Triangle calculator SSA

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Triangle has two solutions with side c=18.59765254097 and with side c=6.677651603

#1 Obtuse scalene triangle.

Sides: a = 12.9   b = 6.5   c = 18.59765254097

Area: T = 24.11988118209
Perimeter: p = 37.99765254097
Semiperimeter: s = 18.99882627048

Angle ∠ A = α = 23.52195736403° = 23°31'10″ = 0.41104939987 rad
Angle ∠ B = β = 11.6° = 11°36' = 0.20224581932 rad
Angle ∠ C = γ = 144.888042636° = 144°52'50″ = 2.52986404617 rad

Height: ha = 3.73993506699
Height: hb = 7.4211172868
Height: hc = 2.59439051828

Median: ma = 12.34765735593
Median: mb = 15.67702864893
Median: mc = 4.22875655728

Inradius: r = 1.27695272297
Circumradius: R = 16.1632888404

Vertex coordinates: A[18.59765254097; 0] B[0; 0] C[12.63765207198; 2.59439051828]
Centroid: CG[10.41110153765; 0.86546350609]
Coordinates of the circumscribed circle: U[9.29882627048; -13.2220486838]
Coordinates of the inscribed circle: I[12.49882627048; 1.27695272297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.488042636° = 156°28'50″ = 0.41104939987 rad
∠ B' = β' = 168.4° = 168°24' = 0.20224581932 rad
∠ C' = γ' = 35.12195736403° = 35°7'10″ = 2.52986404617 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 12.9 ; ; b = 6.5 ; ; beta = 11° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.5**2 = 12.9**2 + c**2 -2 * 12.9 * c * cos (11° 36') ; ; ; ; c**2 -25.273c +124.16 =0 ; ; p=1; q=-25.273; r=124.16 ; ; D = q**2 - 4pr = 25.273**2 - 4 * 1 * 124.16 = 142.086623611 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.27 ± sqrt{ 142.09 } }{ 2 } ; ; c_{1,2} = 12.63652072 ± 5.96000468982 ; ;
c_{1} = 18.5965254098 ; ; c_{2} = 6.67651603018 ; ; ; ; text{ Factored form: } ; ; (c -18.5965254098) (c -6.67651603018) = 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 = 12.9 ; ; b = 6.5 ; ; c = 18.6 ; ;

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

p = a+b+c = 12.9+6.5+18.6 = 38 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-12.9)(19-6.5)(19-18.6) } ; ; T = sqrt{ 581.72 } = 24.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24.12 }{ 12.9 } = 3.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24.12 }{ 6.5 } = 7.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24.12 }{ 18.6 } = 2.59 ; ;

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{ 6.5**2+18.6**2-12.9**2 }{ 2 * 6.5 * 18.6 } ) = 23° 31'10" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.9**2+18.6**2-6.5**2 }{ 2 * 12.9 * 18.6 } ) = 11° 36' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.9**2+6.5**2-18.6**2 }{ 2 * 12.9 * 6.5 } ) = 144° 52'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24.12 }{ 19 } = 1.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.9 }{ 2 * sin 23° 31'10" } = 16.16 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 18.6**2 - 12.9**2 } }{ 2 } = 12.347 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.6**2+2 * 12.9**2 - 6.5**2 } }{ 2 } = 15.67 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 12.9**2 - 18.6**2 } }{ 2 } = 4.228 ; ;







#2 Obtuse scalene triangle.

Sides: a = 12.9   b = 6.5   c = 6.677651603

Area: T = 8.65991247666
Perimeter: p = 26.077651603
Semiperimeter: s = 13.0388258015

Angle ∠ A = α = 156.488042636° = 156°28'50″ = 2.73110986549 rad
Angle ∠ B = β = 11.6° = 11°36' = 0.20224581932 rad
Angle ∠ C = γ = 11.92195736403° = 11°55'10″ = 0.20880358055 rad

Height: ha = 1.34224999638
Height: hb = 2.6644346082
Height: hc = 2.59439051828

Median: ma = 1.34655233738
Median: mb = 9.74332249871
Median: mc = 9.6533291326

Inradius: r = 0.66441320303
Circumradius: R = 16.1632888404

Vertex coordinates: A[6.677651603; 0] B[0; 0] C[12.63765207198; 2.59439051828]
Centroid: CG[6.43876789166; 0.86546350609]
Coordinates of the circumscribed circle: U[3.3388258015; 15.81443920208]
Coordinates of the inscribed circle: I[6.5388258015; 0.66441320303]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 23.52195736403° = 23°31'10″ = 2.73110986549 rad
∠ B' = β' = 168.4° = 168°24' = 0.20224581932 rad
∠ C' = γ' = 168.088042636° = 168°4'50″ = 0.20880358055 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.9 ; ; b = 6.5 ; ; beta = 11° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.5**2 = 12.9**2 + c**2 -2 * 12.9 * c * cos (11° 36') ; ; ; ; c**2 -25.273c +124.16 =0 ; ; p=1; q=-25.273; r=124.16 ; ; D = q**2 - 4pr = 25.273**2 - 4 * 1 * 124.16 = 142.086623611 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.27 ± sqrt{ 142.09 } }{ 2 } ; ; c_{1,2} = 12.63652072 ± 5.96000468982 ; ; : Nr. 1
c_{1} = 18.5965254098 ; ; c_{2} = 6.67651603018 ; ; ; ; text{ Factored form: } ; ; (c -18.5965254098) (c -6.67651603018) = 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 = 12.9 ; ; b = 6.5 ; ; c = 6.68 ; ;

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

p = a+b+c = 12.9+6.5+6.68 = 26.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.08 }{ 2 } = 13.04 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.04 * (13.04-12.9)(13.04-6.5)(13.04-6.68) } ; ; T = sqrt{ 74.98 } = 8.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.66 }{ 12.9 } = 1.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.66 }{ 6.5 } = 2.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.66 }{ 6.68 } = 2.59 ; ;

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{ 6.5**2+6.68**2-12.9**2 }{ 2 * 6.5 * 6.68 } ) = 156° 28'50" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.9**2+6.68**2-6.5**2 }{ 2 * 12.9 * 6.68 } ) = 11° 36' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 12.9**2+6.5**2-6.68**2 }{ 2 * 12.9 * 6.5 } ) = 11° 55'10" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.66 }{ 13.04 } = 0.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.9 }{ 2 * sin 156° 28'50" } = 16.16 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 6.68**2 - 12.9**2 } }{ 2 } = 1.346 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.68**2+2 * 12.9**2 - 6.5**2 } }{ 2 } = 9.743 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 12.9**2 - 6.68**2 } }{ 2 } = 9.653 ; ;
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