Triangle calculator SSA

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Triangle has two solutions with side c=28.65444440559 and with side c=4.58774210557

#1 Acute scalene triangle.

Sides: a = 60.3   b = 59.2   c = 28.65444440559

Area: T = 830.4644247603
Perimeter: p = 148.1544444056
Semiperimeter: s = 74.07772220279

Angle ∠ A = α = 78.27218264562° = 78°16'19″ = 1.36661010832 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 27.72881735438° = 27°43'41″ = 0.48439479239 rad

Height: ha = 27.54444194893
Height: hb = 28.05662245812
Height: hc = 57.96440802651

Median: ma = 35.41095478943
Median: mb = 36.77553121275
Median: mc = 58.00994449979

Inradius: r = 11.21107909134
Circumradius: R = 30.79328633015

Vertex coordinates: A[28.65444440559; 0] B[0; 0] C[16.62109325558; 57.96440802651]
Centroid: CG[15.09217922039; 19.32113600884]
Coordinates of the circumscribed circle: U[14.32772220279; 27.2576763184]
Coordinates of the inscribed circle: I[14.87772220279; 11.21107909134]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.7288173544° = 101°43'41″ = 1.36661010832 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 152.2721826456° = 152°16'19″ = 0.48439479239 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 60.3 ; ; b = 59.2 ; ; beta = 74° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 59.2**2 = 60.3**2 + c**2 -2 * 60.3 * c * cos (74° ) ; ; ; ; c**2 -33.242c +131.45 =0 ; ; p=1; q=-33.242; r=131.45 ; ; D = q**2 - 4pr = 33.242**2 - 4 * 1 * 131.45 = 579.221596093 ; ; D>0 ; ; ; ;
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 33.24 ± sqrt{ 579.22 } }{ 2 } ; ; c_{1,2} = 16.62093256 ± 12.0335115001 ; ; c_{1} = 28.6544440559 ; ; c_{2} = 4.58742105565 ; ; ; ; text{ Factored form: } ; ; (c -28.6544440559) (c -4.58742105565) = 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 = 60.3 ; ; b = 59.2 ; ; c = 28.65 ; ;

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

p = a+b+c = 60.3+59.2+28.65 = 148.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 148.15 }{ 2 } = 74.08 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 74.08 * (74.08-60.3)(74.08-59.2)(74.08-28.65) } ; ; T = sqrt{ 689670.87 } = 830.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 830.46 }{ 60.3 } = 27.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 830.46 }{ 59.2 } = 28.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 830.46 }{ 28.65 } = 57.96 ; ;

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{ 59.2**2+28.65**2-60.3**2 }{ 2 * 59.2 * 28.65 } ) = 78° 16'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60.3**2+28.65**2-59.2**2 }{ 2 * 60.3 * 28.65 } ) = 74° ; ;
 gamma = 180° - alpha - beta = 180° - 78° 16'19" - 74° = 27° 43'41" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 830.46 }{ 74.08 } = 11.21 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60.3 }{ 2 * sin 78° 16'19" } = 30.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.2**2+2 * 28.65**2 - 60.3**2 } }{ 2 } = 35.41 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.65**2+2 * 60.3**2 - 59.2**2 } }{ 2 } = 36.775 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.2**2+2 * 60.3**2 - 28.65**2 } }{ 2 } = 58.009 ; ;



#2 Obtuse scalene triangle.

Sides: a = 60.3   b = 59.2   c = 4.58774210557

Area: T = 132.953282114
Perimeter: p = 124.0877421056
Semiperimeter: s = 62.04437105278

Angle ∠ A = α = 101.7288173544° = 101°43'41″ = 1.77554915704 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 4.27218264562° = 4°16'19″ = 0.07545574367 rad

Height: ha = 4.41097121439
Height: hb = 4.49216493628
Height: hc = 57.96440802651

Median: ma = 29.22201936334
Median: mb = 30.86110955083
Median: mc = 59.70884909541

Inradius: r = 2.14328895856
Circumradius: R = 30.79328633015

Vertex coordinates: A[4.58774210557; 0] B[0; 0] C[16.62109325558; 57.96440802651]
Centroid: CG[7.06994512038; 19.32113600884]
Coordinates of the circumscribed circle: U[2.29437105278; 30.70773170811]
Coordinates of the inscribed circle: I[2.84437105278; 2.14328895856]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 78.27218264562° = 78°16'19″ = 1.77554915704 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 175.7288173544° = 175°43'41″ = 0.07545574367 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 60.3 ; ; b = 59.2 ; ; beta = 74° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 59.2**2 = 60.3**2 + c**2 -2 * 60.3 * c * cos (74° ) ; ; ; ; c**2 -33.242c +131.45 =0 ; ; p=1; q=-33.242; r=131.45 ; ; D = q**2 - 4pr = 33.242**2 - 4 * 1 * 131.45 = 579.221596093 ; ; D>0 ; ; ; ; : Nr. 1
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 33.24 ± sqrt{ 579.22 } }{ 2 } ; ; c_{1,2} = 16.62093256 ± 12.0335115001 ; ; c_{1} = 28.6544440559 ; ; c_{2} = 4.58742105565 ; ; ; ; text{ Factored form: } ; ; (c -28.6544440559) (c -4.58742105565) = 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 = 60.3 ; ; b = 59.2 ; ; c = 4.59 ; ;

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

p = a+b+c = 60.3+59.2+4.59 = 124.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.09 }{ 2 } = 62.04 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.04 * (62.04-60.3)(62.04-59.2)(62.04-4.59) } ; ; T = sqrt{ 17676.45 } = 132.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 132.95 }{ 60.3 } = 4.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 132.95 }{ 59.2 } = 4.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 132.95 }{ 4.59 } = 57.96 ; ;

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{ 59.2**2+4.59**2-60.3**2 }{ 2 * 59.2 * 4.59 } ) = 101° 43'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60.3**2+4.59**2-59.2**2 }{ 2 * 60.3 * 4.59 } ) = 74° ; ;
 gamma = 180° - alpha - beta = 180° - 101° 43'41" - 74° = 4° 16'19" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 132.95 }{ 62.04 } = 2.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60.3 }{ 2 * sin 101° 43'41" } = 30.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.2**2+2 * 4.59**2 - 60.3**2 } }{ 2 } = 29.22 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.59**2+2 * 60.3**2 - 59.2**2 } }{ 2 } = 30.861 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.2**2+2 * 60.3**2 - 4.59**2 } }{ 2 } = 59.708 ; ;
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