Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 42.1   b = 36.23767988226   c = 64.25878119967

Area: T = 716.7833074168
Perimeter: p = 142.5954610819
Semiperimeter: s = 71.29773054097

Angle ∠ A = α = 38° = 0.66332251158 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 110° = 1.92198621772 rad

Height: ha = 34.05114524545
Height: hb = 39.56110593351
Height: hc = 22.31096010242

Median: ma = 47.72882253573
Median: mb = 51.21099775832
Median: mc = 22.59440521776

Inradius: r = 10.05334384862
Circumradius: R = 34.19108676174

Vertex coordinates: A[64.25878119967; 0] B[0; 0] C[35.70328248482; 22.31096010242]
Centroid: CG[33.32202122816; 7.43765336747]
Coordinates of the circumscribed circle: U[32.12989059983; -11.69439654429]
Coordinates of the inscribed circle: I[35.0610506587; 10.05334384862]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142° = 0.66332251158 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 70° = 1.92198621772 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 38° ; ; beta = 32° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 38° - 32° = 110° ; ;

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

a = 42.1 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 42.1 * fraction{ sin(32° ) }{ sin (38° ) } = 36.24 ; ;

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 = 42.1 * fraction{ sin(110° ) }{ sin (38° ) } = 64.26 ; ;


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 = 42.1 ; ; b = 36.24 ; ; c = 64.26 ; ;

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

p = a+b+c = 42.1+36.24+64.26 = 142.59 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 142.59 }{ 2 } = 71.3 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 71.3 * (71.3-42.1)(71.3-36.24)(71.3-64.26) } ; ; T = sqrt{ 513777.98 } = 716.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 716.78 }{ 42.1 } = 34.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 716.78 }{ 36.24 } = 39.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 716.78 }{ 64.26 } = 22.31 ; ;

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{ 42.1**2-36.24**2-64.26**2 }{ 2 * 36.24 * 64.26 } ) = 38° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 36.24**2-42.1**2-64.26**2 }{ 2 * 42.1 * 64.26 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 64.26**2-42.1**2-36.24**2 }{ 2 * 36.24 * 42.1 } ) = 110° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 716.78 }{ 71.3 } = 10.05 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 42.1 }{ 2 * sin 38° } = 34.19 ; ;




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