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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 28.81**2+33.46**2-43**2 }{ 2 * 28.81 * 33.46 } ) = 87° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43**2+33.46**2-28.81**2 }{ 2 * 43 * 33.46 } ) = 42° ; ; gamma = 180° - alpha - beta = 180° - 87° - 42° = 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 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.81**2+2 * 33.46**2 - 43**2 } }{ 2 } = 22.643 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.46**2+2 * 43**2 - 28.81**2 } }{ 2 } = 35.733 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.81**2+2 * 43**2 - 33.46**2 } }{ 2 } = 32.552 ; ;
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