Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 50   b = 25.38656652971   c = 38.89330956715

Area: T = 486.1643695894
Perimeter: p = 114.2798760969
Semiperimeter: s = 57.13993804843

Angle ∠ A = α = 100° = 1.7455329252 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 19.44765478358
Height: hb = 38.30222221559
Height: hc = 25

Median: ma = 21.29767708056
Median: mb = 42.95661223205
Median: mc = 34.55549964341

Inradius: r = 8.50883823411
Circumradius: R = 25.38656652971

Vertex coordinates: A[38.89330956715; 0] B[0; 0] C[43.30112701892; 25]
Centroid: CG[27.39881219536; 8.33333333333]
Coordinates of the circumscribed circle: U[19.44765478358; 16.31875911167]
Coordinates of the inscribed circle: I[31.75437151872; 8.50883823411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 80° = 1.7455329252 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 100° ; ; beta = 30° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 100° - 30° = 50° ; ;

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

a = 50 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 50 * fraction{ sin(30° ) }{ sin (100° ) } = 25.39 ; ;

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 = 50 * fraction{ sin(50° ) }{ sin (100° ) } = 38.89 ; ;


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 = 50 ; ; b = 25.39 ; ; c = 38.89 ; ;

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

p = a+b+c = 50+25.39+38.89 = 114.28 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.28 }{ 2 } = 57.14 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.14 * (57.14-50)(57.14-25.39)(57.14-38.89) } ; ; T = sqrt{ 236355.14 } = 486.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 486.16 }{ 50 } = 19.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 486.16 }{ 25.39 } = 38.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 486.16 }{ 38.89 } = 25 ; ;

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{ 50**2-25.39**2-38.89**2 }{ 2 * 25.39 * 38.89 } ) = 100° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25.39**2-50**2-38.89**2 }{ 2 * 50 * 38.89 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.89**2-50**2-25.39**2 }{ 2 * 25.39 * 50 } ) = 50° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 486.16 }{ 57.14 } = 8.51 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 100° } = 25.39 ; ;




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