Triangle calculator SSA

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Triangle has two solutions with side c=15.39220292999 and with side c=4.78110459925

#1 Acute scalene triangle.

Sides: a = 12.8   b = 9.5   c = 15.39220292999

Area: T = 60.64881885889
Perimeter: p = 37.69220292999
Semiperimeter: s = 18.84660146499

Angle ∠ A = α = 56.05497522588° = 56°2'59″ = 0.97882527218 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 85.95502477412° = 85°57'1″ = 1.5500114816 rad

Height: ha = 9.4766279467
Height: hb = 12.76880397029
Height: hc = 7.88804668842

Median: ma = 11.07334946148
Median: mb = 13.33547209564
Median: mc = 8.23550688223

Inradius: r = 3.21880909182
Circumradius: R = 7.7155278916

Vertex coordinates: A[15.39220292999; 0] B[0; 0] C[10.08765376462; 7.88804668842]
Centroid: CG[8.49328556487; 2.62768222947]
Coordinates of the circumscribed circle: U[7.69660146499; 0.54548736187]
Coordinates of the inscribed circle: I[9.34660146499; 3.21880909182]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.9550247741° = 123°57'1″ = 0.97882527218 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 94.05497522588° = 94°2'59″ = 1.5500114816 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 = 12.8 ; ; b = 9.5 ; ; c = 15.39 ; ;

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

p = a+b+c = 12.8+9.5+15.39 = 37.69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.69 }{ 2 } = 18.85 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.85 * (18.85-12.8)(18.85-9.5)(18.85-15.39) } ; ; T = sqrt{ 3678.2 } = 60.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.65 }{ 12.8 } = 9.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.65 }{ 9.5 } = 12.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.65 }{ 15.39 } = 7.88 ; ;

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{ 12.8**2-9.5**2-15.39**2 }{ 2 * 9.5 * 15.39 } ) = 56° 2'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.5**2-12.8**2-15.39**2 }{ 2 * 12.8 * 15.39 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.39**2-12.8**2-9.5**2 }{ 2 * 9.5 * 12.8 } ) = 85° 57'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.65 }{ 18.85 } = 3.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 56° 2'59" } = 7.72 ; ;





#2 Obtuse scalene triangle.

Sides: a = 12.8   b = 9.5   c = 4.78110459925

Area: T = 18.83884373076
Perimeter: p = 27.08110459925
Semiperimeter: s = 13.54105229962

Angle ∠ A = α = 123.9550247741° = 123°57'1″ = 2.16333399317 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 18.05497522588° = 18°2'59″ = 0.31550276061 rad

Height: ha = 2.94435058293
Height: hb = 3.96659868016
Height: hc = 7.88804668842

Median: ma = 3.94989492768
Median: mb = 8.41334832496
Median: mc = 11.01550079348

Inradius: r = 1.39112636397
Circumradius: R = 7.7155278916

Vertex coordinates: A[4.78110459925; 0] B[0; 0] C[10.08765376462; 7.88804668842]
Centroid: CG[4.95658612129; 2.62768222947]
Coordinates of the circumscribed circle: U[2.39105229962; 7.33655932655]
Coordinates of the inscribed circle: I[4.04105229962; 1.39112636397]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.05497522588° = 56°2'59″ = 2.16333399317 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 161.9550247741° = 161°57'1″ = 0.31550276061 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.8 ; ; b = 9.5 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 9.5**2 = 12.8**2 + c**2 -2 * 9.5 * c * cos (38° ) ; ; ; ; c**2 -20.173c +73.59 =0 ; ; p=1; q=-20.1730752923; r=73.59 ; ; D = q**2 - 4pr = 20.173**2 - 4 * 1 * 73.59 = 112.59296675 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 20.17 ± sqrt{ 112.59 } }{ 2 } ; ; c_{1,2} = 10.0865376462 ± 5.3054916537 ; ; c_{1} = 15.3920292999 ; ;
c_{2} = 4.78104599246 ; ; ; ; (c -15.3920292999) (c -4.78104599246) = 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 = 12.8 ; ; b = 9.5 ; ; c = 4.78 ; ;

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

p = a+b+c = 12.8+9.5+4.78 = 27.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.08 }{ 2 } = 13.54 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.54 * (13.54-12.8)(13.54-9.5)(13.54-4.78) } ; ; T = sqrt{ 354.89 } = 18.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.84 }{ 12.8 } = 2.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.84 }{ 9.5 } = 3.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.84 }{ 4.78 } = 7.88 ; ;

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{ 12.8**2-9.5**2-4.78**2 }{ 2 * 9.5 * 4.78 } ) = 123° 57'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.5**2-12.8**2-4.78**2 }{ 2 * 12.8 * 4.78 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.78**2-12.8**2-9.5**2 }{ 2 * 9.5 * 12.8 } ) = 18° 2'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.84 }{ 13.54 } = 1.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 123° 57'1" } = 7.72 ; ;




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