Triangle calculator SSA

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Triangle has two solutions with side c=10.38435436849 and with side c=4.68552983409

#1 Obtuse scalene triangle.

Sides: a = 8.7   b = 5.2   c = 10.38435436849

Area: T = 22.58442075147
Perimeter: p = 24.28435436849
Semiperimeter: s = 12.14217718425

Angle ∠ A = α = 56.77663748076° = 56°46'35″ = 0.99109346777 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 93.22436251924° = 93°13'25″ = 1.62770592003 rad

Height: ha = 5.19217718425
Height: hb = 8.68662336595
Height: hc = 4.35

Median: ma = 6.96546600583
Median: mb = 9.21992184988
Median: mc = 4.9410698851

Inradius: r = 1.86600421592
Circumradius: R = 5.2

Vertex coordinates: A[10.38435436849; 0] B[0; 0] C[7.53444210129; 4.35]
Centroid: CG[5.97326548993; 1.45]
Coordinates of the circumscribed circle: U[5.19217718425; -0.29224126124]
Coordinates of the inscribed circle: I[6.94217718425; 1.86600421592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.2243625192° = 123°13'25″ = 0.99109346777 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 86.77663748076° = 86°46'35″ = 1.62770592003 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 = 8.7 ; ; b = 5.2 ; ; c = 10.38 ; ;

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

p = a+b+c = 8.7+5.2+10.38 = 24.28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.28 }{ 2 } = 12.14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.14 * (12.14-8.7)(12.14-5.2)(12.14-10.38) } ; ; T = sqrt{ 510.05 } = 22.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.58 }{ 8.7 } = 5.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.58 }{ 5.2 } = 8.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.58 }{ 10.38 } = 4.35 ; ;

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{ 8.7**2-5.2**2-10.38**2 }{ 2 * 5.2 * 10.38 } ) = 56° 46'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.2**2-8.7**2-10.38**2 }{ 2 * 8.7 * 10.38 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.38**2-8.7**2-5.2**2 }{ 2 * 5.2 * 8.7 } ) = 93° 13'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.58 }{ 12.14 } = 1.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.7 }{ 2 * sin 56° 46'35" } = 5.2 ; ;





#2 Obtuse scalene triangle.

Sides: a = 8.7   b = 5.2   c = 4.68552983409

Area: T = 10.19105238916
Perimeter: p = 18.58552983409
Semiperimeter: s = 9.29326491705

Angle ∠ A = α = 123.2243625192° = 123°13'25″ = 2.15106579759 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 26.77663748076° = 26°46'35″ = 0.46773359021 rad

Height: ha = 2.34326491705
Height: hb = 3.9199432266
Height: hc = 4.35

Median: ma = 2.36108283021
Median: mb = 6.48554460349
Median: mc = 6.7733255854

Inradius: r = 1.09766220401
Circumradius: R = 5.2

Vertex coordinates: A[4.68552983409; 0] B[0; 0] C[7.53444210129; 4.35]
Centroid: CG[4.07332397846; 1.45]
Coordinates of the circumscribed circle: U[2.34326491705; 4.64224126124]
Coordinates of the inscribed circle: I[4.09326491705; 1.09766220401]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.77663748076° = 56°46'35″ = 2.15106579759 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 153.2243625192° = 153°13'25″ = 0.46773359021 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.7 ; ; b = 5.2 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 5.2**2 = 8.7**2 + c**2 -2 * 5.2 * c * cos (30° ) ; ; ; ; c**2 -15.069c +48.65 =0 ; ; p=1; q=-15.0688420258; r=48.65 ; ; D = q**2 - 4pr = 15.069**2 - 4 * 1 * 48.65 = 32.47 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.07 ± sqrt{ 32.47 } }{ 2 } ; ; c_{1,2} = 7.53442101292 ± 2.84912267198 ; ; c_{1} = 10.3835436849 ; ;
c_{2} = 4.68529834094 ; ; ; ; (c -10.3835436849) (c -4.68529834094) = 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 = 8.7 ; ; b = 5.2 ; ; c = 4.69 ; ;

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

p = a+b+c = 8.7+5.2+4.69 = 18.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.59 }{ 2 } = 9.29 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.29 * (9.29-8.7)(9.29-5.2)(9.29-4.69) } ; ; T = sqrt{ 103.85 } = 10.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.19 }{ 8.7 } = 2.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.19 }{ 5.2 } = 3.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.19 }{ 4.69 } = 4.35 ; ;

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{ 8.7**2-5.2**2-4.69**2 }{ 2 * 5.2 * 4.69 } ) = 123° 13'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.2**2-8.7**2-4.69**2 }{ 2 * 8.7 * 4.69 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.69**2-8.7**2-5.2**2 }{ 2 * 5.2 * 8.7 } ) = 26° 46'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.19 }{ 9.29 } = 1.1 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.7 }{ 2 * sin 123° 13'25" } = 5.2 ; ;




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