Triangle calculator SSA

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Triangle has two solutions with side c=14.37553280721 and with side c=5.43108325771

#1 Acute scalene triangle.

Sides: a = 12.4   b = 8.7   c = 14.37553280721

Area: T = 53.63879880585
Perimeter: p = 35.47553280721
Semiperimeter: s = 17.73876640361

Angle ∠ A = α = 59.06659253438° = 59°3'57″ = 1.0310894873 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 83.93440746562° = 83°56'3″ = 1.46549259574 rad

Height: ha = 8.65112883965
Height: hb = 12.33105719675
Height: hc = 7.46325062871

Median: ma = 10.13655823015
Median: mb = 12.76997058466
Median: mc = 7.94111891871

Inradius: r = 3.02439600857
Circumradius: R = 7.22881346139

Vertex coordinates: A[14.37553280721; 0] B[0; 0] C[9.90330803246; 7.46325062871]
Centroid: CG[8.09328027989; 2.48875020957]
Coordinates of the circumscribed circle: U[7.18876640361; 0.76438165364]
Coordinates of the inscribed circle: I[9.03876640361; 3.02439600857]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.9344074656° = 120°56'3″ = 1.0310894873 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 96.06659253438° = 96°3'57″ = 1.46549259574 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 = 12.4 ; ; b = 8.7 ; ; c = 14.38 ; ;

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

p = a+b+c = 12.4+8.7+14.38 = 35.48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.48 }{ 2 } = 17.74 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.74 * (17.74-12.4)(17.74-8.7)(17.74-14.38) } ; ; T = sqrt{ 2877.03 } = 53.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.64 }{ 12.4 } = 8.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.64 }{ 8.7 } = 12.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.64 }{ 14.38 } = 7.46 ; ;

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{ 12.4**2-8.7**2-14.38**2 }{ 2 * 8.7 * 14.38 } ) = 59° 3'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.7**2-12.4**2-14.38**2 }{ 2 * 12.4 * 14.38 } ) = 37° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.38**2-12.4**2-8.7**2 }{ 2 * 8.7 * 12.4 } ) = 83° 56'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.64 }{ 17.74 } = 3.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.4 }{ 2 * sin 59° 3'57" } = 7.23 ; ;





#2 Obtuse scalene triangle.

Sides: a = 12.4   b = 8.7   c = 5.43108325771

Area: T = 20.26438111252
Perimeter: p = 26.53108325771
Semiperimeter: s = 13.26554162885

Angle ∠ A = α = 120.9344074656° = 120°56'3″ = 2.11106977806 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 22.06659253438° = 22°3'57″ = 0.38551230497 rad

Height: ha = 3.26883566331
Height: hb = 4.65883473851
Height: hc = 7.46325062871

Median: ma = 3.76219105837
Median: mb = 8.52766916937
Median: mc = 10.36110575898

Inradius: r = 1.52875669217
Circumradius: R = 7.22881346139

Vertex coordinates: A[5.43108325771; 0] B[0; 0] C[9.90330803246; 7.46325062871]
Centroid: CG[5.11113043006; 2.48875020957]
Coordinates of the circumscribed circle: U[2.71554162885; 6.69986897507]
Coordinates of the inscribed circle: I[4.56554162885; 1.52875669217]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 59.06659253438° = 59°3'57″ = 2.11106977806 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 157.9344074656° = 157°56'3″ = 0.38551230497 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.4 ; ; b = 8.7 ; ; beta = 37° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 8.7**2 = 12.4**2 + c**2 -2 * 8.7 * c * cos (37° ) ; ; ; ; c**2 -19.806c +78.07 =0 ; ; p=1; q=-19.8061606492; r=78.07 ; ; D = q**2 - 4pr = 19.806**2 - 4 * 1 * 78.07 = 80.0039996608 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 19.81 ± sqrt{ 80 } }{ 2 } ; ; c_{1,2} = 9.90308032459 ± 4.47224774752 ; ; c_{1} = 14.3753280721 ; ;
c_{2} = 5.43083257706 ; ; ; ; (c -14.3753280721) (c -5.43083257706) = 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 = 12.4 ; ; b = 8.7 ; ; c = 5.43 ; ;

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

p = a+b+c = 12.4+8.7+5.43 = 26.53 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.53 }{ 2 } = 13.27 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.27 * (13.27-12.4)(13.27-8.7)(13.27-5.43) } ; ; T = sqrt{ 410.62 } = 20.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.26 }{ 12.4 } = 3.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.26 }{ 8.7 } = 4.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.26 }{ 5.43 } = 7.46 ; ;

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{ 12.4**2-8.7**2-5.43**2 }{ 2 * 8.7 * 5.43 } ) = 120° 56'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.7**2-12.4**2-5.43**2 }{ 2 * 12.4 * 5.43 } ) = 37° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.43**2-12.4**2-8.7**2 }{ 2 * 8.7 * 12.4 } ) = 22° 3'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.26 }{ 13.27 } = 1.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.4 }{ 2 * sin 120° 56'3" } = 7.23 ; ;




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