Triangle calculator SSA

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Triangle has two solutions with side c=14.88109791193 and with side c=0.80330385571

#1 Acute scalene triangle.

Sides: a = 12.2   b = 11.7   c = 14.88109791193

Area: T = 69.53768973111
Perimeter: p = 38.78109791193
Semiperimeter: s = 19.39904895596

Angle ∠ A = α = 53.01438902672° = 53°50″ = 0.92552669345 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 76.98661097328° = 76°59'10″ = 1.34436610931 rad

Height: ha = 11.39994913625
Height: hb = 11.8876649113
Height: hc = 9.34657422061

Median: ma = 11.91545612497
Median: mb = 12.2854920422
Median: mc = 9.35443634264

Inradius: r = 3.58661341766
Circumradius: R = 7.63766326426

Vertex coordinates: A[14.88109791193; 0] B[0; 0] C[7.84220088382; 9.34657422061]
Centroid: CG[7.57443293191; 3.1155247402]
Coordinates of the circumscribed circle: U[7.44404895596; 1.72196724197]
Coordinates of the inscribed circle: I[7.69904895596; 3.58661341766]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.9866109733° = 126°59'10″ = 0.92552669345 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 103.0143890267° = 103°50″ = 1.34436610931 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 12.2 ; ; b = 11.7 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.7**2 = 12.2**2 + c**2 -2 * 12.2 * c * cos (50° ) ; ; ; ; c**2 -15.684c +11.95 =0 ; ; p=1; q=-15.684; r=11.95 ; ; D = q**2 - 4pr = 15.684**2 - 4 * 1 * 11.95 = 198.188410472 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.68 ± sqrt{ 198.19 } }{ 2 } ; ; c_{1,2} = 7.84200884 ± 7.03897028109 ; ; c_{1} = 14.8809791211 ; ; c_{2} = 0.803038558913 ; ; ; ; text{ Factored form: } ; ; (c -14.8809791211) (c -0.803038558913) = 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 = 12.2 ; ; b = 11.7 ; ; c = 14.88 ; ;

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

p = a+b+c = 12.2+11.7+14.88 = 38.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.78 }{ 2 } = 19.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.39 * (19.39-12.2)(19.39-11.7)(19.39-14.88) } ; ; T = sqrt{ 4835.38 } = 69.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.54 }{ 12.2 } = 11.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.54 }{ 11.7 } = 11.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.54 }{ 14.88 } = 9.35 ; ;

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{ 11.7**2+14.88**2-12.2**2 }{ 2 * 11.7 * 14.88 } ) = 53° 50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.2**2+14.88**2-11.7**2 }{ 2 * 12.2 * 14.88 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 53° 50" - 50° = 76° 59'10" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.54 }{ 19.39 } = 3.59 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.2 }{ 2 * sin 53° 50" } = 7.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.7**2+2 * 14.88**2 - 12.2**2 } }{ 2 } = 11.915 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.88**2+2 * 12.2**2 - 11.7**2 } }{ 2 } = 12.285 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.7**2+2 * 12.2**2 - 14.88**2 } }{ 2 } = 9.354 ; ;







#2 Obtuse scalene triangle.

Sides: a = 12.2   b = 11.7   c = 0.80330385571

Area: T = 3.7522495668
Perimeter: p = 24.70330385571
Semiperimeter: s = 12.35215192785

Angle ∠ A = α = 126.9866109733° = 126°59'10″ = 2.21663257191 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 3.01438902672° = 3°50″ = 0.05326023085 rad

Height: ha = 0.61551632243
Height: hb = 0.64114522509
Height: hc = 9.34657422061

Median: ma = 5.61876005075
Median: mb = 6.36655271158
Median: mc = 11.94658688369

Inradius: r = 0.3043808429
Circumradius: R = 7.63766326426

Vertex coordinates: A[0.80330385571; 0] B[0; 0] C[7.84220088382; 9.34657422061]
Centroid: CG[2.88216824651; 3.1155247402]
Coordinates of the circumscribed circle: U[0.40215192785; 7.62660697864]
Coordinates of the inscribed circle: I[0.65215192785; 0.3043808429]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 53.01438902672° = 53°50″ = 2.21663257191 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 176.9866109733° = 176°59'10″ = 0.05326023085 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.2 ; ; b = 11.7 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.7**2 = 12.2**2 + c**2 -2 * 12.2 * c * cos (50° ) ; ; ; ; c**2 -15.684c +11.95 =0 ; ; p=1; q=-15.684; r=11.95 ; ; D = q**2 - 4pr = 15.684**2 - 4 * 1 * 11.95 = 198.188410472 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.68 ± sqrt{ 198.19 } }{ 2 } ; ; c_{1,2} = 7.84200884 ± 7.03897028109 ; ; c_{1} = 14.8809791211 ; ; c_{2} = 0.803038558913 ; ; ; ; text{ Factored form: } ; ; (c -14.8809791211) (c -0.803038558913) = 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 = 12.2 ; ; b = 11.7 ; ; c = 0.8 ; ;

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

p = a+b+c = 12.2+11.7+0.8 = 24.7 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.7 }{ 2 } = 12.35 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.35 * (12.35-12.2)(12.35-11.7)(12.35-0.8) } ; ; T = sqrt{ 14.08 } = 3.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.75 }{ 12.2 } = 0.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.75 }{ 11.7 } = 0.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.75 }{ 0.8 } = 9.35 ; ;

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{ 11.7**2+0.8**2-12.2**2 }{ 2 * 11.7 * 0.8 } ) = 126° 59'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.2**2+0.8**2-11.7**2 }{ 2 * 12.2 * 0.8 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 126° 59'10" - 50° = 3° 50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.75 }{ 12.35 } = 0.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.2 }{ 2 * sin 126° 59'10" } = 7.64 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.7**2+2 * 0.8**2 - 12.2**2 } }{ 2 } = 5.618 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.8**2+2 * 12.2**2 - 11.7**2 } }{ 2 } = 6.366 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.7**2+2 * 12.2**2 - 0.8**2 } }{ 2 } = 11.946 ; ;
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