Triangle calculator SSA

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Triangle has two solutions with side c=69.10109804877 and with side c=3.95107398893

#1 Obtuse scalene triangle.

Sides: a = 47   b = 44   c = 69.10109804877

Area: T = 1021.936641747
Perimeter: p = 160.1010980488
Semiperimeter: s = 80.05504902438

Angle ∠ A = α = 42.23993030744° = 42°14'21″ = 0.73772149124 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 98.76106969256° = 98°45'39″ = 1.72436993329 rad

Height: ha = 43.48766560624
Height: hb = 46.45216553394
Height: hc = 29.57880583793

Median: ma = 52.94554696096
Median: mb = 54.84549883962
Median: mc = 29.64439475089

Inradius: r = 12.76661481442
Circumradius: R = 34.95883460394

Vertex coordinates: A[69.10109804877; 0] B[0; 0] C[36.52658601885; 29.57880583793]
Centroid: CG[35.20989468921; 9.85993527931]
Coordinates of the circumscribed circle: U[34.55504902438; -5.3244432526]
Coordinates of the inscribed circle: I[36.05504902438; 12.76661481442]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.7610696926° = 137°45'39″ = 0.73772149124 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 81.23993030744° = 81°14'21″ = 1.72436993329 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 47 ; ; b = 44 ; ; beta = 39° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 44**2 = 47**2 + c**2 -2 * 47 * c * cos (39° ) ; ; ; ; c**2 -73.052c +273 =0 ; ; p=1; q=-73.052; r=273 ; ; D = q**2 - 4pr = 73.052**2 - 4 * 1 * 273 = 4244.55385003 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 73.05 ± sqrt{ 4244.55 } }{ 2 } ; ; c_{1,2} = 36.52586019 ± 32.5751202992 ; ; c_{1} = 69.1009804892 ; ;
c_{2} = 3.95073989079 ; ; ; ; text{ Factored form: } ; ; (c -69.1009804892) (c -3.95073989079) = 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 = 47 ; ; b = 44 ; ; c = 69.1 ; ;

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

p = a+b+c = 47+44+69.1 = 160.1 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 160.1 }{ 2 } = 80.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 80.05 * (80.05-47)(80.05-44)(80.05-69.1) } ; ; T = sqrt{ 1044354.04 } = 1021.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1021.94 }{ 47 } = 43.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1021.94 }{ 44 } = 46.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1021.94 }{ 69.1 } = 29.58 ; ;

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{ 44**2+69.1**2-47**2 }{ 2 * 44 * 69.1 } ) = 42° 14'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+69.1**2-44**2 }{ 2 * 47 * 69.1 } ) = 39° ; ; gamma = 180° - alpha - beta = 180° - 42° 14'21" - 39° = 98° 45'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1021.94 }{ 80.05 } = 12.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 47 }{ 2 * sin 42° 14'21" } = 34.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 69.1**2 - 47**2 } }{ 2 } = 52.945 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.1**2+2 * 47**2 - 44**2 } }{ 2 } = 54.845 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 47**2 - 69.1**2 } }{ 2 } = 29.644 ; ;







#2 Obtuse scalene triangle.

Sides: a = 47   b = 44   c = 3.95107398893

Area: T = 58.42876075431
Perimeter: p = 94.95107398893
Semiperimeter: s = 47.47553699446

Angle ∠ A = α = 137.7610696926° = 137°45'39″ = 2.40443777412 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 3.23993030744° = 3°14'21″ = 0.05765365041 rad

Height: ha = 2.4866281172
Height: hb = 2.65658003429
Height: hc = 29.57880583793

Median: ma = 20.58804317942
Median: mb = 25.06659963464
Median: mc = 45.48218415808

Inradius: r = 1.23106930438
Circumradius: R = 34.95883460394

Vertex coordinates: A[3.95107398893; 0] B[0; 0] C[36.52658601885; 29.57880583793]
Centroid: CG[13.49222000259; 9.85993527931]
Coordinates of the circumscribed circle: U[1.97553699446; 34.90224909053]
Coordinates of the inscribed circle: I[3.47553699446; 1.23106930438]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.23993030744° = 42°14'21″ = 2.40443777412 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 176.7610696926° = 176°45'39″ = 0.05765365041 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 47 ; ; b = 44 ; ; beta = 39° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 44**2 = 47**2 + c**2 -2 * 47 * c * cos (39° ) ; ; ; ; c**2 -73.052c +273 =0 ; ; p=1; q=-73.052; r=273 ; ; D = q**2 - 4pr = 73.052**2 - 4 * 1 * 273 = 4244.55385003 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 73.05 ± sqrt{ 4244.55 } }{ 2 } ; ; c_{1,2} = 36.52586019 ± 32.5751202992 ; ; c_{1} = 69.1009804892 ; ; : Nr. 1
c_{2} = 3.95073989079 ; ; ; ; text{ Factored form: } ; ; (c -69.1009804892) (c -3.95073989079) = 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 = 47 ; ; b = 44 ; ; c = 3.95 ; ;

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

p = a+b+c = 47+44+3.95 = 94.95 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 94.95 }{ 2 } = 47.48 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 47.48 * (47.48-47)(47.48-44)(47.48-3.95) } ; ; T = sqrt{ 3413.79 } = 58.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.43 }{ 47 } = 2.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.43 }{ 44 } = 2.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.43 }{ 3.95 } = 29.58 ; ;

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{ 44**2+3.95**2-47**2 }{ 2 * 44 * 3.95 } ) = 137° 45'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+3.95**2-44**2 }{ 2 * 47 * 3.95 } ) = 39° ; ; gamma = 180° - alpha - beta = 180° - 137° 45'39" - 39° = 3° 14'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.43 }{ 47.48 } = 1.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 47 }{ 2 * sin 137° 45'39" } = 34.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 3.95**2 - 47**2 } }{ 2 } = 20.58 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.95**2+2 * 47**2 - 44**2 } }{ 2 } = 25.066 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 47**2 - 3.95**2 } }{ 2 } = 45.482 ; ;
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