Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 50   b = 76.60444443119   c = 98.48107753012

Area: T = 1886.016626684
Perimeter: p = 225.0855219613
Semiperimeter: s = 112.5432609807

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

Height: ha = 75.44106506735
Height: hb = 49.24403876506
Height: hc = 38.30222221559

Median: ma = 84.60770446011
Median: mb = 68.06600567872
Median: mc = 41.94664500069

Inradius: r = 16.75882417902
Circumradius: R = 50

Vertex coordinates: A[98.48107753012; 0] B[0; 0] C[32.13993804843; 38.30222221559]
Centroid: CG[43.54400519285; 12.76774073853]
Coordinates of the circumscribed circle: U[49.24403876506; -8.68224088833]
Coordinates of the inscribed circle: I[35.93881654947; 16.75882417902]

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

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

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

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(50° ) }{ sin (30° ) } = 76.6 ; ;

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(100° ) }{ sin (30° ) } = 98.48 ; ;


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 = 76.6 ; ; c = 98.48 ; ;

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

p = a+b+c = 50+76.6+98.48 = 225.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 225.09 }{ 2 } = 112.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 112.54 * (112.54-50)(112.54-76.6)(112.54-98.48) } ; ; T = sqrt{ 3557057.36 } = 1886.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1886.02 }{ 50 } = 75.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1886.02 }{ 76.6 } = 49.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1886.02 }{ 98.48 } = 38.3 ; ;

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**2-76.6**2-98.48**2 }{ 2 * 76.6 * 98.48 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 76.6**2-50**2-98.48**2 }{ 2 * 50 * 98.48 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 98.48**2-50**2-76.6**2 }{ 2 * 76.6 * 50 } ) = 100° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1886.02 }{ 112.54 } = 16.76 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 30° } = 50 ; ;




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