Triangle calculator SSA

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Triangle has two solutions with side c=5.73222225734 and with side c=1.32440937355

#1 Acute scalene triangle.

Sides: a = 12.8   b = 12.5   c = 5.73222225734

Area: T = 35.26550623514
Perimeter: p = 31.03222225734
Semiperimeter: s = 15.51661112867

Angle ∠ A = α = 79.84442169226° = 79°50'39″ = 1.3943544474 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 26.15657830774° = 26°9'21″ = 0.45765045331 rad

Height: ha = 5.51101659924
Height: hb = 5.64224099762
Height: hc = 12.3044149708

Median: ma = 7.32108051344
Median: mb = 7.76997849201
Median: mc = 12.32219481452

Inradius: r = 2.27328028757
Circumradius: R = 6.50218714741

Vertex coordinates: A[5.73222225734; 0] B[0; 0] C[3.52881581545; 12.3044149708]
Centroid: CG[3.0876793576; 4.1011383236]
Coordinates of the circumscribed circle: U[2.86661112867; 5.8366072203]
Coordinates of the inscribed circle: I[3.01661112867; 2.27328028757]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.1565783077° = 100°9'21″ = 1.3943544474 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 153.8444216923° = 153°50'39″ = 0.45765045331 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 = 12.5 ; ; c = 5.73 ; ;

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

p = a+b+c = 12.8+12.5+5.73 = 31.03 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.03 }{ 2 } = 15.52 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.52 * (15.52-12.8)(15.52-12.5)(15.52-5.73) } ; ; T = sqrt{ 1243.62 } = 35.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35.27 }{ 12.8 } = 5.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35.27 }{ 12.5 } = 5.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.27 }{ 5.73 } = 12.3 ; ;

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**2-12.5**2-5.73**2 }{ 2 * 12.5 * 5.73 } ) = 79° 50'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.5**2-12.8**2-5.73**2 }{ 2 * 12.8 * 5.73 } ) = 74° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.73**2-12.8**2-12.5**2 }{ 2 * 12.5 * 12.8 } ) = 26° 9'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.27 }{ 15.52 } = 2.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 79° 50'39" } = 6.5 ; ;





#2 Obtuse scalene triangle.

Sides: a = 12.8   b = 12.5   c = 1.32440937355

Area: T = 8.14659237746
Perimeter: p = 26.62440937355
Semiperimeter: s = 13.31220468678

Angle ∠ A = α = 100.1565783077° = 100°9'21″ = 1.74880481796 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 5.84442169226° = 5°50'39″ = 0.10220008275 rad

Height: ha = 1.27328005898
Height: hb = 1.30333478039
Height: hc = 12.3044149708

Median: ma = 6.16877882673
Median: mb = 6.6133177157
Median: mc = 12.63435542879

Inradius: r = 0.61219212061
Circumradius: R = 6.50218714741

Vertex coordinates: A[1.32440937355; 0] B[0; 0] C[3.52881581545; 12.3044149708]
Centroid: CG[1.61774172967; 4.1011383236]
Coordinates of the circumscribed circle: U[0.66220468678; 6.4688077505]
Coordinates of the inscribed circle: I[0.81220468678; 0.61219212061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 79.84442169226° = 79°50'39″ = 1.74880481796 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 174.1565783077° = 174°9'21″ = 0.10220008275 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.8 ; ; b = 12.5 ; ; beta = 74° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 12.5**2 = 12.8**2 + c**2 -2 * 12.5 * c * cos (74° ) ; ; ; ; c**2 -7.056c +7.59 =0 ; ; p=1; q=-7.05631630892; r=7.59 ; ; D = q**2 - 4pr = 7.056**2 - 4 * 1 * 7.59 = 19.4315998515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7.06 ± sqrt{ 19.43 } }{ 2 } ; ; c_{1,2} = 3.52815815446 ± 2.20406441895 ; ; c_{1} = 5.7322225734 ; ;
c_{2} = 1.32409373551 ; ; ; ; (c -5.7322225734) (c -1.32409373551) = 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 = 12.5 ; ; c = 1.32 ; ;

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

p = a+b+c = 12.8+12.5+1.32 = 26.62 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.62 }{ 2 } = 13.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.31 * (13.31-12.8)(13.31-12.5)(13.31-1.32) } ; ; T = sqrt{ 66.36 } = 8.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.15 }{ 12.8 } = 1.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.15 }{ 12.5 } = 1.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.15 }{ 1.32 } = 12.3 ; ;

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**2-12.5**2-1.32**2 }{ 2 * 12.5 * 1.32 } ) = 100° 9'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.5**2-12.8**2-1.32**2 }{ 2 * 12.8 * 1.32 } ) = 74° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.32**2-12.8**2-12.5**2 }{ 2 * 12.5 * 12.8 } ) = 5° 50'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.15 }{ 13.31 } = 0.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 100° 9'21" } = 6.5 ; ;




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