Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 37   b = 36.64399945706   c = 54.94401840398

Area: T = 673.4832646853
Perimeter: p = 128.588017861
Semiperimeter: s = 64.29900893052

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 41.5° = 41°30' = 0.72443116396 rad
Angle ∠ C = γ = 96.5° = 96°30' = 1.68442427282 rad

Height: ha = 36.40444673974
Height: hb = 36.762215866
Height: hc = 24.5176941784

Median: ma = 42.87443106327
Median: mb = 43.10655635694
Median: mc = 24.51881289148

Inradius: r = 10.47656837972
Circumradius: R = 27.64878161725

Vertex coordinates: A[54.94401840398; 0] B[0; 0] C[27.71113616692; 24.5176941784]
Centroid: CG[27.55105152363; 8.1722313928]
Coordinates of the circumscribed circle: U[27.47700920199; -3.13298216444]
Coordinates of the inscribed circle: I[27.65500947346; 10.47656837972]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 138.5° = 138°30' = 0.72443116396 rad
∠ C' = γ' = 83.5° = 83°30' = 1.68442427282 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 42° ; ; beta = 41° 30' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 42° - 41° 30' = 96° 30' ; ;

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

a = 37 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 37 * fraction{ sin(41° 30') }{ sin (42° ) } = 36.64 ; ;

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 = 37 * fraction{ sin(96° 30') }{ sin (42° ) } = 54.94 ; ;


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 = 37 ; ; b = 36.64 ; ; c = 54.94 ; ;

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

p = a+b+c = 37+36.64+54.94 = 128.58 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 128.58 }{ 2 } = 64.29 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.29 * (64.29-37)(64.29-36.64)(64.29-54.94) } ; ; T = sqrt{ 453578.88 } = 673.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 673.48 }{ 37 } = 36.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 673.48 }{ 36.64 } = 36.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 673.48 }{ 54.94 } = 24.52 ; ;

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{ 37**2-36.64**2-54.94**2 }{ 2 * 36.64 * 54.94 } ) = 42° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 36.64**2-37**2-54.94**2 }{ 2 * 37 * 54.94 } ) = 41° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54.94**2-37**2-36.64**2 }{ 2 * 36.64 * 37 } ) = 96° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 673.48 }{ 64.29 } = 10.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 42° } = 27.65 ; ;




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