Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 1.414   b = 21.9996979772   c = 2.73216382367

Area: T = 1.36656128641
Perimeter: p = 6.14553362139
Semiperimeter: s = 3.0732668107

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 105° = 1.83325957146 rad

Height: ha = 1.9321559921
Height: hb = 1.36658191184
Height: hc = 10.9998489886

Median: ma = 2.28770222404
Median: mb = 1.9321559921
Median: mc = 1.06547216237

Inradius: r = 0.44444387798
Circumradius: R = 1.414

Vertex coordinates: A[2.73216382367; 0] B[0; 0] C[10.9998489886; 10.9998489886]
Centroid: CG[1.24438290751; 0.33332829962]
Coordinates of the circumscribed circle: U[1.36658191184; -0.36659701298]
Coordinates of the inscribed circle: I[1.07329701298; 0.44444387798]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 75° = 1.83325957146 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 30° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 30° - 45° = 105° ; ;

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

a = 1.41 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 1.41 * fraction{ sin(45° ) }{ sin (30° ) } = 2 ; ;

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 = 1.41 * fraction{ sin(105° ) }{ sin (30° ) } = 2.73 ; ;


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 = 1.41 ; ; b = 2 ; ; c = 2.73 ; ;

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

p = a+b+c = 1.41+2+2.73 = 6.15 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.15 }{ 2 } = 3.07 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.07 * (3.07-1.41)(3.07-2)(3.07-2.73) } ; ; T = sqrt{ 1.86 } = 1.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.37 }{ 1.41 } = 1.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.37 }{ 2 } = 1.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.37 }{ 2.73 } = 1 ; ;

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{ 1.41**2-2**2-2.73**2 }{ 2 * 2 * 2.73 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2**2-1.41**2-2.73**2 }{ 2 * 1.41 * 2.73 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.73**2-1.41**2-2**2 }{ 2 * 2 * 1.41 } ) = 105° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.37 }{ 3.07 } = 0.44 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.41 }{ 2 * sin 30° } = 1.41 ; ;




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