50219428826
Median: mb = 20.09326384256
Median: mc = 36.48882746995

Inradius: r = 0.88774618276
Circumradius: R = 29.88545760677

Vertex coordinates: A[3.03303857514; 0] B[0; 0] C[30.42224037678; 22.26658336693]
Centroid: CG[11.15109298397; 7.42219445564]
Coordinates of the circumscribed circle: U[1.51551928757; 29.84661400736]
Coordinates of the inscribed circle: I[2.71551928757; 0.88774618276]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 39.10662280994° = 39°6'22″ = 2.45990602152 rad
∠ B' = β' = 143.8° = 143°48' = 0.63218091892 rad
∠ C' = γ' = 177.0943771901° = 177°5'38″ = 0.05107232491 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 37.7 ; ; b = 35.3 ; ; beta = 36° 12' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 35.3**2 = 37.7**2 + c**2 -2 * 35.3 * c * cos (36° 12') ; ; ; ; c**2 -60.845c +175.2 =0 ; ; p=1; q=-60.8448075356; r=175.2 ; ; D = q**2 - 4pr = 60.845**2 - 4 * 1 * 175.2 = 3001.29060405 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.84 ± sqrt{ 3001.29 } }{ 2 } ; ; c_{1,2} = 30.4224037678 ± 27.3920180164 ; ;
c_{1} = 57.8144217842 ; ; c_{2} = 3.03038575139 ; ; ; ; (c -57.8144217842) (c -3.03038575139) = 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 = 37.7 ; ; b = 35.3 ; ; c = 3.03 ; ;

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

p = a+b+c = 37.7+35.3+3.03 = 76.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.03 }{ 2 } = 38.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.02 * (38.02-37.7)(38.02-35.3)(38.02-3.03) } ; ; T = sqrt{ 1138.19 } = 33.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 33.74 }{ 37.7 } = 1.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 33.74 }{ 35.3 } = 1.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 33.74 }{ 3.03 } = 22.27 ; ;

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{ 37.7**2-35.3**2-3.03**2 }{ 2 * 35.3 * 3.03 } ) = 140° 53'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35.3**2-37.7**2-3.03**2 }{ 2 * 37.7 * 3.03 } ) = 36° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.03**2-37.7**2-35.3**2 }{ 2 * 35.3 * 37.7 } ) = 2° 54'22" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 33.74 }{ 38.02 } = 0.89 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.7 }{ 2 * sin 140° 53'38" } = 29.88 ; ;




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