Triangle calculator SSA

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Triangle has two solutions with side c=15.21552242202 and with side c=6.03442521853

#1 Obtuse scalene triangle.

Sides: a = 10.95   b = 5.3   c = 15.21552242202

Area: T = 20.15329049721
Perimeter: p = 31.46552242202
Semiperimeter: s = 15.73326121101

Angle ∠ A = α = 29.98880764935° = 29°59'17″ = 0.52333906712 rad
Angle ∠ B = β = 14° = 0.24443460953 rad
Angle ∠ C = γ = 136.0121923506° = 136°43″ = 2.37438558872 rad

Height: ha = 3.68108958853
Height: hb = 7.60548698008
Height: hc = 2.64990447568

Median: ma = 9.99110409385
Median: mb = 12.98876970258
Median: mc = 4.01550327499

Inradius: r = 1.2810963697
Circumradius: R = 10.95439485603

Vertex coordinates: A[15.21552242202; 0] B[0; 0] C[10.62547382027; 2.64990447568]
Centroid: CG[8.61333208076; 0.88330149189]
Coordinates of the circumscribed circle: U[7.60876121101; -7.88111945188]
Coordinates of the inscribed circle: I[10.43326121101; 1.2810963697]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0121923506° = 150°43″ = 0.52333906712 rad
∠ B' = β' = 166° = 0.24443460953 rad
∠ C' = γ' = 43.98880764935° = 43°59'17″ = 2.37438558872 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 10.95 ; ; b = 5.3 ; ; beta = 14° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.3**2 = 10.95**2 + c**2 -2 * 10.95 * c * cos (14° ) ; ; ; ; c**2 -21.249c +91.813 =0 ; ; p=1; q=-21.249; r=91.813 ; ; D = q**2 - 4pr = 21.249**2 - 4 * 1 * 91.813 = 84.2902475055 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 21.25 ± sqrt{ 84.29 } }{ 2 } ; ; c_{1,2} = 10.6247382 ± 4.59048601745 ; ; c_{1} = 15.2152242174 ; ;
c_{2} = 6.03425218255 ; ; ; ; text{ Factored form: } ; ; (c -15.2152242174) (c -6.03425218255) = 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 = 10.95 ; ; b = 5.3 ; ; c = 15.22 ; ;

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

p = a+b+c = 10.95+5.3+15.22 = 31.47 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.47 }{ 2 } = 15.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.73 * (15.73-10.95)(15.73-5.3)(15.73-15.22) } ; ; T = sqrt{ 406.14 } = 20.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.15 }{ 10.95 } = 3.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.15 }{ 5.3 } = 7.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.15 }{ 15.22 } = 2.65 ; ;

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{ 5.3**2+15.22**2-10.95**2 }{ 2 * 5.3 * 15.22 } ) = 29° 59'17" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.95**2+15.22**2-5.3**2 }{ 2 * 10.95 * 15.22 } ) = 14° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 10.95**2+5.3**2-15.22**2 }{ 2 * 10.95 * 5.3 } ) = 136° 43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.15 }{ 15.73 } = 1.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.95 }{ 2 * sin 29° 59'17" } = 10.95 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 15.22**2 - 10.95**2 } }{ 2 } = 9.991 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.22**2+2 * 10.95**2 - 5.3**2 } }{ 2 } = 12.988 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 10.95**2 - 15.22**2 } }{ 2 } = 4.015 ; ;







#2 Obtuse scalene triangle.

Sides: a = 10.95   b = 5.3   c = 6.03442521853

Area: T = 7.99325020564
Perimeter: p = 22.28442521853
Semiperimeter: s = 11.14221260926

Angle ∠ A = α = 150.0121923506° = 150°43″ = 2.61882019824 rad
Angle ∠ B = β = 14° = 0.24443460953 rad
Angle ∠ C = γ = 15.98880764935° = 15°59'17″ = 0.27990445759 rad

Height: ha = 1.46598177272
Height: hb = 3.01660385118
Height: hc = 2.64990447568

Median: ma = 1.50884676721
Median: mb = 8.43441478359
Median: mc = 8.05656315793

Inradius: r = 0.7177322887
Circumradius: R = 10.95439485603

Vertex coordinates: A[6.03442521853; 0] B[0; 0] C[10.62547382027; 2.64990447568]
Centroid: CG[5.5532996796; 0.88330149189]
Coordinates of the circumscribed circle: U[3.01771260926; 10.53302392756]
Coordinates of the inscribed circle: I[5.84221260926; 0.7177322887]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.98880764935° = 29°59'17″ = 2.61882019824 rad
∠ B' = β' = 166° = 0.24443460953 rad
∠ C' = γ' = 164.0121923506° = 164°43″ = 0.27990445759 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 10.95 ; ; b = 5.3 ; ; beta = 14° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.3**2 = 10.95**2 + c**2 -2 * 10.95 * c * cos (14° ) ; ; ; ; c**2 -21.249c +91.813 =0 ; ; p=1; q=-21.249; r=91.813 ; ; D = q**2 - 4pr = 21.249**2 - 4 * 1 * 91.813 = 84.2902475055 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 21.25 ± sqrt{ 84.29 } }{ 2 } ; ; c_{1,2} = 10.6247382 ± 4.59048601745 ; ; c_{1} = 15.2152242174 ; ; : Nr. 1
c_{2} = 6.03425218255 ; ; ; ; text{ Factored form: } ; ; (c -15.2152242174) (c -6.03425218255) = 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 = 10.95 ; ; b = 5.3 ; ; c = 6.03 ; ;

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

p = a+b+c = 10.95+5.3+6.03 = 22.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.28 }{ 2 } = 11.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.14 * (11.14-10.95)(11.14-5.3)(11.14-6.03) } ; ; T = sqrt{ 63.88 } = 7.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.99 }{ 10.95 } = 1.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.99 }{ 5.3 } = 3.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.99 }{ 6.03 } = 2.65 ; ;

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{ 5.3**2+6.03**2-10.95**2 }{ 2 * 5.3 * 6.03 } ) = 150° 43" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.95**2+6.03**2-5.3**2 }{ 2 * 10.95 * 6.03 } ) = 14° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 10.95**2+5.3**2-6.03**2 }{ 2 * 10.95 * 5.3 } ) = 15° 59'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.99 }{ 11.14 } = 0.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.95 }{ 2 * sin 150° 43" } = 10.95 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 6.03**2 - 10.95**2 } }{ 2 } = 1.508 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.03**2+2 * 10.95**2 - 5.3**2 } }{ 2 } = 8.434 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.3**2+2 * 10.95**2 - 6.03**2 } }{ 2 } = 8.056 ; ;
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