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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 12.6**2 = 13.88**2 + c**2 -2 * 12.6 * c * cos (27° ) ; ; ; ; c**2 -24.734c +33.894 =0 ; ; p=1; q=-24.7343411115; r=33.8944 ; ; 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} = 12.3671705557 ± 10.911118529 ; ;
c_{1} = 23.2782890848 ; ; c_{2} = 1.4560520267 ; ; ; ; (c -23.2782890848) (c -1.4560520267) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13.88**2-12.6**2-1.46**2 }{ 2 * 12.6 * 1.46 } ) = 149° 59'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.6**2-13.88**2-1.46**2 }{ 2 * 13.88 * 1.46 } ) = 27° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.46**2-13.88**2-12.6**2 }{ 2 * 12.6 * 13.88 } ) = 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 ; ;




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