Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 14.2   b = 6.72877614589   c = 19.30769296431

Area: T = 35.94106829789
Perimeter: p = 40.2354691102
Semiperimeter: s = 20.1177345551

Angle ∠ A = α = 33.6° = 33°36' = 0.58664306287 rad
Angle ∠ B = β = 15.2° = 15°12' = 0.26552900463 rad
Angle ∠ C = γ = 131.2° = 131°12' = 2.29898719786 rad

Height: ha = 5.06220680252
Height: hb = 10.68442917063
Height: hc = 3.72330863367

Median: ma = 12.59436552774
Median: mb = 16.61097282506
Median: mc = 5.50110911703

Inradius: r = 1.78765519528
Circumradius: R = 12.83299754662

Vertex coordinates: A[19.30769296431; 0] B[0; 0] C[13.70332342215; 3.72330863367]
Centroid: CG[11.00333879549; 1.24110287789]
Coordinates of the circumscribed circle: U[9.65334648215; -8.45109696132]
Coordinates of the inscribed circle: I[13.39895840921; 1.78765519528]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4° = 146°24' = 0.58664306287 rad
∠ B' = β' = 164.8° = 164°48' = 0.26552900463 rad
∠ C' = γ' = 48.8° = 48°48' = 2.29898719786 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 33° 36' ; ; beta = 15° 12' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 33° 36' - 15° 12' = 131° 12' ; ;

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

a = 14.2 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 14.2 * fraction{ sin(15° 12') }{ sin (33° 36') } = 6.73 ; ;

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 = 14.2 * fraction{ sin(131° 12') }{ sin (33° 36') } = 19.31 ; ;


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 = 14.2 ; ; b = 6.73 ; ; c = 19.31 ; ;

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

p = a+b+c = 14.2+6.73+19.31 = 40.23 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.23 }{ 2 } = 20.12 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.12 * (20.12-14.2)(20.12-6.73)(20.12-19.31) } ; ; T = sqrt{ 1291.73 } = 35.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35.94 }{ 14.2 } = 5.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35.94 }{ 6.73 } = 10.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.94 }{ 19.31 } = 3.72 ; ;

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{ 14.2**2-6.73**2-19.31**2 }{ 2 * 6.73 * 19.31 } ) = 33° 36' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.73**2-14.2**2-19.31**2 }{ 2 * 14.2 * 19.31 } ) = 15° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19.31**2-14.2**2-6.73**2 }{ 2 * 6.73 * 14.2 } ) = 131° 12' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.94 }{ 20.12 } = 1.79 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.2 }{ 2 * sin 33° 36' } = 12.83 ; ;




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