Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 50   b = 359.2654826716   c = 355.7688486119

Area: T = 8894.212215298
Perimeter: p = 765.0333312836
Semiperimeter: s = 382.5176656418

Angle ∠ A = α = 8° = 0.14396263402 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 82° = 1.43111699866 rad

Height: ha = 355.7688486119
Height: hb = 49.51334034371
Height: hc = 50

Median: ma = 356.646578466
Median: mb = 179.6322413358
Median: mc = 184.7787714914

Inradius: r = 23.25218297014
Circumradius: R = 179.6322413358

Vertex coordinates: A[355.7688486119; 0] B[0; 0] C[0; 50]
Centroid: CG[118.5899495373; 16.66766666667]
Coordinates of the circumscribed circle: U[177.884424306; 25]
Coordinates of the inscribed circle: I[23.25218297014; 23.25218297014]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172° = 0.14396263402 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 98° = 1.43111699866 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 8° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 8° - 90° = 82° ; ;

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(90° ) }{ sin (8° ) } = 359.26 ; ;

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(82° ) }{ sin (8° ) } = 355.77 ; ;


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 = 359.26 ; ; c = 355.77 ; ;

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

p = a+b+c = 50+359.26+355.77 = 765.03 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 765.03 }{ 2 } = 382.52 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 382.52 * (382.52-50)(382.52-359.26)(382.52-355.77) } ; ; T = sqrt{ 79107009.82 } = 8894.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8894.21 }{ 50 } = 355.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8894.21 }{ 359.26 } = 49.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8894.21 }{ 355.77 } = 50 ; ;

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**2-359.26**2-355.77**2 }{ 2 * 359.26 * 355.77 } ) = 8° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 359.26**2-50**2-355.77**2 }{ 2 * 50 * 355.77 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 355.77**2-50**2-359.26**2 }{ 2 * 359.26 * 50 } ) = 82° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8894.21 }{ 382.52 } = 23.25 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 8° } = 179.63 ; ;

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