Triangle calculator AAS

Please enter two angles and one opposite side
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Right isosceles triangle.

Sides: a = 300   b = 212.1322034356   c = 212.1322034356

Area: T = 22500
Perimeter: p = 724.2644068712
Semiperimeter: s = 362.1322034356

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 150
Height: hb = 212.1322034356
Height: hc = 212.1322034356

Median: ma = 150
Median: mb = 237.1710824513
Median: mc = 237.1710824513

Inradius: r = 62.1322034356
Circumradius: R = 150

Vertex coordinates: A[212.1322034356; 0] B[0; 0] C[212.1322034356; 212.1322034356]
Centroid: CG[141.4211356237; 70.71106781187]
Coordinates of the circumscribed circle: U[106.0666017178; 106.0666017178]
Coordinates of the inscribed circle: I[150; 62.1322034356]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° = 45° ; ;

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

a = 300 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 300 * fraction{ sin(45° ) }{ sin (90° ) } = 212.13 ; ;

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 = 300 * fraction{ sin(45° ) }{ sin (90° ) } = 212.13 ; ;


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 = 300 ; ; b = 212.13 ; ; c = 212.13 ; ;

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

p = a+b+c = 300+212.13+212.13 = 724.26 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 724.26 }{ 2 } = 362.13 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 362.13 * (362.13-300)(362.13-212.13)(362.13-212.13) } ; ; T = sqrt{ 506250000 } = 22500 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22500 }{ 300 } = 150 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22500 }{ 212.13 } = 212.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22500 }{ 212.13 } = 212.13 ; ;

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{ 300**2-212.13**2-212.13**2 }{ 2 * 212.13 * 212.13 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 212.13**2-300**2-212.13**2 }{ 2 * 300 * 212.13 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 212.13**2-300**2-212.13**2 }{ 2 * 212.13 * 300 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22500 }{ 362.13 } = 62.13 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 300 }{ 2 * sin 90° } = 150 ; ;




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