Triangle calculator SSA

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Triangle has two solutions with side c=6.36992024538 and with side c=1.4444450866

#1 Acute scalene triangle.

Sides: a = 5.1   b = 4.1   c = 6.36992024538

Area: T = 10.44398132733
Perimeter: p = 15.56992024538
Semiperimeter: s = 7.78546012269

Angle ∠ A = α = 53.08985901677° = 53°5'19″ = 0.92765706937 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 86.91114098323° = 86°54'41″ = 1.51768902591 rad

Height: ha = 4.09440444209
Height: hb = 5.09325918407
Height: hc = 3.27882168094

Median: ma = 4.71101878889
Median: mb = 5.39331317385
Median: mc = 3.35768310987

Inradius: r = 1.34110851717
Circumradius: R = 3.18992338451

Vertex coordinates: A[6.36992024538; 0] B[0; 0] C[3.90768266599; 3.27882168094]
Centroid: CG[3.42553430379; 1.09327389365]
Coordinates of the circumscribed circle: U[3.18546012269; 0.17218358053]
Coordinates of the inscribed circle: I[3.68546012269; 1.34110851717]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.9111409832° = 126°54'41″ = 0.92765706937 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 93.08985901677° = 93°5'19″ = 1.51768902591 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 5.1 ; ; b = 4.1 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.1**2 = 5.1**2 + c**2 -2 * 5.1 * c * cos (40° ) ; ; ; ; c**2 -7.814c +9.2 =0 ; ; p=1; q=-7.814; r=9.2 ; ; D = q**2 - 4pr = 7.814**2 - 4 * 1 * 9.2 = 24.2531782022 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7.81 ± sqrt{ 24.25 } }{ 2 } ; ; c_{1,2} = 3.90682666 ± 2.46237579394 ; ; c_{1} = 6.36920245394 ; ;
c_{2} = 1.44445086606 ; ; ; ; text{ Factored form: } ; ; (c -6.36920245394) (c -1.44445086606) = 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 = 5.1 ; ; b = 4.1 ; ; c = 6.37 ; ;

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

p = a+b+c = 5.1+4.1+6.37 = 15.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.57 }{ 2 } = 7.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.78 * (7.78-5.1)(7.78-4.1)(7.78-6.37) } ; ; T = sqrt{ 108.99 } = 10.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.44 }{ 5.1 } = 4.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.44 }{ 4.1 } = 5.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.44 }{ 6.37 } = 3.28 ; ;

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{ 4.1**2+6.37**2-5.1**2 }{ 2 * 4.1 * 6.37 } ) = 53° 5'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.1**2+6.37**2-4.1**2 }{ 2 * 5.1 * 6.37 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 53° 5'19" - 40° = 86° 54'41" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.44 }{ 7.78 } = 1.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.1 }{ 2 * sin 53° 5'19" } = 3.19 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.1**2+2 * 6.37**2 - 5.1**2 } }{ 2 } = 4.71 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.37**2+2 * 5.1**2 - 4.1**2 } }{ 2 } = 5.393 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.1**2+2 * 5.1**2 - 6.37**2 } }{ 2 } = 3.357 ; ;







#2 Obtuse scalene triangle.

Sides: a = 5.1   b = 4.1   c = 1.4444450866

Area: T = 2.36876115546
Perimeter: p = 10.6444450866
Semiperimeter: s = 5.3222225433

Angle ∠ A = α = 126.9111409832° = 126°54'41″ = 2.21550219599 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 13.08985901677° = 13°5'19″ = 0.22884389929 rad

Height: ha = 0.92884751194
Height: hb = 1.15549324657
Height: hc = 3.27882168094

Median: ma = 1.71663097483
Median: mb = 3.13877888954
Median: mc = 4.57703818685

Inradius: r = 0.44548536772
Circumradius: R = 3.18992338451

Vertex coordinates: A[1.4444450866; 0] B[0; 0] C[3.90768266599; 3.27882168094]
Centroid: CG[1.78437591753; 1.09327389365]
Coordinates of the circumscribed circle: U[0.7222225433; 3.10663810041]
Coordinates of the inscribed circle: I[1.2222225433; 0.44548536772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 53.08985901677° = 53°5'19″ = 2.21550219599 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 166.9111409832° = 166°54'41″ = 0.22884389929 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 5.1 ; ; b = 4.1 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.1**2 = 5.1**2 + c**2 -2 * 5.1 * c * cos (40° ) ; ; ; ; c**2 -7.814c +9.2 =0 ; ; p=1; q=-7.814; r=9.2 ; ; D = q**2 - 4pr = 7.814**2 - 4 * 1 * 9.2 = 24.2531782022 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7.81 ± sqrt{ 24.25 } }{ 2 } ; ; c_{1,2} = 3.90682666 ± 2.46237579394 ; ; c_{1} = 6.36920245394 ; ; : Nr. 1
c_{2} = 1.44445086606 ; ; ; ; text{ Factored form: } ; ; (c -6.36920245394) (c -1.44445086606) = 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 = 5.1 ; ; b = 4.1 ; ; c = 1.44 ; ;

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

p = a+b+c = 5.1+4.1+1.44 = 10.64 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.64 }{ 2 } = 5.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.32 * (5.32-5.1)(5.32-4.1)(5.32-1.44) } ; ; T = sqrt{ 5.61 } = 2.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.37 }{ 5.1 } = 0.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.37 }{ 4.1 } = 1.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.37 }{ 1.44 } = 3.28 ; ;

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{ 4.1**2+1.44**2-5.1**2 }{ 2 * 4.1 * 1.44 } ) = 126° 54'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.1**2+1.44**2-4.1**2 }{ 2 * 5.1 * 1.44 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 126° 54'41" - 40° = 13° 5'19" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.37 }{ 5.32 } = 0.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.1 }{ 2 * sin 126° 54'41" } = 3.19 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.1**2+2 * 1.44**2 - 5.1**2 } }{ 2 } = 1.716 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.44**2+2 * 5.1**2 - 4.1**2 } }{ 2 } = 3.138 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.1**2+2 * 5.1**2 - 1.44**2 } }{ 2 } = 4.57 ; ;
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