Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 56.3   b = 80.2   c = 84.43656582107

Area: T = 2171.37331084
Perimeter: p = 220.9365658211
Semiperimeter: s = 110.4687829105

Angle ∠ A = α = 39.88991508634° = 39°53'21″ = 0.69661970184 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 74.11108491366° = 74°6'39″ = 1.29334783289 rad

Height: ha = 77.13658120215
Height: hb = 54.14989553218
Height: hc = 51.43326092653

Median: ma = 77.38440273489
Median: mb = 59.5110714907
Median: mc = 54.94110584684

Inradius: r = 19.65661580506
Circumradius: R = 43.89549147681

Vertex coordinates: A[84.43656582107; 0] B[0; 0] C[22.89992730052; 51.43326092653]
Centroid: CG[35.77883104053; 17.14442030884]
Coordinates of the circumscribed circle: U[42.21878291054; 12.01774226908]
Coordinates of the inscribed circle: I[30.26878291054; 19.65661580506]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.1110849137° = 140°6'39″ = 0.69661970184 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 105.8899150863° = 105°53'21″ = 1.29334783289 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 56.3 ; ; b = 80.2 ; ; beta = 66° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 80.2**2 = 56.3**2 + c**2 -2 * 80.2 * c * cos (66° ) ; ; ; ; c**2 -45.799c -3262.35 =0 ; ; p=1; q=-45.7985460103; r=-3262.35 ; ; D = q**2 - 4pr = 45.799**2 - 4 * 1 * (-3262.35) = 15146.9068167 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.8 ± sqrt{ 15146.91 } }{ 2 } ; ; c_{1,2} = 22.8992730052 ± 61.5363852055 ; ;
c_{1} = 84.4356582107 ; ; c_{2} = -38.6371122004 ; ; ; ; (c -84.4356582107) (c +38.6371122004) = 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 = 56.3 ; ; b = 80.2 ; ; c = 84.44 ; ;

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

p = a+b+c = 56.3+80.2+84.44 = 220.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 220.94 }{ 2 } = 110.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.47 * (110.47-56.3)(110.47-80.2)(110.47-84.44) } ; ; T = sqrt{ 4714861.18 } = 2171.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2171.37 }{ 56.3 } = 77.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2171.37 }{ 80.2 } = 54.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2171.37 }{ 84.44 } = 51.43 ; ;

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{ 56.3**2-80.2**2-84.44**2 }{ 2 * 80.2 * 84.44 } ) = 39° 53'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 80.2**2-56.3**2-84.44**2 }{ 2 * 56.3 * 84.44 } ) = 66° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 84.44**2-56.3**2-80.2**2 }{ 2 * 80.2 * 56.3 } ) = 74° 6'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2171.37 }{ 110.47 } = 19.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 56.3 }{ 2 * sin 39° 53'21" } = 43.89 ; ;




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