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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 6.5**2 = 12.9**2 + c**2 -2 * 6.5 * c * cos (11° 36') ; ; ; ; c**2 -25.273c +124.16 =0 ; ; p=1; q=-25.2730414397; 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} = 12.6365207198 ± 5.96000468982 ; ;
c_{1} = 18.5965254097 ; ; c_{2} = 6.67651603001 ; ; ; ; (c -18.5965254097) (c -6.67651603001) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12.9**2-6.5**2-6.68**2 }{ 2 * 6.5 * 6.68 } ) = 156° 28'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.5**2-12.9**2-6.68**2 }{ 2 * 12.9 * 6.68 } ) = 11° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.68**2-12.9**2-6.5**2 }{ 2 * 6.5 * 12.9 } ) = 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 ; ;




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