Triangle calculator AAS

Please enter two angles and one opposite side
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Acute scalene triangle.

Sides: a = 43   b = 28.81221020579   c = 33.46331364081

Area: T = 481.4110988242
Perimeter: p = 105.2755238466
Semiperimeter: s = 52.6387619233

Angle ∠ A = α = 87° = 1.51884364492 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 51° = 0.89901179185 rad

Height: ha = 22.39112087554
Height: hb = 33.41772763426
Height: hc = 28.77326160734

Median: ma = 22.64330863981
Median: mb = 35.7333128087
Median: mc = 32.55218546005

Inradius: r = 9.14657591596
Circumradius: R = 21.5329505439

Vertex coordinates: A[33.46331364081; 0] B[0; 0] C[31.95552274955; 28.77326160734]
Centroid: CG[21.80661213012; 9.59108720245]
Coordinates of the circumscribed circle: U[16.73215682041; 13.5498956782]
Coordinates of the inscribed circle: I[23.82655171751; 9.14657591596]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 93° = 1.51884364492 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 129° = 0.89901179185 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 87° ; ; beta = 42° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 87° - 42° = 51° ; ;

2. By using the law of sines, we calculate unknown side b

a = 43 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 43 * fraction{ sin(42° ) }{ sin (87° ) } = 28.81 ; ;

3. By using the law of sines, we calculate last unknown side c

 fraction{ c }{ a } = fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = a * fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = 43 * fraction{ sin(51° ) }{ sin (87° ) } = 33.46 ; ;


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 = 43 ; ; b = 28.81 ; ; c = 33.46 ; ;

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

p = a+b+c = 43+28.81+33.46 = 105.28 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 105.28 }{ 2 } = 52.64 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52.64 * (52.64-43)(52.64-28.81)(52.64-33.46) } ; ; T = sqrt{ 231756.54 } = 481.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 481.41 }{ 43 } = 22.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 481.41 }{ 28.81 } = 33.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 481.41 }{ 33.46 } = 28.77 ; ;

8. 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{ 43**2-28.81**2-33.46**2 }{ 2 * 28.81 * 33.46 } ) = 87° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.81**2-43**2-33.46**2 }{ 2 * 43 * 33.46 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 33.46**2-43**2-28.81**2 }{ 2 * 28.81 * 43 } ) = 51° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 481.41 }{ 52.64 } = 9.15 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 43 }{ 2 * sin 87° } = 21.53 ; ;




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