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?

1. Use Law of Cosines

a = 8.7 ; ; b = 5.2 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.2**2 = 8.7**2 + c**2 -2 * 8.7 * c * cos (30° ) ; ; ; ; c**2 -15.069c +48.65 =0 ; ; p=1; q=-15.069; 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} = 7.53442101 ± 2.84912267198 ; ; c_{1} = 10.3835436849 ; ; c_{2} = 4.68529834094 ; ; ; ; text{ Factored form: } ; ; (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 = 10.38 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 5.2**2+10.38**2-8.7**2 }{ 2 * 5.2 * 10.38 } ) = 56° 46'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.7**2+10.38**2-5.2**2 }{ 2 * 8.7 * 10.38 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 56° 46'35" - 30° = 93° 13'25" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 10.38**2 - 8.7**2 } }{ 2 } = 6.965 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.38**2+2 * 8.7**2 - 5.2**2 } }{ 2 } = 9.219 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 8.7**2 - 10.38**2 } }{ 2 } = 4.941 ; ;



#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

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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 - 2ac cos beta ; ; 5.2**2 = 8.7**2 + c**2 -2 * 8.7 * c * cos (30° ) ; ; ; ; c**2 -15.069c +48.65 =0 ; ; p=1; q=-15.069; 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 } ; ; : Nr. 1
c_{1,2} = 7.53442101 ± 2.84912267198 ; ; c_{1} = 10.3835436849 ; ; c_{2} = 4.68529834094 ; ; ; ; text{ Factored form: } ; ; (c -10.3835436849) (c -4.68529834094) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.2**2+4.69**2-8.7**2 }{ 2 * 5.2 * 4.69 } ) = 123° 13'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.7**2+4.69**2-5.2**2 }{ 2 * 8.7 * 4.69 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 123° 13'25" - 30° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 4.69**2 - 8.7**2 } }{ 2 } = 2.361 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.69**2+2 * 8.7**2 - 5.2**2 } }{ 2 } = 6.485 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 8.7**2 - 4.69**2 } }{ 2 } = 6.773 ; ;
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