Triangle calculator SSA

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Triangle has two solutions with side c=232.55501256 and with side c=14.99546167133

#1 Obtuse scalene triangle.

Sides: a = 164   b = 153   c = 232.55501256

Area: T = 12510.46219866
Perimeter: p = 549.55501256
Semiperimeter: s = 274.77550628

Angle ∠ A = α = 44.68664570052° = 44°41'11″ = 0.78799258058 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 94.31435429948° = 94°18'49″ = 1.64660818545 rad

Height: ha = 152.5676609593
Height: hb = 163.5355450805
Height: hc = 107.5943680754

Median: ma = 178.9422114825
Median: mb = 186.1066234335
Median: mc = 107.8554576958

Inradius: r = 45.53298303242
Circumradius: R = 116.6055361133

Vertex coordinates: A[232.55501256; 0] B[0; 0] C[123.7722371157; 107.5943680754]
Centroid: CG[118.7744165585; 35.86545602515]
Coordinates of the circumscribed circle: U[116.27550628; -8.77704056827]
Coordinates of the inscribed circle: I[121.77550628; 45.53298303242]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.3143542995° = 135°18'49″ = 0.78799258058 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 85.68664570052° = 85°41'11″ = 1.64660818545 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 164 ; ; b = 153 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 153**2 = 164**2 + c**2 -2 * 164 * c * cos (41° ) ; ; ; ; c**2 -247.545c +3487 =0 ; ; p=1; q=-247.545; r=3487 ; ; D = q**2 - 4pr = 247.545**2 - 4 * 1 * 3487 = 47330.3994468 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 247.54 ± sqrt{ 47330.4 } }{ 2 } ; ;
c_{1,2} = 123.77237116 ± 108.777754443 ; ; c_{1} = 232.5501256 ; ; c_{2} = 14.9946167133 ; ; ; ; text{ Factored form: } ; ; (c -232.5501256) (c -14.9946167133) = 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 = 164 ; ; b = 153 ; ; c = 232.55 ; ;

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

p = a+b+c = 164+153+232.55 = 549.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 549.55 }{ 2 } = 274.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 274.78 * (274.78-164)(274.78-153)(274.78-232.55) } ; ; T = sqrt{ 156511659.12 } = 12510.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12510.46 }{ 164 } = 152.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12510.46 }{ 153 } = 163.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12510.46 }{ 232.55 } = 107.59 ; ;

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{ 153**2+232.55**2-164**2 }{ 2 * 153 * 232.55 } ) = 44° 41'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 164**2+232.55**2-153**2 }{ 2 * 164 * 232.55 } ) = 41° ; ;
 gamma = 180° - alpha - beta = 180° - 44° 41'11" - 41° = 94° 18'49" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12510.46 }{ 274.78 } = 45.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 164 }{ 2 * sin 44° 41'11" } = 116.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 153**2+2 * 232.55**2 - 164**2 } }{ 2 } = 178.942 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 232.55**2+2 * 164**2 - 153**2 } }{ 2 } = 186.106 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 153**2+2 * 164**2 - 232.55**2 } }{ 2 } = 107.855 ; ;



#2 Obtuse scalene triangle.

Sides: a = 164   b = 153   c = 14.99546167133

Area: T = 806.6633001843
Perimeter: p = 331.9954616713
Semiperimeter: s = 165.9977308357

Angle ∠ A = α = 135.3143542995° = 135°18'49″ = 2.36216668478 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 3.68664570052° = 3°41'11″ = 0.06443408125 rad

Height: ha = 9.8377353681
Height: hb = 10.54546144032
Height: hc = 107.5943680754

Median: ma = 71.36546920065
Median: mb = 87.79661802426
Median: mc = 158.4188087248

Inradius: r = 4.85994944691
Circumradius: R = 116.6055361133

Vertex coordinates: A[14.99546167133; 0] B[0; 0] C[123.7722371157; 107.5943680754]
Centroid: CG[46.25656626233; 35.86545602515]
Coordinates of the circumscribed circle: U[7.49773083567; 116.3644086437]
Coordinates of the inscribed circle: I[12.99773083567; 4.85994944691]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 44.68664570052° = 44°41'11″ = 2.36216668478 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 176.3143542995° = 176°18'49″ = 0.06443408125 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 164 ; ; b = 153 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 153**2 = 164**2 + c**2 -2 * 164 * c * cos (41° ) ; ; ; ; c**2 -247.545c +3487 =0 ; ; p=1; q=-247.545; r=3487 ; ; D = q**2 - 4pr = 247.545**2 - 4 * 1 * 3487 = 47330.3994468 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 247.54 ± sqrt{ 47330.4 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 123.77237116 ± 108.777754443 ; ; c_{1} = 232.5501256 ; ; c_{2} = 14.9946167133 ; ; ; ; text{ Factored form: } ; ; (c -232.5501256) (c -14.9946167133) = 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 = 164 ; ; b = 153 ; ; c = 14.99 ; ;

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

p = a+b+c = 164+153+14.99 = 331.99 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.99 }{ 2 } = 166 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 166 * (166-164)(166-153)(166-14.99) } ; ; T = sqrt{ 650705.2 } = 806.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 806.66 }{ 164 } = 9.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 806.66 }{ 153 } = 10.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 806.66 }{ 14.99 } = 107.59 ; ;

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{ 153**2+14.99**2-164**2 }{ 2 * 153 * 14.99 } ) = 135° 18'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 164**2+14.99**2-153**2 }{ 2 * 164 * 14.99 } ) = 41° ; ;
 gamma = 180° - alpha - beta = 180° - 135° 18'49" - 41° = 3° 41'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 806.66 }{ 166 } = 4.86 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 164 }{ 2 * sin 135° 18'49" } = 116.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 153**2+2 * 14.99**2 - 164**2 } }{ 2 } = 71.365 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.99**2+2 * 164**2 - 153**2 } }{ 2 } = 87.796 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 153**2+2 * 164**2 - 14.99**2 } }{ 2 } = 158.418 ; ;
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