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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 11.7**2 = 12.2**2 + c**2 -2 * 11.7 * c * cos (50° ) ; ; ; ; c**2 -15.684c +11.95 =0 ; ; p=1; q=-15.6840176764; 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} = 7.84200883818 ± 7.03897028109 ; ;
c_{1} = 14.8809791193 ; ; c_{2} = 0.803038557089 ; ; ; ; (c -14.8809791193) (c -0.803038557089) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12.2**2-11.7**2-0.8**2 }{ 2 * 11.7 * 0.8 } ) = 126° 59'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.7**2-12.2**2-0.8**2 }{ 2 * 12.2 * 0.8 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.8**2-12.2**2-11.7**2 }{ 2 * 11.7 * 12.2 } ) = 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 ; ;




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