Triangle calculator SSA

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Triangle has two solutions with side c=4.38109934777 and with side c=0.74986886289

#1 Acute scalene triangle.

Sides: a = 8.3   b = 8.1   c = 4.38109934777

Area: T = 17.29112754386
Perimeter: p = 20.78109934777
Semiperimeter: s = 10.39904967389

Angle ∠ A = α = 77.04331990626° = 77°2'36″ = 1.34546574899 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 30.95768009374° = 30°57'24″ = 0.54402981022 rad

Height: ha = 4.16765723949
Height: hb = 4.26994507256
Height: hc = 7.89437690852

Median: ma = 5.01878732473
Median: mb = 5.25772856044
Median: mc = 7.90326403206

Inradius: r = 1.6644143291
Circumradius: R = 4.25884220082

Vertex coordinates: A[4.38109934777; 0] B[0; 0] C[2.56548410533; 7.89437690852]
Centroid: CG[2.3155278177; 2.63112563617]
Coordinates of the circumscribed circle: U[2.19904967389; 3.65218326956]
Coordinates of the inscribed circle: I[2.29904967389; 1.6644143291]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.9576800937° = 102°57'24″ = 1.34546574899 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 149.0433199063° = 149°2'36″ = 0.54402981022 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 8.3 ; ; b = 8.1 ; ; beta = 72° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.1**2 = 8.3**2 + c**2 -2 * 8.3 * c * cos (72° ) ; ; ; ; c**2 -5.13c +3.28 =0 ; ; p=1; q=-5.13; r=3.28 ; ; D = q**2 - 4pr = 5.13**2 - 4 * 1 * 3.28 = 13.193638515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 5.13 ± sqrt{ 13.19 } }{ 2 } ; ; c_{1,2} = 2.56484105 ± 1.81615242443 ; ; c_{1} = 4.38099347443 ; ; c_{2} = 0.748688625572 ; ; ; ; text{ Factored form: } ; ; (c -4.38099347443) (c -0.748688625572) = 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 = 8.3 ; ; b = 8.1 ; ; c = 4.38 ; ;

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

p = a+b+c = 8.3+8.1+4.38 = 20.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.78 }{ 2 } = 10.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.39 * (10.39-8.3)(10.39-8.1)(10.39-4.38) } ; ; T = sqrt{ 298.99 } = 17.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.29 }{ 8.3 } = 4.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.29 }{ 8.1 } = 4.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.29 }{ 4.38 } = 7.89 ; ;

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{ 8.1**2+4.38**2-8.3**2 }{ 2 * 8.1 * 4.38 } ) = 77° 2'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+4.38**2-8.1**2 }{ 2 * 8.3 * 4.38 } ) = 72° ; ; gamma = 180° - alpha - beta = 180° - 77° 2'36" - 72° = 30° 57'24" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.29 }{ 10.39 } = 1.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 77° 2'36" } = 4.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 4.38**2 - 8.3**2 } }{ 2 } = 5.018 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.38**2+2 * 8.3**2 - 8.1**2 } }{ 2 } = 5.257 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 8.3**2 - 4.38**2 } }{ 2 } = 7.903 ; ;







#2 Obtuse scalene triangle.

Sides: a = 8.3   b = 8.1   c = 0.74986886289

Area: T = 2.95549875766
Perimeter: p = 17.14986886289
Semiperimeter: s = 8.57443443144

Angle ∠ A = α = 102.9576800937° = 102°57'24″ = 1.79769351637 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 5.04331990626° = 5°2'36″ = 0.08880204285 rad

Height: ha = 0.71220451992
Height: hb = 0.73296265621
Height: hc = 7.89437690852

Median: ma = 3.9832808975
Median: mb = 4.28105101719
Median: mc = 8.19220611774

Inradius: r = 0.34546313174
Circumradius: R = 4.25884220082

Vertex coordinates: A[0.74986886289; 0] B[0; 0] C[2.56548410533; 7.89437690852]
Centroid: CG[1.10545098941; 2.63112563617]
Coordinates of the circumscribed circle: U[0.37443443144; 4.24219363897]
Coordinates of the inscribed circle: I[0.47443443144; 0.34546313174]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 77.04331990626° = 77°2'36″ = 1.79769351637 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 174.9576800937° = 174°57'24″ = 0.08880204285 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 8.3 ; ; b = 8.1 ; ; beta = 72° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.1**2 = 8.3**2 + c**2 -2 * 8.3 * c * cos (72° ) ; ; ; ; c**2 -5.13c +3.28 =0 ; ; p=1; q=-5.13; r=3.28 ; ; D = q**2 - 4pr = 5.13**2 - 4 * 1 * 3.28 = 13.193638515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 5.13 ± sqrt{ 13.19 } }{ 2 } ; ; c_{1,2} = 2.56484105 ± 1.81615242443 ; ; c_{1} = 4.38099347443 ; ; c_{2} = 0.748688625572 ; ; ; ; text{ Factored form: } ; ; (c -4.38099347443) (c -0.748688625572) = 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 = 8.3 ; ; b = 8.1 ; ; c = 0.75 ; ;

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

p = a+b+c = 8.3+8.1+0.75 = 17.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.15 }{ 2 } = 8.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.57 * (8.57-8.3)(8.57-8.1)(8.57-0.75) } ; ; T = sqrt{ 8.73 } = 2.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.95 }{ 8.3 } = 0.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.95 }{ 8.1 } = 0.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.95 }{ 0.75 } = 7.89 ; ;

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{ 8.1**2+0.75**2-8.3**2 }{ 2 * 8.1 * 0.75 } ) = 102° 57'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+0.75**2-8.1**2 }{ 2 * 8.3 * 0.75 } ) = 72° ; ; gamma = 180° - alpha - beta = 180° - 102° 57'24" - 72° = 5° 2'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.95 }{ 8.57 } = 0.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 102° 57'24" } = 4.26 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 0.75**2 - 8.3**2 } }{ 2 } = 3.983 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.75**2+2 * 8.3**2 - 8.1**2 } }{ 2 } = 4.281 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.1**2+2 * 8.3**2 - 0.75**2 } }{ 2 } = 8.192 ; ;
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