Triangle calculator SSA

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Triangle has two solutions with side c=29.30217716025 and with side c=2.6610931259

#1 Obtuse scalene triangle.

Sides: a = 18.1   b = 15.8   c = 29.30217716025

Area: T = 124.4954953985
Perimeter: p = 63.20217716025
Semiperimeter: s = 31.60108858013

Angle ∠ A = α = 32.53548405162° = 32°32'5″ = 0.56878400886 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 119.4655159484° = 119°27'55″ = 2.08550603744 rad

Height: ha = 13.75663485066
Height: hb = 15.75988549348
Height: hc = 8.49774352864

Median: ma = 21.73304949213
Median: mb = 23.03767512797
Median: mc = 8.60109618787

Inradius: r = 3.94396033
Circumradius: R = 16.82774302987

Vertex coordinates: A[29.30217716025; 0] B[0; 0] C[15.98113514307; 8.49774352864]
Centroid: CG[15.09443743444; 2.83224784288]
Coordinates of the circumscribed circle: U[14.65108858013; -8.27773157301]
Coordinates of the inscribed circle: I[15.80108858013; 3.94396033]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.4655159484° = 147°27'55″ = 0.56878400886 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 60.53548405162° = 60°32'5″ = 2.08550603744 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 18.1 ; ; b = 15.8 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 15.8**2 = 18.1**2 + c**2 -2 * 18.1 * c * cos (28° ) ; ; ; ; c**2 -31.963c +77.97 =0 ; ; p=1; q=-31.963; r=77.97 ; ; D = q**2 - 4pr = 31.963**2 - 4 * 1 * 77.97 = 709.734374212 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 31.96 ± sqrt{ 709.73 } }{ 2 } ; ; c_{1,2} = 15.98135143 ± 13.3204201718 ; ; c_{1} = 29.3017716018 ; ;
c_{2} = 2.66093125821 ; ; ; ; text{ Factored form: } ; ; (c -29.3017716018) (c -2.66093125821) = 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 = 18.1 ; ; b = 15.8 ; ; c = 29.3 ; ;

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

p = a+b+c = 18.1+15.8+29.3 = 63.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63.2 }{ 2 } = 31.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.6 * (31.6-18.1)(31.6-15.8)(31.6-29.3) } ; ; T = sqrt{ 15498.99 } = 124.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 124.49 }{ 18.1 } = 13.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 124.49 }{ 15.8 } = 15.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 124.49 }{ 29.3 } = 8.5 ; ;

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{ 15.8**2+29.3**2-18.1**2 }{ 2 * 15.8 * 29.3 } ) = 32° 32'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.1**2+29.3**2-15.8**2 }{ 2 * 18.1 * 29.3 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 32° 32'5" - 28° = 119° 27'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 124.49 }{ 31.6 } = 3.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.1 }{ 2 * sin 32° 32'5" } = 16.83 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.8**2+2 * 29.3**2 - 18.1**2 } }{ 2 } = 21.73 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.3**2+2 * 18.1**2 - 15.8**2 } }{ 2 } = 23.037 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.8**2+2 * 18.1**2 - 29.3**2 } }{ 2 } = 8.601 ; ;







#2 Obtuse scalene triangle.

Sides: a = 18.1   b = 15.8   c = 2.6610931259

Area: T = 11.30655455873
Perimeter: p = 36.5610931259
Semiperimeter: s = 18.28804656295

Angle ∠ A = α = 147.4655159484° = 147°27'55″ = 2.5743752565 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 4.53548405162° = 4°32'5″ = 0.07991478981 rad

Height: ha = 1.24992315566
Height: hb = 1.43110817199
Height: hc = 8.49774352864

Median: ma = 6.81659942475
Median: mb = 10.24437921485
Median: mc = 16.93767901684

Inradius: r = 0.61884495415
Circumradius: R = 16.82774302987

Vertex coordinates: A[2.6610931259; 0] B[0; 0] C[15.98113514307; 8.49774352864]
Centroid: CG[6.21440942299; 2.83224784288]
Coordinates of the circumscribed circle: U[1.33304656295; 16.77547510165]
Coordinates of the inscribed circle: I[2.48804656295; 0.61884495415]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 32.53548405162° = 32°32'5″ = 2.5743752565 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 175.4655159484° = 175°27'55″ = 0.07991478981 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 18.1 ; ; b = 15.8 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 15.8**2 = 18.1**2 + c**2 -2 * 18.1 * c * cos (28° ) ; ; ; ; c**2 -31.963c +77.97 =0 ; ; p=1; q=-31.963; r=77.97 ; ; D = q**2 - 4pr = 31.963**2 - 4 * 1 * 77.97 = 709.734374212 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 31.96 ± sqrt{ 709.73 } }{ 2 } ; ; c_{1,2} = 15.98135143 ± 13.3204201718 ; ; c_{1} = 29.3017716018 ; ; : Nr. 1
c_{2} = 2.66093125821 ; ; ; ; text{ Factored form: } ; ; (c -29.3017716018) (c -2.66093125821) = 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 = 18.1 ; ; b = 15.8 ; ; c = 2.66 ; ;

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

p = a+b+c = 18.1+15.8+2.66 = 36.56 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.56 }{ 2 } = 18.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.28 * (18.28-18.1)(18.28-15.8)(18.28-2.66) } ; ; T = sqrt{ 127.82 } = 11.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.31 }{ 18.1 } = 1.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.31 }{ 15.8 } = 1.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.31 }{ 2.66 } = 8.5 ; ;

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{ 15.8**2+2.66**2-18.1**2 }{ 2 * 15.8 * 2.66 } ) = 147° 27'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.1**2+2.66**2-15.8**2 }{ 2 * 18.1 * 2.66 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 147° 27'55" - 28° = 4° 32'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.31 }{ 18.28 } = 0.62 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.1 }{ 2 * sin 147° 27'55" } = 16.83 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.8**2+2 * 2.66**2 - 18.1**2 } }{ 2 } = 6.816 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.66**2+2 * 18.1**2 - 15.8**2 } }{ 2 } = 10.244 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.8**2+2 * 18.1**2 - 2.66**2 } }{ 2 } = 16.937 ; ;
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