Triangle calculator SSA

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Triangle has two solutions with side c=23.27882890848 and with side c=1.45660520267

#1 Obtuse scalene triangle.

Sides: a = 13.88   b = 12.6   c = 23.27882890848

Area: T = 73.3432767337
Perimeter: p = 49.75882890848
Semiperimeter: s = 24.87991445424

Angle ∠ A = α = 30.00772890268° = 30°26″ = 0.52437259931 rad
Angle ∠ B = β = 27° = 0.4711238898 rad
Angle ∠ C = γ = 122.9932710973° = 122°59'34″ = 2.14766277624 rad

Height: ha = 10.56881220947
Height: hb = 11.64217091011
Height: hc = 6.30113881364

Median: ma = 17.38326284364
Median: mb = 18.09990765333
Median: mc = 6.34333046846

Inradius: r = 2.94879617843
Circumradius: R = 13.87769423669

Vertex coordinates: A[23.27882890848; 0] B[0; 0] C[12.36771705557; 6.30113881364]
Centroid: CG[11.88218198802; 2.11004627121]
Coordinates of the circumscribed circle: U[11.63991445424; -7.55664438578]
Coordinates of the inscribed circle: I[12.27991445424; 2.94879617843]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9932710973° = 149°59'34″ = 0.52437259931 rad
∠ B' = β' = 153° = 0.4711238898 rad
∠ C' = γ' = 57.00772890268° = 57°26″ = 2.14766277624 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 13.88 ; ; b = 12.6 ; ; beta = 27° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 12.6**2 = 13.88**2 + c**2 -2 * 13.88 * c * cos (27° ) ; ; ; ; c**2 -24.734c +33.894 =0 ; ; p=1; q=-24.734; r=33.894 ; ; D = q**2 - 4pr = 24.734**2 - 4 * 1 * 33.894 = 476.210030219 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 24.73 ± sqrt{ 476.21 } }{ 2 } ; ; c_{1,2} = 12.36717056 ± 10.911118529 ; ; c_{1} = 23.278289089 ; ; c_{2} = 1.45605203097 ; ; ; ; text{ Factored form: } ; ; (c -23.278289089) (c -1.45605203097) = 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 = 13.88 ; ; b = 12.6 ; ; c = 23.28 ; ;

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

p = a+b+c = 13.88+12.6+23.28 = 49.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49.76 }{ 2 } = 24.88 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.88 * (24.88-13.88)(24.88-12.6)(24.88-23.28) } ; ; T = sqrt{ 5379.16 } = 73.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 73.34 }{ 13.88 } = 10.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 73.34 }{ 12.6 } = 11.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 73.34 }{ 23.28 } = 6.3 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12.6**2+23.28**2-13.88**2 }{ 2 * 12.6 * 23.28 } ) = 30° 26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.88**2+23.28**2-12.6**2 }{ 2 * 13.88 * 23.28 } ) = 27° ; ; gamma = 180° - alpha - beta = 180° - 30° 26" - 27° = 122° 59'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 73.34 }{ 24.88 } = 2.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.88 }{ 2 * sin 30° 26" } = 13.88 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.6**2+2 * 23.28**2 - 13.88**2 } }{ 2 } = 17.383 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.28**2+2 * 13.88**2 - 12.6**2 } }{ 2 } = 18.099 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.6**2+2 * 13.88**2 - 23.28**2 } }{ 2 } = 6.343 ; ;







#2 Obtuse scalene triangle.

Sides: a = 13.88   b = 12.6   c = 1.45660520267

Area: T = 4.58875744835
Perimeter: p = 27.93660520267
Semiperimeter: s = 13.96880260134

Angle ∠ A = α = 149.9932710973° = 149°59'34″ = 2.61878666605 rad
Angle ∠ B = β = 27° = 0.4711238898 rad
Angle ∠ C = γ = 3.00772890268° = 3°26″ = 0.05224870951 rad

Height: ha = 0.66110337872
Height: hb = 0.7288186426
Height: hc = 6.30113881364

Median: ma = 5.68112361113
Median: mb = 7.59658701774
Median: mc = 13.23554515648

Inradius: r = 0.32884339877
Circumradius: R = 13.87769423669

Vertex coordinates: A[1.45660520267; 0] B[0; 0] C[12.36771705557; 6.30113881364]
Centroid: CG[4.60877408608; 2.11004627121]
Coordinates of the circumscribed circle: U[0.72880260134; 13.85878319941]
Coordinates of the inscribed circle: I[1.36880260134; 0.32884339877]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00772890268° = 30°26″ = 2.61878666605 rad
∠ B' = β' = 153° = 0.4711238898 rad
∠ C' = γ' = 176.9932710973° = 176°59'34″ = 0.05224870951 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 13.88 ; ; b = 12.6 ; ; beta = 27° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 12.6**2 = 13.88**2 + c**2 -2 * 13.88 * c * cos (27° ) ; ; ; ; c**2 -24.734c +33.894 =0 ; ; p=1; q=-24.734; r=33.894 ; ; D = q**2 - 4pr = 24.734**2 - 4 * 1 * 33.894 = 476.210030219 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 24.73 ± sqrt{ 476.21 } }{ 2 } ; ; c_{1,2} = 12.36717056 ± 10.911118529 ; ; c_{1} = 23.278289089 ; ; c_{2} = 1.45605203097 ; ; ; ; text{ Factored form: } ; ; (c -23.278289089) (c -1.45605203097) = 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 = 13.88 ; ; b = 12.6 ; ; c = 1.46 ; ;

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

p = a+b+c = 13.88+12.6+1.46 = 27.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.94 }{ 2 } = 13.97 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.97 * (13.97-13.88)(13.97-12.6)(13.97-1.46) } ; ; T = sqrt{ 21.05 } = 4.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.59 }{ 13.88 } = 0.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.59 }{ 12.6 } = 0.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.59 }{ 1.46 } = 6.3 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12.6**2+1.46**2-13.88**2 }{ 2 * 12.6 * 1.46 } ) = 149° 59'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.88**2+1.46**2-12.6**2 }{ 2 * 13.88 * 1.46 } ) = 27° ; ; gamma = 180° - alpha - beta = 180° - 149° 59'34" - 27° = 3° 26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.59 }{ 13.97 } = 0.33 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.88 }{ 2 * sin 149° 59'34" } = 13.88 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.6**2+2 * 1.46**2 - 13.88**2 } }{ 2 } = 5.681 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.46**2+2 * 13.88**2 - 12.6**2 } }{ 2 } = 7.596 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.6**2+2 * 13.88**2 - 1.46**2 } }{ 2 } = 13.235 ; ;
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