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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 4.1**2 = 5.1**2 + c**2 -2 * 4.1 * c * cos (40° ) ; ; ; ; c**2 -7.814c +9.2 =0 ; ; p=1; q=-7.81365331981; 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} = 3.90682665991 ± 2.46237579394 ; ; c_{1} = 6.36920245384 ; ;
c_{2} = 1.44445086597 ; ; ; ; (c -6.36920245384) (c -1.44445086597) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5.1**2-4.1**2-1.44**2 }{ 2 * 4.1 * 1.44 } ) = 126° 54'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.1**2-5.1**2-1.44**2 }{ 2 * 5.1 * 1.44 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.44**2-5.1**2-4.1**2 }{ 2 * 4.1 * 5.1 } ) = 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 ; ;




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