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?

1. Use Law of Cosines

a = 12.4 ; ; b = 8.7 ; ; beta = 37° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.7**2 = 12.4**2 + c**2 -2 * 12.4 * c * cos (37° ) ; ; ; ; c**2 -19.806c +78.07 =0 ; ; p=1; q=-19.806; 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} = 9.90308032 ± 4.47224774752 ; ; c_{1} = 14.3753280675 ; ;
c_{2} = 5.43083257248 ; ; ; ; (c -14.3753280675) (c -5.43083257248) = 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 = 14.38 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-14.38**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 8.7**2 = 12.4**2 + c**2 -2 * 12.4 * c * cos (37° ) ; ; ; ; c**2 -19.806c +78.07 =0 ; ; p=1; q=-19.806; 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} = 9.90308032 ± 4.47224774752 ; ; c_{1} = 14.3753280675 ; ; : Nr. 1
c_{2} = 5.43083257248 ; ; ; ; (c -14.3753280675) (c -5.43083257248) = 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 = 12.4 ; ; 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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