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?

1. Use Law of Cosines

a = 37.7 ; ; b = 35.3 ; ; beta = 36° 12' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 35.3**2 = 37.7**2 + c**2 -2 * 37.7 * c * cos (36° 12') ; ; ; ; c**2 -60.845c +175.2 =0 ; ; p=1; q=-60.845; 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} = 30.42240377 ± 27.3920180164 ; ; c_{1} = 57.8144217842 ; ; c_{2} = 3.03038575139 ; ; ; ; text{ Factored form: } ; ; (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 = 57.81 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 35.3**2+57.81**2-37.7**2 }{ 2 * 35.3 * 57.81 } ) = 39° 6'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.7**2+57.81**2-35.3**2 }{ 2 * 37.7 * 57.81 } ) = 36° 12' ; ;
 gamma = 180° - alpha - beta = 180° - 39° 6'22" - 36° 12' = 104° 41'38" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.3**2+2 * 57.81**2 - 37.7**2 } }{ 2 } = 44.034 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 57.81**2+2 * 37.7**2 - 35.3**2 } }{ 2 } = 45.501 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.3**2+2 * 37.7**2 - 57.81**2 } }{ 2 } = 22.317 ; ;



#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

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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 - 2ac cos beta ; ; 35.3**2 = 37.7**2 + c**2 -2 * 37.7 * c * cos (36° 12') ; ; ; ; c**2 -60.845c +175.2 =0 ; ; p=1; q=-60.845; r=175.2 ; ; D = q**2 - 4pr = 60.845**2 - 4 * 1 * 175.2 = 3001.29060405 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.84 ± sqrt{ 3001.29 } }{ 2 } ; ; c_{1,2} = 30.42240377 ± 27.3920180164 ; ; c_{1} = 57.8144217842 ; ; c_{2} = 3.03038575139 ; ; ; ; text{ Factored form: } ; ; (c -57.8144217842) (c -3.03038575139) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 35.3**2+3.03**2-37.7**2 }{ 2 * 35.3 * 3.03 } ) = 140° 53'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.7**2+3.03**2-35.3**2 }{ 2 * 37.7 * 3.03 } ) = 36° 12' ; ;
 gamma = 180° - alpha - beta = 180° - 140° 53'38" - 36° 12' = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.3**2+2 * 3.03**2 - 37.7**2 } }{ 2 } = 16.502 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.03**2+2 * 37.7**2 - 35.3**2 } }{ 2 } = 20.093 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.3**2+2 * 37.7**2 - 3.03**2 } }{ 2 } = 36.488 ; ;
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