Triangle calculator SSA

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Triangle has two solutions with side c=9.06771430981 and with side c=1.49222010002

#1 Obtuse scalene triangle.

Sides: a = 6.7   b = 5.6   c = 9.06771430981

Area: T = 18.70106738342
Perimeter: p = 21.36771430981
Semiperimeter: s = 10.68435715491

Angle ∠ A = α = 47.44221621532° = 47°26'32″ = 0.82880219338 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 94.55878378468° = 94°33'28″ = 1.6550345604 rad

Height: ha = 5.58222906968
Height: hb = 6.67988120837
Height: hc = 4.12549318847

Median: ma = 6.75501142198
Median: mb = 7.46440164778
Median: mc = 4.1921864622

Inradius: r = 1.75504140585
Circumradius: R = 4.54879538874

Vertex coordinates: A[9.06771430981; 0] B[0; 0] C[5.28796720492; 4.12549318847]
Centroid: CG[4.78222717158; 1.37549772949]
Coordinates of the circumscribed circle: U[4.53435715491; -0.36114049958]
Coordinates of the inscribed circle: I[5.08435715491; 1.75504140585]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.5587837847° = 132°33'28″ = 0.82880219338 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 85.44221621532° = 85°26'32″ = 1.6550345604 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6.7 ; ; b = 5.6 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.6**2 = 6.7**2 + c**2 -2 * 6.7 * c * cos (38° ) ; ; ; ; c**2 -10.559c +13.53 =0 ; ; p=1; q=-10.559; r=13.53 ; ; D = q**2 - 4pr = 10.559**2 - 4 * 1 * 13.53 = 57.3797477869 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 10.56 ± sqrt{ 57.38 } }{ 2 } ; ; c_{1,2} = 5.27967205 ± 3.78747104896 ; ; c_{1} = 9.06714309896 ; ;
c_{2} = 1.49220100104 ; ; ; ; text{ Factored form: } ; ; (c -9.06714309896) (c -1.49220100104) = 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 = 6.7 ; ; b = 5.6 ; ; c = 9.07 ; ;

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

p = a+b+c = 6.7+5.6+9.07 = 21.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.37 }{ 2 } = 10.68 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.68 * (10.68-6.7)(10.68-5.6)(10.68-9.07) } ; ; T = sqrt{ 349.72 } = 18.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.7 }{ 6.7 } = 5.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.7 }{ 5.6 } = 6.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.7 }{ 9.07 } = 4.12 ; ;

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{ 5.6**2+9.07**2-6.7**2 }{ 2 * 5.6 * 9.07 } ) = 47° 26'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.7**2+9.07**2-5.6**2 }{ 2 * 6.7 * 9.07 } ) = 38° ; ; gamma = 180° - alpha - beta = 180° - 47° 26'32" - 38° = 94° 33'28" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.7 }{ 10.68 } = 1.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.7 }{ 2 * sin 47° 26'32" } = 4.55 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.6**2+2 * 9.07**2 - 6.7**2 } }{ 2 } = 6.75 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.07**2+2 * 6.7**2 - 5.6**2 } }{ 2 } = 7.464 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.6**2+2 * 6.7**2 - 9.07**2 } }{ 2 } = 4.192 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.7   b = 5.6   c = 1.49222010002

Area: T = 3.0787613742
Perimeter: p = 13.79222010002
Semiperimeter: s = 6.89661005001

Angle ∠ A = α = 132.5587837847° = 132°33'28″ = 2.31435707198 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 9.44221621532° = 9°26'32″ = 0.16547968181 rad

Height: ha = 0.91986906693
Height: hb = 1.0999147765
Height: hc = 4.12549318847

Median: ma = 2.3660260984
Median: mb = 3.9654635155
Median: mc = 6.12993012688

Inradius: r = 0.44662831918
Circumradius: R = 4.54879538874

Vertex coordinates: A[1.49222010002; 0] B[0; 0] C[5.28796720492; 4.12549318847]
Centroid: CG[2.25772910165; 1.37549772949]
Coordinates of the circumscribed circle: U[0.74661005001; 4.48663368805]
Coordinates of the inscribed circle: I[1.29661005001; 0.44662831918]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.44221621532° = 47°26'32″ = 2.31435707198 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 170.5587837847° = 170°33'28″ = 0.16547968181 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6.7 ; ; b = 5.6 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.6**2 = 6.7**2 + c**2 -2 * 6.7 * c * cos (38° ) ; ; ; ; c**2 -10.559c +13.53 =0 ; ; p=1; q=-10.559; r=13.53 ; ; D = q**2 - 4pr = 10.559**2 - 4 * 1 * 13.53 = 57.3797477869 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 10.56 ± sqrt{ 57.38 } }{ 2 } ; ; c_{1,2} = 5.27967205 ± 3.78747104896 ; ; c_{1} = 9.06714309896 ; ; : Nr. 1
c_{2} = 1.49220100104 ; ; ; ; text{ Factored form: } ; ; (c -9.06714309896) (c -1.49220100104) = 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 = 6.7 ; ; b = 5.6 ; ; c = 1.49 ; ;

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

p = a+b+c = 6.7+5.6+1.49 = 13.79 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.79 }{ 2 } = 6.9 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.9 * (6.9-6.7)(6.9-5.6)(6.9-1.49) } ; ; T = sqrt{ 9.47 } = 3.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.08 }{ 6.7 } = 0.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.08 }{ 5.6 } = 1.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.08 }{ 1.49 } = 4.12 ; ;

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{ 5.6**2+1.49**2-6.7**2 }{ 2 * 5.6 * 1.49 } ) = 132° 33'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.7**2+1.49**2-5.6**2 }{ 2 * 6.7 * 1.49 } ) = 38° ; ; gamma = 180° - alpha - beta = 180° - 132° 33'28" - 38° = 9° 26'32" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.08 }{ 6.9 } = 0.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.7 }{ 2 * sin 132° 33'28" } = 4.55 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.6**2+2 * 1.49**2 - 6.7**2 } }{ 2 } = 2.36 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.49**2+2 * 6.7**2 - 5.6**2 } }{ 2 } = 3.965 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.6**2+2 * 6.7**2 - 1.49**2 } }{ 2 } = 6.129 ; ;
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